Representations of Elementary Abelian p-Groups and Vector Bundles

Version 2.2.2, 4 Sep 2016.

Dave Benson

Institute of Mathematics, University of Aberdeen, Aberdeen AB24 3UE 2010 Mathematics Subject Classification. 14F05, 20C20

Contents

Preface  vii
Introduction  ix
1. Modular Representations and Elementary Abelian Groups  1
1.1. Introduction  1
1.2. Representation Type  1
1.3. Shifted Subgroups  2
1.4. The Language of π-Points  5
1.5. The Stable Module Category  5
1.6. The Derived Category  7
1.7. Singularity Categories  9
1.8. Cohomology of Elementary Abelian p-Groups  11
1.9. Chouinard’s Theorem, Dade’s Lemma and Rank Varieties  13
1.10. Carlson’s L_{ζ} Modules, and a Matrix Version  15
1.11. Diagrams for Modules  17
1.12. Tensor Products  20
1.13. Duality  22
1.14. Symmetric and Exterior Powers  23
1.15. Schur Functions  24
1.16. Schur Functors  26
1.17. Radical Layers of kE  27
1.18. Twisted Versions of kE  30
2. Cyclic Groups of Order p  33
2.1. Introduction  33
2.2. Modules for ℤ/p  33
2.3. Tensor Products  34
2.4. Gaussian Polynomials  35
2.5. Generalised Gaussian Polynomials and a Hook Formula  37
2.6. λ-Rings and Representations of SL(2, ℂ)  39
2.7. The Representation Ring of ℤ/p  41
2.8. Symmetric and Exterior Powers of Jordan Blocks  43
2.9. Schur Functors for SL(2, ℂ) and ℤ/p  44
3. Background from Algebraic Geometry  47
3.1. Affine Space and Affine Varieties  47
3.2. Generic Points and Closed Points  48
3.3. Projective Space and Projective Varieties  49

3.4. Tangent Spaces  51
3.5. Presheaves and Sheaves  52
3.6. Stalks and Sheafification  53
3.7. The Language of Schemes  54
3.8. Sheaves of Modules  55
3.9. Coherent Sheaves on Projective Varieties  56
3.10. Cohomology of Sheaves  59
4. Jordan Type  63
4.1. Nilvarieties  63
4.2. Matrices and Tangent Spaces  65
4.3. A Theorem of Gerstenhaber  66
4.4. Dominance Order and Nilpotent Jordan Types  68
4.5. Generic and Maximal Jordan Type  72
4.6. Tensor Products  75
5. Modules of Constant Jordan Type  77
5.1. Introduction and Definitions  77
5.2. Homogeneous Modules  81
5.3. An Exact Category  85
5.4. Endotrivial Modules  87
5.5. Wild Representation Type  88
5.6. The Constant Image Property  91
5.7. The Generic Kernel  94
5.8. The Subquotient Rad^{−1}𝔎(M)/Rad^{2}𝔎(M)  96
5.9. The Constant Kernel Property  97
5.10. The Generic Image  99
5.11. W -Modules  100
5.12. Constant Jordan type with One Non-Projective Block  102
5.13. Rickard’s Conjecture  102
5.14. Consequences and Variations  105
5.15. Further Conjectures  108
6. Vector Bundles on Projective Space  111
6.1. Definitions and First Properties  111
6.2. Tests for Vector Bundles  112
6.3. Vector Bundles on Projective Space  115
6.4. The Tangent Bundle and the Euler Sequence  116
6.5. Homogeneity and Uniformity  117
6.6. Monads and Subquotients  118
6.7. The Null Correlation Bundle  119
6.8. The Examples of Tango  120
6.9. Cohomology of Projective Space  121
6.10. Differential Forms and Bott’s Theorem  122
6.11. Simplicity  123
6.12. Hilbert’s Syzygy Theorem  125

7. Chern Classes  129
7.1. Chern Classes of Graded Modules  129
7.2. Chern Classes of Coherent Sheaves on ℙ^{r−1}  131
7.3. Some Computations  133
7.4. Restriction of Vector Bundles  134
7.5. Chern Numbers of Twists and Duals  136
7.6. Chern Roots  137
7.7. Power Sums  138
7.8. The Hirzebruch–Riemann–Roch Theorem  143
7.9. Chern Numbers and the Frobenius Map  146
8. Modules of Constant Jordan Type and Vector Bundles  149
8.1. Introduction  149
8.2. The Operator θ  150
8.3. The Action of θ on Fibres  151
8.4. The Functors ℱ_{i} and ℱ_{i,j}  152
8.5. Twists and Syzygies  157
8.6. Chern Numbers of ℱ_{i}(M)  158
8.7. The Construction: p = 2  160
8.8. The Construction: p Odd  162
8.9. Proof of the Realisation Theorem  164
8.10. Functoriality  166
8.11. Tensor Products  167
8.12. Negative Tate Cohomology  168
8.13. The BGG Correspondence  169
9. Examples  173
9.1. Modules for (ℤ/2)^{2}  173
9.2. Modules for (ℤ/p)^{2}  175
9.3. Larger Rank  177
9.4. Nilvarieties  178
9.5. The Tangent and Cotangent Bundles  179
9.6. The Null Correlation Bundle, p = 2  180
9.7. The Null Correlation Bundle, p Odd  181
9.8. Instanton Bundles  183
9.9. Schwarzenberger’s Bundles  185
9.10. The Examples of Tango  186
9.11. The Horrocks–Mumford Bundle  187
9.12. Automorphisms of the Horrocks–Mumford Bundle  190
9.13. Realising the Horrocks–Mumford Bundle  194
9.14. The Horrocks Parent Bundle and the Tango Bundle  197
10. Restrictions Coming from Chern Numbers  203
10.1. Matrices of Constant Rank  203
10.2. Congruences on Chern Numbers  205
10.3. Restrictions on Stable Jordan Type, p Odd  207

10.4. Eliminating More Stable Jordan Types  208
10.5. Restrictions on Jordan Type for p = 2  210
10.6. Applying Hirzebruch–Riemann–Roch for p = 2: The Case m = 0  213
10.7. Bypassing Hirzebruch–Riemann–Roch  214
10.8. Applying and Bypassing Hirzebruch–Riemann–Roch for p = 2:The Case 1 ≤ m ≤ r − 3215
10.9. Nilvarieties of Constant Jordan Type [p]^{n} for p ≥ 3  217
10.10. Nilvarieties with a Single Jordan Block  219
10.11. Babylonian Towers  221
11. Orlov’s Correspondence  223
11.1. Introduction  223
11.2. Maximal Cohen–Macaulay Modules  225
11.3. The Orlov Correspondence  226
11.4. The Functors  228
11.5. An Example  229
11.6. The Bidirectional Koszul Complex  230
11.7. A Bimodule Resolution  233
11.8. The Adjunction  234
11.9. The Equivalence  235
11.10. The Trivial Module  235
11.11. Computer Algebra  236
11.12. Cohomology  237
11.13. Twisted Versions of kE  239
12. Phenomenology of Modules over Elementary Abelian p-Groups  241
12.1. Introduction  241
12.2. Module Constructions  243
12.3. Odd Primes Are More Difficult  244
12.4. Relative Cohomology  245
12.5. Small Modules for Quadrics, p = 2  247
12.6. Small Modules for Quadrics, p Odd  248
12.7. Trying to Understand the Specht Module S^{(3^{3})}  249
12.8. Modules with Small Loewy Length  250
12.9. Small Modules for (ℤ/p)^{2}  257
12.10. The Bound is Close to Sharp  260
Appendix A. Modules for ℤ/p  263
Appendix B. Problems  269
References  273
Index  283

Preface

The origins of this book lie in an extended visit that I made in the Spring of 2008 to MSRI in Berkeley, California as the Simons Professor for the programme on Representation Theory of Finite Groups and Related Topics. Jon Carlson and Julia Pevtsova were there for a large part of this time, and Eric Friedlander passed through town several times. Through numerous conversations and lectures, they ignited my interest in the theory of modules of constant Jordan type. At first, I was reluctant to be drawn in. But then I managed to prove one of the conjectures from a paper of Carlson, Friedlander and Pevtsova [81], and I was hooked. This work has been published [42] and appears here as Section 5.12.

After that initial success, I started working with Julia Pevtsova on the vector bundles on projective space associated to modules of constant Jordan type. We refined a previous definition of Friedlander and Pevtsova, and proved a realisation theorem. This work has also been published [53] and appears here as Theorem 8.1.1, Sections 8.2–8.9 and Section 10.2.

I wrote several other papers soon after that, and soon it was getting to the point where I had so much material that it made sense to make a book based on the resulting series of papers as well as a great deal of other unpublished work. This is that book.

My thanks go to MSRI for its hospitality in 2008 and 2013 while parts of this work were being written; David Eisenbud for sharing his knowledge of vector bundles on projective spaces; Jon Carlson, Eric Friedlander and Julia Pevtsova for numerous conversations; Serge Bouc, Radu Stancu and Jon Carlson for their extensive feedback on earlier versions of the text; Jeremy Rickard for various interesting comments and questions, and particularly for formulating Conjecture 5.13.1; Andrew Granville for helping me with some number theoretic questions, and especially for formulating and supplying a proof of a statement similar to Lemma 12.10.1. I’d also like to thank Jesse Burke, Jon Carlson, Claudia Miller, Julia Pevtsova and Greg Stevenson for enlightening conversations and feedback related to Chapter 11, and Mark Walker for giving a talk at a conference in Seattle that stimulated my interest in Orlov’s correspondence.

Introduction

Let us begin with an example. Consider the following 30 × 30 matrices with entries in an algebraically closed field k of characteristic two.

( \begin{matrix}a \\ b \\ c \\ d \\ e \\ f \\ & a & & & & & f & e \\ & b & & & & f & & d \\ & c & & & & e & d \\ & d & & c & b \\ & e & c & & a \\ & f & b & a \\ & & d & & f & a & & c \\ & & e & & & & a \\ & & f & & & & & a \\ & & & d & & b \\ & & & e & f & & b & c \\ & & & f & & & & b \\ & & & & d & c \\ & & & & e & & c \\ & & & & & & & & & & & & f & e & & b & c & & a \\ & & & & & & & & & & & f & & d & b & & & a & & c \\ & & & & & & & & & & & e & d & & c & & & & c & & a & b \\ & & & & & & & & & c & b & & & & & & & e & d & & f \\ & & & & & & & & c & & a & & & & e & d & & & & & & f \\ & & & & & & & & b & a & & & & & f & & d & & f & e \\ & & & & & & & & d & e & f & a & b & c \\ & & & & & & & & & & & & & & & & & & & & & & d & e & f & a & b & c \end{matrix}  )

Blank entries are taken to be zero (there are two blank rows at the top and two blank columns on the right). These matrices, for different values of a, b, c, d, e, f ∈ k, all commute and square to zero. Furthermore, as long as a, b, c, d, e, f are not all zero, the matrix has rank exactly 14. So the kernel modulo the image is a two-dimensional vector space. This means that we have assigned to each point (a:b:c:d:e:f) ∈ ℙ^{5} a vector space of dimension two. This gives us a rank two algebraic vector bundle on projective 5-space, which happens to be the indecomposable Tango bundle.

On the other hand, if we add the identity to the six matrices obtained by setting one of the variables equal to 1 and the rest equal to 0, we obtain six commuting matrices which square to the identity. In other words, we have a kE-module, where E ≅ (ℤ/2)^{6} is an elementary abelian 2-group of rank six. This module has constant Jordan type [2]^{14}[1]^{2}, meaning that the matrices above all have the same Jordan canonical form with 14 blocks of length two and two blocks of length one.

This illustrates the connection between modules of constant Jordan type for elementary abelian p-groups and vector bundles on projective space in characteristic p investigated in this book. Because the people studying these two subjects are almost disjoint, I have tried to include plenty of introductory material. The reader should feel free to skip this if appropriate.

So what are modules of constant Jordan type? Among all modules, they are analogous to the vector bundles among the sheaves.

As Dade put it in [97], “There are just too many modules over p-groups!” More explicitly, the group algebra of a finite p-group in characteristic p usually has wild representation type, as we explain in Section 1.2. For a more general finite group, the Sylow p-subgroup controls the representation type. It follows that in general we do not hope to classify all the finite dimensional indecomposable representations of a finite group.

The theory of varieties for modules was developed by Carlson and others [24, 73, 74, 75] as a way of getting at module structure without making such a classification. Many aspects of this theory are controlled by the elementary abelian p-subgroups. So it makes sense to study modules for an elementary abelian p-group as a subject in its own right. We concentrate on modules of constant Jordan type. For these modules the variety gives essentially no information, so the theory supplements the now well established variety theory. Modules of constant Jordan type are still wild whenever the representation type of the group is wild, as we show in Section 5.5. These modules are much more rigidly behaved than the general module, but the theory is nonetheless surprisingly rich.

The formal definition of constant Jordan type is as follows. Let E = ⟨g_{1}, …, g_{r}⟩ ≅ (ℤ/p)^{r} be an elementary abelian p-group and let k be an algebraically closed field of characteristic p. We set

X_{i} = g_{i} − 1 ∈ kE

for 1 ≤ i ≤ r, so that X_{i}^{p} = 0. If α = (λ_{1}, …, λ_{r}) ∈ 𝔸^{r}(k), we define

X_{α} = λ_{1}X_{1} + ⋯ + λ_{r}X_{r} ∈ kE.

If α ≠ 0 then g_{α} = 1 + X_{α} is a unit of order p in kE.

A cyclic shifted subgroup of E is a subgroup of the group algebra kE of the form E_{α} = ⟨g_{α}⟩ for 0 ≠ α ∈ 𝔸^{r}(k). A finitely generated kE-module is said to have constant Jordan type if the Jordan canonical form of X_{α} on M is independent of α for 0 ≠ α ∈ 𝔸^{r}(k).

We are far from understanding what Jordan types occur for a module M of constant Jordan type. Of course if r = 1 then the problem is trivial. So let us assume that r ≥ 2. We write [a_{1}] … [a_{t}] for a Jordan type with blocks of lengths a_{1}, …, a_{t}, each of which is an integer between 1 and p. We often wish to ignore Jordan blocks of length p, and the stable Jordan type is the same list with the length p blocks omitted. The first important theorem in the subject is Dade’s lemma, from his 1978 paper [98], which states that if the stable Jordan type is empty, in other words if M has constant Jordan type [p]^{n} for some n ≥ 0, then M is projective. In particular, n is divisible by p^{r−1}. Using this, it is not hard to show that a module M of stable constant Jordan type [1] or [p − 1] is endotrivial, in the sense that M ⊗_{k} M^{∗} is trivial plus projective. Dade’s classification of endotrivial modules for an elementary abelian p-group then implies that M is isomorphic to Ω^{n}(k) plus a projective for some n ∈ ℤ.

In the paper of Carlson, Friedlander and Pevtsova [81], it is conjectured that for r ≥ 2 and p ≥ 5 there is no module of stable constant Jordan type [2]. In other words, there is no module with the property that every X_{α} acts with Jordan blocks all of length p except for a single block of length two. In section 5.12 we prove the more general statement that for r ≥ 2 and 2 ≤ a ≤ p − 2 there is no kE-module of stable constant Jordan type [a]. This completes the analysis of modules of constant Jordan type with one non-projective Jordan block. For larger stable Jordan types, our knowledge is much more limited. The following conjectures appear in Sections 5.13 and 5.15. We continue to assume that E has rank r ≥ 2.

Conjecture (Rickard). If a kE-module of constant Jordan type has no Jordan blocks of length j then the total number of Jordan blocks of length larger than j (including the blocks of length p) is divisible by p.

In Section 5.13 we prove the cases j = 1 and j = p−1 of Rickard’s conjecture. The proof involves the notion of generic kernel for modules over rank two elementary abelian groups, developed by Carlson, Friedlander and Suslin [82].

Conjecture (Suslin). If a kE-module of constant Jordan type has Jordan blocks of length j then it also has to have Jordan blocks of length either j +1 or j −1. In other words, there are no isolated lengths.

Conjecture (Carlson, Friedlander and Pevtsova). If M is a kE-module of stable constant Jordan type [2][1]^{j} then j ≥ r − 1.

In Section 10.4 we prove a weak version of the conjecture of Carlson, Friedlander and Pevtsova, on modules of stable constant Jordan type [2][1]^{j}. Namely we prove that j ≥ r − 2 if p is large enough. The proof of this uses the theory of Chern classes for vector bundles on projective space, a subject which we discuss in detail in Chapter 7.

When we talk of vector bundles, we are referring to algebraic vector bundles, namely locally free sheaves of modules over the structure sheaf. In the case of projective space ℙ^{r−1}, the only rank one vector bundles (line bundles) are twists of the structure sheaf 𝒪(a). There are plenty of indecomposable vector bundles of every rank at least r − 2 if r ≥ 3, but very little is known about vector bundles whose rank s is in the range 2 ≤ s ≤ r − 3. The only values of r and s in this range for which we know of indecomposable vector bundles are r = 5, s = 2 (Horrocks–Mumford), r = 6, s = 3 (Horrocks), and in characteristic two r = 6, s = 2 (Tango). A vector bundle in this range with other values of r and s will be referred to as a new low rank vector bundle on projective space.

Part of the point of Chapters 6 and 7 is to give an account of the theory of vector bundles on projective space, leading quickly and efficiently to the definition of Chern classes and a proof of the Hirzebruch–Riemann–Roch theorem in this case. This will greatly facilitate the discussion of restrictions on Jordan type coming from Chern classes.

If M is a kE-module of constant Jordan type, then we associate to M vector bundles ℱ_{1}(M), …, ℱ_{p}(M) on projective space ℙ^{r−1} in such a way that the rank of ℱ_{i}(M) is equal to the number of Jordan blocks of length i on a cyclic shifted subgroup of kE. The way that projective space enters the game is that points in ℙ^{r−1} correspond to cyclic shifted subgroups of kE (up to a scalar). In some sense, ℱ_{i}(M) associates to each point of projective space the socle of the sum of the Jordan blocks of length i of the restriction of M to the corresponding cyclic shifted subgroup. The twists ℱ_{i}(M), ℱ_{i}(M)(1), …, ℱ_{i}(M)(i − 1) associate to each point on projective space the successive socle layers of the length i blocks of the restriction.

At this point, an interesting question arises. What vector bundles occur this way? The answer to this question is quite different for p = 2 and p odd, as we shall see in Section 8.9. For p = 2, given any vector bundle ℱ of rank s on ℙ^{r−1}, there exists a kE-module M of stable constant Jordan type [1]^{s} such that ℱ_{1}(M) ≅ ℱ. The same construction with p odd only shows that given ℱ, there is a kE-module M of stable constant Jordan type [1]^{s} such that ℱ_{1}(M) ≅ F^{∗}(ℱ), the pullback of ℱ through the Frobenius map F on ℙ^{r−1}. And indeed, it turns out that without pulling back through the Frobenius map, there are restrictions coming from Chern classes. In Section 10.3 we show that if M has stable constant Jordan type [1]^{s} then the Chern numbers c_{1}(ℱ_{1}(M)), …, c_{p−2}(ℱ_{1}(M)) are divisible by p.

Further congruences on Chern numbers come from the Hirzebruch–Riemann–Roch theorem. For example, in Section 10.8 we prove the following. Let E ≅ (ℤ/2)^{r}. If M is a kE-module of constant Jordan type [2]^{n}[1]^{m} and n is not congruent to one of the integers 0, −1, …, −m modulo 2^{r−1} then ℱ_{1}(M) is a vector bundle on ℙ^{r−1} which is not a sum of line bundles. In particular, if r > 6 and m ≤ r−3 then ℱ_{1}(M) is a new low rank vector bundle on projective space. The Tango bundle example from the beginning of the introduction shows why the restriction r > 6 is necessary.

Finally, the last chapter is a bit more speculative. We investigate the general question of how to construct small modules with interesting properties. The constructions are basically the same as those used to construct modules of constant Jordan type, but we see that the applicability of the methods is much wider. Again, p = 2 behaves quite differently from p odd. So, for example, we shall see that a module of Loewy length p can have an arbitrary hypersurface as its variety if p = 2 but only finite unions of hyperplanes can be realised when p is odd.

1 Modular Representations and Elementary Abelian Groups

1.1. Introduction

This chapter introduces background material on the relevant aspects of modular representation theory of finite elementary abelian p-groups. We work over an algebraically closed field k of characteristic p, and we only consider finitely generated modules, even when some statements are true more generally.

Since the group algebra of an elementary abelian 2-group is just an exterior algebra, many of the methods we describe apply equally well to modules over exterior algebras in any characteristic. So to some extent we develop the theory of exterior algebras in parallel to that of elementary abelian p-groups.

In the representation theory of elementary abelian p-groups, we often have to describe separately the cases p = 2 and p odd. In the case of exterior algebras, usually no such separation of cases is necessary.

1.2. Representation Type

The trichotomy theorem (Drozd [107], Crawley–Boevey [96]) partitions finite dimensional algebras over an algebraically closed field into three mutually disjoint classes:

(i) Finite representation type: in this case there are only a finite number of isomorphism classes of finitely generated indecomposable modules.

(ii) Tame representation type: in this case there are infinitely many isomorphism classes of finitely generated indecomposables, but in any given dimension they come in one-parameter families, with finitely many exceptions. For algebras of tame representation type, one usually hopes to write down a complete classification of the finitely generated indecomposable modules.

(iii) Wild representation type: an algebra A has wild representation type if there is a finitely generated A-k⟨X, Y ⟩-bimodule B, free as a right k⟨X, Y ⟩-module, such that the functor B⊗_{k⟨X,Y ⟩}− from finite dimensional k⟨X, Y ⟩-modules to finite dimensional A-modules preserves indecomposability and reflects isomorphisms.

Broadly speaking, this means that a classification of finite dimensional indecomposable A-modules would entail the classification of pairs of square matrices under simultaneous conjugation. This problem is thought to be unsolvable.

Group algebras of finite groups are almost always of wild representation type.

Theorem 1.2.1 (Bondarenko and Drozd [57]; see also Ringel [220]). Let G be a finite group and k have characteristic p.

(i) If the Sylow p-subgroups of G are cyclic then kG has finite representation type.

(ii) If p = 2 and the Sylow p-subgroups of G are dihedral, semidihedral or generalised quaternion, then kG has tame representation type.

(iii) In all other cases kG has wild representation type.

Looking in particular at an elementary abelian p-group E, this says that finite representation type happens when E has rank one, tame representation type happens only for (ℤ/2)^{2}, and otherwise the representation type is wild.

In the rank one case, the classification follows easily from the theory of the Jordan canonical form. There is one indecomposable module for each size of Jordan block between 1 and p. In the case of (ℤ/2)^{2}, the classification follows from Kronecker’s classification of matrix pencils, see for example Section 4.3 of [39].

Next we discuss exterior algebras. Let Λ = Λ(X_{1}, …, X_{r}) be an exterior algebra over a field k on generators X_{1}, …, X_{r}. We can either consider this as an ungraded algebra, and look at ungraded modules over it, or we can consider it as a graded algebra, and look at graded modules. In the latter case, we put the generators X_{i} in degree one. The relations satisfied by the X_{i} are X_{i}^{2} = 0 (1 ≤ i ≤ r) and X_{i}X_{j} = −X_{j}X_{i} (1 ≤ i < j ≤ r).

Theorem 1.2.2. If Λ = Λ(X_{1}, …, X_{r}) is an exterior algebra over a field k on generators X_{1}, …, X_{r}, then the representation type of Λ either as an ungraded or as a graded algebra is as follows:

(i) finite if r = 1, (ii) tame if r = 2, and (iii) wild if r ≥ 3.

In case (i), if Λ is graded, we really mean that there are finitely many isomorphism classes of indecomposable modules up to shift in degree. In fact, there are only two isomorphism classes of indecomposables, of dimensions one and two, and shifts of them in the graded situation. If r = 2 then again the classification of the indecomposables follows from Kronecker’s classification. For r ≥ 3, wildness follows for example from Ringel [220], or from the wildness of the quiver \xymatrix{• \ar[r] \ar[r] \ar[r] & •}.

Given that we usually do not hope to classify the indecomposable kE-modules or the indecomposable Λ-modules, there are several approaches to making progress. One approach is to make classifications that are less refined. For example, the classification of thick subcategories of the stable module category is achieved in [49]. Another is to find general properties of modules, short of their classification. Examples of theorems in this direction are Dade’s lemma 1.9.5 and Carlson’s theory of rank varieties, discussed in Section 1.9, both of which we shall make use of in these notes. A third way to make progress is to restrict the class of modules under consideration. That is the approach taken in these notes. We shall concentrate largely on modules of constant Jordan type. There are many other interesting subclasses of the class of all finitely generated kE-modules, and we are in no way suggesting that this is the most important class.

1.3. Shifted Subgroups

We begin with a discussion of shifted subgroups of an elementary abelian p-group E. These are certain subgroups of the group algebra kE.

The notation that we use throughout this book is as follows. A finite group E is said to be an elementary abelian p-group if it is abelian and has exponent p. Equivalently, E is isomorphic to a direct product of r copies of a cyclic group of order p for some r ≥ 0:

E = ⟨g_{1}, …, g_{r}⟩ ≅ (ℤ/p)^{r}

where [g_{i}, g_{j}] = 1 for 1 ≤ i, j ≤ r and g_{i}^{p} = 1 for 1 ≤ i ≤ r.

We begin with the definition of shifted subgroup. Let

X_{i} = g_{i} − 1 ∈ kE

for 1 ≤ i ≤ r. Since we are in characteristic p we have

X_{i}^{p} = (g_{i} − 1)^{p} = g_{i}^{p} − 1^{p} = 0

in kE. The images of X_{1}, …, X_{r} form a basis for J(kE)/J^{2}(kE), where J(kE) denotes the Jacobson radical of kE. If α = (λ_{1}, …, λ_{r}) is an element of affine space 𝔸^{r}(k), we define

X_{α} = λ_{1}X_{1} + ⋯ + λ_{r}X_{r} ∈ kE.

This is an element of J(kE) which again satisfies X_{α}^{p} = 0. It follows that (1 + X_{α})^{p} = 1. So if α ≠ 0 then g_{α} = 1 + X_{α} is an element of order p in the group of units of the group algebra kE.

Definition 1.3.1. A cyclic shifted subgroup of E is a subgroup E_{α} of the group of units of kE generated by such an element g_{α} = 1 + X_{α} with α ≠ 0. More generally, a shifted subgroup is a subgroup E′ of the group of units in kE generated by elements g_{α_{1}}, …, g_{α_{s}} where α_{1}, …, α_{s} are linearly independent elements of 𝔸^{r}(k). This linear independence condition is equivalent to the statement that the induced map ρ: kE′ → kE is injective. If this is the case then kE is free as a module over the image of ρ.

A maximal shifted subgroup is a shifted subgroup E′ of the same rank as E. The induced map ρ: kE′ → kE is then an isomorphism. Choosing a basis of E′ allows us to identify it with E and obtain an automorphism ρ of kE.

This notion of a cyclic shifted subgroup is dependent on the choice of generators for E. For a different choice of generators, if we make the corresponding linear transformation on 𝔸^{r}(k), the resulting element X_{α} with respect to the new basis differs from the old one by an element of J^{2}(kE).

Let us illustrate this by example. If E = ⟨g_{1}, g_{2}⟩ ≅ (ℤ/p)^{2} and X_{1} = g_{1} − 1, X_{2} = g_{2} − 1, then a multiplicative basis change on E using the matrix

(\begin{matrix}1 & 0 \\ 1 & 1 \end{matrix})

gives new generators E = ⟨g_{1}g_{2}, g_{2}⟩. Now we have

g_{1}g_{2} − 1 = X_{1} + X_{2} + X_{1}X_{2}.

So modulo the square of the radical, we have the additive base change on X_{1} and X_{2} given by the same matrix. However, the linear span of g_{1}g_{2} − 1 and g_{2} − 1 is not the same as the linear span of X_{1} and X_{2}, so the definition of a shifted subgroup is basis dependent. In Chapter 4, we shall make a systematic study of the effect of adding an element of J^{2}(kE).

The following is immediate from the definitions.

Theorem 1.3.2. The group algebra kE is a truncated polynomial ring:

kE = k[X_{1}, …, X_{r}]/(X_{1}^{p}, …, X_{r}^{p}).

We define the rank variety of E to be the affine space V_{E}^{rk} ≅ 𝔸^{r}(k) obtained by taking the linear span in J(kE) of the elements

X_{1} = g_{1} − 1, …, X_{r} = g_{r} − 1.

Thus V_{E}^{rk} is a complement in J(kE) of J^{2}(kE). This is the home for the theory of rank varieties, as described in Section 1.9.

Warning 1.3.3. It is easy fall into the trap of thinking that the definitions

X_{i} = log(g_{i}) = (g_{i} − 1) − \frac{1}{2}(g_{i} − 1)^{2} + ⋯
X_{α} = λ_{1}X_{1} + ⋯ + λ_{r}X_{r}
g_{α} = exp(X_{α}) = 1 + X_{α} + \frac{1}{2!}X_{α}^{2} + ⋯

give a more invariant definition of shifted subgroups. The problem is that it is not true that

log(gh) = log(g) + log(h).

For example, in characteristic two we have log(g) = 1 + g, log(g) + log(h) = g + h and log(gh) = 1 + gh.

The existence of an invariant definition of shifted subgroups would imply the existence of a non-zero GL(r, 𝔽_{p})-invariant homomorphism ϕ: E → J(kE). Such a homomorphism exists if and only if E is either cyclic or isomorphic to (ℤ/2)^{2}.

To see this, we use the following argument, due to Serge Bouc. Let ϕ be such a homomorphism and write

ϕ(g) = ∑_{\begin{matrix}h∈E \\ h≠1 \end{matrix}} ϕ_{h}(g)(h − 1).

Invariance amounts to the statement that for α ∈ GL(r, 𝔽_{p}) we have ϕ_{h}(α(g)) = ϕ_{α^{−1}(h)}(g). The group GL(r, 𝔽_{p}) is transitive on pairs of elements g, h such that g is not in ⟨h⟩. So ϕ_{h}(g) is a constant a, independent of h and g, provided g ∉ ⟨h⟩. If E is neither cyclic nor isomorphic to (ℤ/2)^{2} then we can find linearly independent elements g and h such that ⟨g⟩, ⟨h⟩ and ⟨gh⟩ do not exhaust E. Then the statement that ϕ(gh) = ϕ(g) + ϕ(h) implies first that 2a = a, so that a = 0, and then that ϕ = 0 since ϕ(gh), ϕ(g) and ϕ(h) have supports intersecting in the identity.

The discussion of rank varieties for exterior algebras is essentially the same. Let Λ = Λ(X_{1}, …, X_{r}) be an exterior algebra on generators X_{1}, …, X_{r}. If α = (λ_{1}, …, λ_{r}) ∈ 𝔸^{r}(k), we define

X_{α} = λ_{1}X_{1} + ⋯ + λ_{r}X_{r} ∈ Λ.

This is an element of J(Λ) which satisfies X_{α}^{2} = 0. The rank variety of Λ is the affine space V_{Λ}^{rk} ≅ 𝔸^{r}(k) obtained by taking the linear span of X_{1}, …, X_{r} in J(Λ). It is a complement to J^{2}(Λ) in J(Λ). Details can be found in the paper of Aramova, Avramov and Herzog [12].

1.4. The Language of π-Points

Much of the modern literature on Jordan type for finite group schemes [80, 81, 82, 118, 124, 125, 126, 127, 128] is written in the language of π-points. It is worth making a few remarks on the translation between this and the language of cyclic shifted subgroups, in the case of an elementary abelian p-group. Understanding this section is not logically necessary for the rest of the book, but will help reconcile it with the rest of the literature.

Giving a finite group scheme G over k is equivalent to giving a finite dimensional cocommutative Hopf algebra kG over k. For example, if G is a finite group then kG is just its group algebra in the normal sense. Other examples include the restricted universal enveloping algebra of a finite dimensional p-restricted Lie algebra. In this example the underlying variety of the group scheme has only one point, but the local ring at that point contains all the information – it is just the dual Hopf algebra.

If G is a finite group scheme then a π-point of G is defined to be a flat homomorphism of algebras K[t]/(t^{p}) → KG over some extension field K of k, which factors through some unipotent abelian subgroup scheme of KG.

To say that a homomorphism K[t]/(t^{p}) → KG is flat means that KG is flat as a module over K[t]/(t^{p}), which is equivalent to each one of the following adjectives: free, projective, injective.

An equivalence relation is put on π-points as follows. Two π-points are said to be equivalent if for every finitely generated kG-module M, the restriction of K ⊗_{k} M along one is free if and only if the restriction along the other is free. This condition is difficult to work with in practice, since it is expressed in terms of all finitely generated modules, but can be reduced to an easier cohomological condition. See for example Theorem 3.6 of [125].

In the case where G = E is a finite elementary abelian p-group, KE is unipotent, so the second condition is automatically satisfied. The flatness condition is equivalent to the statement that the image of t lies in J(KE) but not in J^{2}(KE). The equivalence relation in this case is as follows. By Lemma 6.4 of Carlson [76], two such homomorphisms give equivalent π-points if and only if the differences between the images of t lies in J^{2}(KE). Thus the set of equivalence classes of π-points over K is in bijection with 𝔸^{r}(K) \ {0}. With respect to a given choice of generators of E, in each equivalence class there is a unique representative which is given by a cyclic shifted subgroup.

Note that an exterior algebra in characteristic not equal to two is not an example of a finite group scheme. Nonetheless, the notion of π-point makes perfect sense in this context, if defined as a flat embedding K[t]/(t^{2}) → K ⊗_{k} Λ.

1.5. The Stable Module Category

In general, if G is a finite group the cohomology ring H^{∗}(G, k) is defined to be Ext_{kG}^{∗}(k, k). By a theorem of Evens [115] and Venkov [242] (or an earlier theorem of Golod [136] in the case of finite p-groups), this is a finitely generated graded commutative algebra over k.

The module category mod(kG) is the category of finitely generated kG-modules and module homomorphisms. The stable module category stmod(kG) has the same objects as mod(kG) but its arrows are given by

\underline{Hom}_{kG}(M, N) = Hom_{kG}(M, N)/P Hom_{kG}(M, N),

where P Hom_{kG}(M, N) is the linear subspace of Hom_{kG}(M, N) consisting of homomorphisms that factor through some projective kG-module. Whereas mod(kG) is an abelian category, stmod(kG) is a triangulated category.

If M is a finitely generated kG-module, we define Ω(M) to be the kernel of the projective cover of M and Ω^{−1}(M) to be the cokernel of the injective hull of M. It follows from the fact that kG is self-injective that Ω(Ω^{−1}(M)) and Ω^{−1}(Ω(M)) are naturally isomorphic to M in the stable module category stmod(kG). If M has no projective summands, this implies that they are isomorphic to M in the module category mod(kG), but the isomorphism is not natural. If n > 0 we have

(1.5.1)  Ext_{kG}^{n}(M, N) ≅ \underline{Hom}_{kG}(Ω^{n}(M), N),

and in particular since P Hom_{kG}(Ω^{n}(k), k) = 0 we have

H^{n}(G, k) ≅ Ext_{kG}^{n}(k, k) ≅ Hom_{kG}(Ω^{n}(k), k).

If ζ ∈ H^{n}(G, k), we write ζ̂ for the corresponding homomorphism Ω^{n}(k) → k. If ζ ≠ 0 then ζ̂ is surjective, and the Carlson module L_{ζ} is defined to be the kernel of ζ̂. We shall study these modules further in Sections 1.10 and 12.4.

Tate cohomology is defined by extending (1.5.1) from positive degrees to all degrees:

Ex̂t_{kG}^{n}(M, N) ≅ \underline{Hom}_{kG}(Ω^{n}(M), N) (n ∈ ℤ).

The Tate cohomology ring of G is defined to be

Ĥ^{∗}(G, k) = Ex̂t_{kG}^{∗}(k, k)

with multiplication given by shifting and composing. This makes it into a graded commutative k-algebra with the ordinary cohomology H^{∗}(G, k) as the subalgebra of non-negative degree elements. The k-algebra Ĥ^{∗}(G, k) is finitely generated if and only if it is periodic, which in turn happens if and only if all abelian p-subgroups of G are cyclic.

Tate duality (see for example Cartan and Eilenberg [83], section XII.6) states that for n ∈ ℤ there are natural vector space isomorphisms

(Ĥ^{−n−1}(G, M))^{∗} ≅ Ex̂t_{kG}^{n}(M, k).

In particular, Ĥ^{−n−1}(G, k) is the vector space dual of Ĥ^{n}(G, k).

If H is a subgroup of G, then transfer in Tate cohomology from H to G is Tate dual to restriction from G to H.

Exactly the same definitions make sense in the case of an exterior algebra. If Λ = Λ(X_{1}, …, X_{r}) is an exterior algebra over k then the cohomology ring H^{∗}(Λ, k) = Ext_{Λ}^{∗}(k, k) is a polynomial ring k[Y_{1}, …, Y_{r}]. If Λ is regarded here as an ungraded algebra, then the Y_{i} are in cohomological degree one. So if k does not have characteristic two and r > 1 then H^{∗}(Λ, k) is not a graded commutative ring, because the Y_{i} do not commute with each other in the graded sense. On the other hand, if Λ is regarded as a graded algebra with the X_{i} in degree one, then H^{∗}(Λ, k) is bigraded, with the Y_{i} in degree (1, 1). The total degree is two, so the cohomology ring is graded commutative in this case.

Tate cohomology and Tate duality for exterior algebras works in the same way as for group algebras.

1.6. The Derived Category

If 𝒜 is an abelian category, we write D^{b}(𝒜) for the bounded derived category of 𝒜. Its objects are the bounded chain complexes of objects and arrows in 𝒜. The morphisms are the homotopy classes of degree preserving chain maps, with the quasi-isomorphisms inverted; a quasi-isomorphism is a morphism that induces an isomorphism in homology. Thus a morphism C → D in D^{b}(𝒜) is given by an equivalence class of diagrams of chain complexes

\xymatrix{ & C′ \ar[dl]_{qi} \ar[dr] \\ C &  & D.}

where “qi” denotes a quasi-isomorphism. Two such diagrams C ⟵^{qi} C′ → D and C ⟵^{qi} C′′ → D are equivalent if there is a third C ⟵^{qi} C′′′ → D that maps to both of them to make a commutative diagram

\xymatrix{ &  & C′′′ \ar@{-->}[dl]_{qi} \ar@{-->}[dr] \\  & C′ \ar[dl]_{qi} \ar[drrr] &  & C′′ \ar[dlll]^{qi} \ar[dr] \\ C &  &  &  & D.}

To compose arrows we form a pullback:

\xymatrix{ &  & C′′ \ar@{.>}[dl]_{qi} \ar@{.>}[dr] \\  & C′ \ar[dl]_{qi} \ar@{-->}[dr] &  & D′ \ar@{-->}[dl]^{qi} \ar[dr] \\ C &  & D &  & E.}

The distinguished triangles in D^{b}(𝒜) are the triangles isomorphic to

C ⟶^{f} D → M_{f} → C[1]

where M_{f} is the mapping cone of f.

The derived categories we shall be interested in are D^{b}(kG) = D^{b}(mod(kG)), the derived category of kG-modules, and D^{b}(Coh(X)), the derived category of coherent 𝒪_{V} -modules for a variety V.

Next we describe the construction of a functor

ℛ: D^{b}(kG) → stmod(kG).

This functor appeared in the late 1980s in the work of several people, see Buchweitz [66], Keller and Vossieck [176], Rickard [219, Theorem 2.1]. We shall use this construction in Section 8.9 as part of the proof of the realisation theorem.

Definition 1.6.1. A perfect complex is an object in D^{b}(kG) which is isomorphic to a bounded complex of projective modules. We write perf(kG) for the full subcategory of D^{b}(kG) whose objects are the perfect complexes.

The quotient category D^{b}(kG)/perf(kG) has the same objects as D^{b}(kG). The arrows are formed by inverting arrows whose mapping cone is in perf(kG).

Theorem 1.6.2. The quotient category D^{b}(kG)/perf(kG) is equivalent to the stable module category stmod(kG).

Proof (sketch). We begin with the functor mod(kG) → D^{b}(kG) which takes a module M to the complex which consists of M in degree zero and the zero module in all other degrees. The composite of this with the quotient functor

D^{b}(mod(kG)) → D^{b}(kG)/perf(kG)

takes projective modules to zero, and so it factors through mod(kG) → stmod(kG) to give a well defined functor

stmod(kG) → D^{b}(kG)/perf(kG).

One can check that this functor takes distinguished triangles to distinguished triangles. It is also full, and takes non-zero objects to non-zero objects. To prove that it is an equivalence, it therefore remains to prove that every object in D^{b}(kG)/perf(kG) is isomorphic to an object in the image of this functor.

Given a bounded chain complex of finitely generated kG-modules C, we can resolve it to give a quasi-isomorphism

\xymatrix{0 \ar[r] & K \ar[r] & P_{s} \ar[r] \ar[d] & ⋯ \ar[r] & P_{t} \ar[r] \ar[d] & 0 \\  & 0 \ar[r] & M_{s} \ar[r] & ⋯ \ar[r] & M_{t} \ar[r] & 0}

where the P_{i} are projective modules, and without loss of generality we can take s > 0. Now take an injective resolution of K. Then there is a quasi-isomorphism

(1.6.1)  \xymatrix{0 \ar[r] & K \ar[r] \ar[d] & Q_{s} \ar[r] \ar[d] & ⋯ \ar[r] & Q_{1} \ar[r] \ar[d] & Q_{0} \ar[r] \ar[d] & 0 \\ 0 \ar[r] & 0 \ar[r] & 0 \ar[r] & ⋯ \ar[r] & 0 \ar[r] & Ω^{−s−1}(K) \ar[r] & 0.}

Then modulo perf(kG) the original complex is isomorphic to K in degree s + 1, which is in turn isomorphic to the top row and then the bottom row of (1.6.1). The latter is in the image of stmod(kG). □

Definition 1.6.3. We write ℛ for the functor

ℛ: D^{b}(kG) → stmod(kG)

whose existence is guaranteed by the theorem. This functor takes distinguished triangles in D^{b}(kG) to distinguished triangles in stmod(kG).

The proof of the theorem shows that the image of a complex under this functor is obtained by resolving, taking the kernel, and then shifting it back to degree zero. It is possible, but tedious to check directly that this process is functorial.

It follows from the theorem that an object in D^{b}(kG) goes to zero in stmod(kG) if and only if it is a perfect complex.

In the case of the exterior algebra Λ, there are two versions of the derived category and of the stable module category, according to whether we are talking about ungraded modules or graded modules. In both cases, the discussion of the derived category and its relationship with the stable module category works in the same way as above. More generally, it works for any finite dimensional ungraded or graded self-injective algebra.

1.7. Singularity Categories

As a generalisation of the equivalence stmod(kG) ≃ D^{b}(kG)/perf(kG) of Theorem 1.6.2, we have the following definition, which will become important in Chapter 11.

Definition 1.7.1. Let R be a ring. Then the singularity category of R is the Verdier quotient

D_{sg}(R) = D^{b}(R)/perf(R).

Likewise, if R is a graded ring, we denote by D^{b}(R) the bounded derived category of finitely generated graded R-modules and D_{sg}(R) the quotient by the the perfect complexes of graded modules.

Warning 1.7.2. In commutative algebra, this definition is much better behaved for Gorenstein rings than for more general commutative Noetherian rings. For a Gorenstein ring, the singularity category is equivalent to the stable category of maximal Cohen–Macaulay modules (Buchweitz [66]), but the following example is typical of the behaviour for non-Gorenstein rings.

Example 1.7.3. Let R be the ring

k[X, Y]/(X^{2}, XY, Y^{2}).

Then the radical of R is isomorphic to k ⊕ k, so there is a short exact sequence of R-modules

0 → k ⊕ k → R → k → 0.

This means that in D_{sg}(R) the connecting homomorphism of this short exact sequence gives an isomorphism k ≅ k[1] ⊕ k[1]. We have

k ≅ k[1]^{⊕2} ≅ k[2]^{⊕4} ≅ k[3]^{⊕8} ≅ ⋯

and so k is an infinitely divisible module. Its endomorphism ring End_{D_{sg}(R)}(k) is the colimit of

k → Mat_{2}(k) → Mat_{4}(k) → Mat_{8}(k) → ⋯

where each matrix ring is embedded diagonally into a product of two copies, sitting in the next matrix ring. In fact, this endomorphism ring is an example of a von Neumann regular ring. For a generalisation of this example to finite dimensional algebras with radical square zero, see Chen [88].

The reason why D_{sg}(R) is called the “singularity category” is that it only “sees” the singular locus of R.

Example 1.7.4. If R is a regular ring then R has finite global dimension. So we have D^{b}(R) = perf(R) and hence D_{sg}(R) = 0.

More generally, we have the following definition and theorem.

Definition 1.7.5. Let R be a [graded] Noetherian commutative ring. Then the singular locus of R is the set of [homogeneous] prime ideals 𝔭 of R such that the [homogeneous] localisation R_{𝔭} is not regular.

Remark 1.7.6. Provided that R satisfies a mild technical condition known as “excellence,” the singular locus is a Zariski closed set, so that it is of the form V (I) for some [homogeneous] radical ideal I of R. Thus a ∈ I if and only if R[a^{−1}] is regular. Quotients of polynomial rings, for example, are excellent.

Theorem 1.7.7. Let R be a [graded] Noetherian commutative ring of finite Krull dimension whose singular locus is a Zariski closed set. Then D_{sg}(R) is generated by [the graded shifts of] the modules R/𝔭 where 𝔭 is a [homogeneous] prime ideal in the singular locus of R.

Remark 1.7.8. In the ungraded case, the theorem of Schoutens [223] implies the above theorem, but it is stronger, and the proof is more complicated. Schoutens’ theorem also holds in the graded case, with minor adjustments to the proof.

I’d like to thank Srikanth Iyengar for suggesting the simple proof presented here. The idea behind this argument also appears in Lemma 2.2 of Herzog and Popescu [151], and in Iyengar and Takahashi [168]. This theorem will be used in the proof of Proposition 11.9.1.

Proof of Theorem 1.7.7. Let d = dim R and let V (I) be the singular locus of R. It suffices to show that if M is a finitely generated [graded] R-module then M is in the thick subcategory of D^{b}(R) generated by [graded shifts of] R and of R/𝔭 with 𝔭 ∈ V (I), i.e., with 𝔭 ⊇ I.

The first step is to replace M by its dth syzygy Ω^{d}(M), i.e., the dth kernel in any [graded] resolution of M by finitely generated free [graded] R-modules (we allow free graded modules to be sums of shifts of R). Thus we may assume that M is a dth syzygy.

We claim that if a is a [homogeneous] element of I then for some n > 0, a^{n} annihilates Ext_{R}^{1}(M, Ω(M)). This is because R[a^{−1}] has global dimension at most d, so the fact that M is a dth syzygy implies that M[a^{−1}] is also a dth syzygy and is hence projective as an R[a^{−1}]-module. So Ext_{R}^{1}(M, Ω(M))[a^{−1}] = Ext_{R[a^{−1}]}^{1}(M[a^{−1}], Ω(M[a^{−1}])) = 0.

Apply this to the extension 0 → Ω(M) → F → M → 0, with F a finitely generated free [graded] R-module. Multiplying this extension by a^{n} amounts to forming the pullback X in the following diagram:

[figure: 0; Ω(M); X; M; a^{n}; 0; 0; Ω(M); F; M; 0]

The resulting extension splits, so X ≅ M ⊕Ω(M). The snake lemma implies that the middle vertical arrow gives rise to an exact sequence

0 → Ker (a^{n}, M) → X → F → M/a^{n}M → 0.

Now Ker (a^{n}, M) and M/a^{n}M are annihilated by a^{n}. So X, and hence M, is in the thick subcategory of D^{b}(R) generated by R and modules supported on R/aR. Now inducting on a finite set of generators for the ideal I, we see that M is in the thick subcategory of D^{b}(R) generated by R and modules supported on V (I). The latter are in turn generated by the R/𝔭 with 𝔭 ∈ V (I). □

1.8. Cohomology of Elementary Abelian p-Groups

The ring structure of the cohomology of E ≅ (ℤ/p)^{r} depends on whether p = 2 or p is odd.

Case 1. If p = 2 then the cohomology is a polynomial ring

H^{∗}(E, k) = k[y_{1}, …, y_{r}]

where the generators y_{i} are in degree one.

Case 2. If p is odd then the cohomology is the tensor product of an exterior algebra and a polynomial ring

H^{∗}(E, k) = Λ(y_{1}, …, y_{r}) ⊗_{k} k[x_{1}, …, x_{r}]

where the exterior generators y_{i} are in degree one and the polynomial generators x_{i} are in degree two. The Bockstein map β: H^{1}(E, k) → H^{2}(E, k) is the semilinear map given by

(1.8.1)  β(∑_{i=1}^{r} λ_{i}y_{i}) = ∑_{i} λ_{i}^{p}x_{i}.

It is semilinear with respect to the Frobenius map, in the sense that for y ∈ H^{1}(E, k) and λ ∈ k we have β(λy) = λ^{p}β(y). The Bockstein map is related to the p-fold Massey product by the formula

β(y) = −⟨y, y, …, y⟩ (p terms)

for y ∈ H^{1}(E, k) (Kraines [180], Theorem 14). The Frobenius twist involved in the Bockstein map will manifest itself in Theorem 8.1.1, where it explains the difference in behaviour between p = 2 and p odd for the realisation of vector bundles from modules of constant Jordan type.

In the proof of Theorem 8.1.1, we shall need to use the nature of explicit maps representing the elements y_{i} (p = 2), x_{i} (p odd). We begin by discussing the case of a cyclic group of order p. Let C = ⟨g⟩ ≅ ℤ/p, and let C be the periodic projective resolution of k for kC given by

⋯ kC ⟶^{1−g} kC ⟶^{1+g+···+g^{p−1}} kC ⟶^{1−g} kC → 0.

If p = 2 this is periodic of period one, and there is a periodicity map ŷ: C → C of degree one representing y ∈ H^{1}(C, k). If p is odd it has period two, and there is a periodicity map x̂: C → C of degree two representing x ∈ H^{2}(C, k).

Now if E = ⟨g_{1}, …, g_{r}⟩ ≅ (ℤ/p)^{r}, we let P = C ⊗ ⋯ ⊗ C (r factors). Then P is a minimal resolution of k as a kE-module. Let

ŷ_{i} = 1 ⊗ ⋯ ⊗ ŷ ⊗ ⋯ ⊗ 1: P → P (p = 2)
x̂_{i} = 1 ⊗ ⋯ ⊗ x̂ ⊗ ⋯ ⊗ 1: P → P (p odd)

Here, all terms in the tensor product of maps are equal to 1 except the ith term, which is ŷ, respectively x̂. The elements ŷ_{i} and x̂_{i} represent the elements y_{i} ∈ H^{1}(E, k) (p = 2), respectively x_{i} ∈ H^{2}(E, k) (p odd). These maps commute, and so we get a ring homomorphism

k[y_{1}, …, y_{r}] → Hom_{kE}^{∗}(P, P) (p = 2)
k[x_{1}, …, x_{r}] → Hom_{kE}^{∗}(P, P) (p odd).

Letting Ω^{n}(k) be the nth kernel in the complex P, we get induced maps which by abuse of notation we write as

y_{i}: Ω^{n+1}(k) → Ω^{n}(k) (p = 2)
x_{i}: Ω^{n+2}(k) → Ω^{n}(k) (p odd)

for all n ≥ 0. They strictly commute, in the sense that y_{i}y_{j} and y_{j}y_{i} give the same map Ω^{n+2}(k) → Ω^{n}(k) (p = 2); and x_{i}x_{j} and x_{j}x_{i} give the same map Ω^{n+4}(k) → Ω^{n}(k) (p odd).

Example 1.8.1. It is easy to compute explicit minimal resolutions over elementary abelian p-groups, either by hand or in larger cases using computer algebra packages such as Magma or Macaulay2. For example, if E = (ℤ/p)^{2}, with X_{1} = g_{1} − 1 and X_{2} = g_{2} − 1, here is the beginning of the minimal resolution of k as a module over kE

6kE ⟶^{( \begin{matrix}X_{1} & X_{2} & 0 & 0 & 0 & 0 \\ 0 & X_{1}^{p−1} & X_{2}^{p−1} & 0 & 0 & 0 \\ 0 & 0 & X_{1} & X_{2} & 0 & 0 \\ 0 & 0 & 0 & X_{1}^{p−1} & X_{2}^{p−1} & 0 \\ 0 & 0 & 0 & 0 & X_{1} & X_{2} \end{matrix} )} 5kE ⟶^{( \begin{matrix}X_{1}^{p−1} & −X_{2} & 0 & 0 & 0 \\ 0 & X_{1} & −X_{2}^{p−1} & 0 & 0 \\ 0 & 0 & X_{1}^{p−1} & −X_{2} & 0 \\ 0 & 0 & 0 & X_{1} & −X_{2}^{p−1} \end{matrix} )} 4kE

⟶^{( \begin{matrix}X_{1} & X_{2} & 0 & 0 \\ 0 & X_{1}^{p−1} & X_{2}^{p−1} & 0 \\ 0 & 0 & X_{1} & X_{2} \end{matrix} )} 3kE \underrightarrow{( \begin{matrix}X_{1}^{p−1} & −X_{2} & 0 \\ 0 & X_{1} & −X_{2}^{p−1} \end{matrix} )} 2kE ⟶^{( X_{1} X_{2} )} kE ⟶ k

and here is the beginning of the minimal resolution of the three dimensional kE-module kE/J^{2}(kE)

9kE ⟶^{( \begin{matrix}X_{1}^{2} & X_{1}X_{2} & X_{2}^{2} & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & X_{1}^{p−1} & X_{1}^{p−2}X_{2} & X_{2}^{p−1} & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & X_{1}^{p−1} & X_{1}X_{2}^{p−2} & X_{2}^{p−1} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & X_{1}^{2} & X_{1}X_{2} & X_{2}^{2} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & X_{1}^{p−1} & X_{1}^{p−2}X_{2} & X_{2}^{p−1} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & X_{1}^{p−1} & X_{1}X_{2}^{p−2} & X_{2}^{p−1} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & X_{1}^{2} & X_{1}X_{2} & X_{2}^{2} \end{matrix} )} 7kE

⟶^{( \begin{matrix}X_{1}^{p−2} & −X_{2} & 0 & 0 & 0 & 0 & 0 \\ 0 & X_{1} & −X_{2} & 0 & 0 & 0 & 0 \\ 0 & 0 & X_{1} & −X_{2}^{p−2} & 0 & 0 & 0 \\ 0 & 0 & 0 & X_{1}^{p−2} & −X_{2} & 0 & 0 \\ 0 & 0 & 0 & 0 & X_{1} & −X_{2} & 0 \\ 0 & 0 & 0 & 0 & 0 & X_{1} & −X_{2}^{p−1} \end{matrix} )} 6kE ⟶^{( \begin{matrix}X_{1}^{2} & X_{1}X_{2} & X_{2}^{2} & 0 & 0 & 0 \\ 0 & X_{1}^{p−1} & X_{1}^{p−2}X_{2} & X_{2}^{p−1} & 0 & 0 \\ 0 & 0 & X_{1}^{p−1} & X_{1}X_{2}^{p−2} & X_{2}^{p−1} & 0 \\ 0 & 0 & 0 & X_{1}^{2} & X_{1}X_{2} & X_{2}^{2} \end{matrix} )} 4kE

⟶^{( \begin{matrix}X_{1}^{p−2} & −X_{2} & 0 & 0 \\ 0 & X_{1} & −X_{2} & 0 \\ 0 & 0 & X_{1} & −X_{2}^{p−2} \end{matrix} )} 3kE \underrightarrow{(_{X_{1}^{2}} _{X_{1}X_{2}} _{X_{2}^{2}} )} kE ⟶ kE/J^{2}(kE).

The number of copies of kE at the nth stage of the minimal resolution of a module M is dim_{k} Ext_{kE}^{n}(M, k). The Poincaré series

∑_{n=0}^{∞} dim_{k} Ext_{kE}^{n}(M, k)t^{n}

is a rational function of t of the form (polynomial)/(1 − t^{2})^{c} where c is the complexity of M. The value of the polynomial in the numerator at t = 1 is a positive integer which we shall denote d(M). David Eisenbud asks whether it is always true that d(M) ≥ 2^{r} if c = r. The example of kE/J^{2}(kE) above shows that it does not have to be an integer multiple of 2^{r} (in this case c = r = 2 and the degree is 6).

1.9. Chouinard’s Theorem, Dade’s Lemma and Rank Varieties

Part of the importance of elementary abelian groups in the modular representation theory of finite groups is explained by the following theorem of Chouinard. First recall that if G is a finite group and k is a field then the conditions that a kG-module is projective, injective, and flat are all equivalent, and that if G is a finite p-group then these conditions are also equivalent to the kG-module being free.

Theorem 1.9.1 (Chouinard [89]). Let G be a finite group. A kG-module M is projective if and only if the restriction of M to every elementary abelian p-subgroup of G is projective.

Remark 1.9.2. This theorem was proved by Chouinard in the generality of an arbitrary commutative ring k of coefficients and arbitrary modules without the restriction of finite generation. In this context one needs to use the elementary abelian p-groups for all p dividing the group order.

Example 1.9.3. To see how remarkable this theorem is, let us look at the example of the generalised quaternion group of order 2^{n} (n ≥ 3),

G = Q_{8} = ⟨g, h | g^{2^{n−1}} = 1, h^{2} = g^{2^{n−2}}, hg = g^{−1}h^{−1}⟩

with k a field of characteristic two. Set z = g^{2^{n−2}}, so that Z(G) = {1, z}. The subgroup Z(G) is the only non-trivial elementary abelian subgroup of G, so the theorem states that a kGmodule M is projective if and only if M↓_{Z(G)} is projective. Since the only indecomposable modules for Z(G) are the trivial module of dimension one and the regular representation of dimension two, it follows that if M is finite dimensional then it is projective if and only if the rank of the matrix representing 1 + z ∈ kG is as large as it can be, namely one half of the dimension of M.

Remark 1.9.4. There are many other related aspects of group cohomology and modular representation theory where the elementary abelian p-subgroups play a crucial role. See the papers of Quillen [206, 207, 210], Carlson [72, 73, 74, 76, 77, 79], Alperin and Evens [7, 8], Avrunin and Scott [24], Jackowski and McClure [169], Kroll [181], Linckelmann [188], Symonds [232], Webb [243], etc.

Chouinard’s theorem suggests that it is important to be able to decide whether a kEmodule is projective. The following test only works for finitely generated kE-modules with k algebraically closed.

Lemma 1.9.5 (Dade’s lemma; [98], lemma 11.8). A finitely generated kE-module M is projective if and only if the restriction of M to each cyclic shifted subgroup (with respect to a chosen system of generators for E) is projective.

Remark 1.9.6. Dade’s lemma in the form stated is not true for infinitely generated modules, nor for finitely generated modules over a field that is not algebraically closed. For further information see Benson, Carlson and Rickard [48].

Dade’s lemma motivates the following definition, introduced by Carlson.

Definition 1.9.7. Recall that V_{E}^{rk} denotes the affine space 𝔸^{r}(k) of dimension equal to the rank of E. The rank variety V_{E}^{rk}(M) of a finitely generated kE-module M is defined to be

V_{E}^{rk}(M) = {0} ∪ {0 ≠ α ∈ V_{E}^{rk} | M↓_{⟨g_{α}⟩} is not projective}.

To test whether M↓_{⟨g_{α}⟩} is projective, look at the rank of the linear transformation effected by X_{α} on M. This is at most (p − 1)/p times the dimension of M, with equality if and only if the restriction is projective.

Lemma 1.9.8 (Carlson [74, §6]). Let x, y be elements of J(kE) \ J^{2}(kE) with x − y ∈ J^{2}(kE) and let M be a finitely generated kE-module. Then M↓_{⟨1+x⟩} is projective if and only if M↓_{⟨1+y⟩} is projective.

This may be seen as a consequence of the following, which appears as Proposition 2.2 in Friedlander and Pevtsova [124].

Proposition 1.9.9. Let M be a k-vector space (recall that k has characteristic p), and let α, β, γ be pairwise commuting endomorphisms of M with α^{p} = 0, β^{p} = 0 and γ nilpotent. Then M is free as a k[t]/(t^{p})-module with t acting via α if and only if it is free with t acting via α + βγ.

Proof. A k[t]/(t^{p})-module is free if and only if Ker (t^{p−1}) = Im (t), or equivalently Ker (t) = Im (t^{p−1}). So we need to check that if Ker ((α + βγ)^{p−1}, M) = Im (α + βγ, M) then Ker (α^{p−1}, M) = Im (α, M). Let V = Ker (α^{p−1}, M)/Im (α, M). Since γ is nilpotent, it suffices to show that the map V → V induced by γ is injective. Suppose that x ∈ Ker (α^{p−1}, M) with γx ∈ Im (α, M). Then α^{p−1}x = 0 and there exists y ∈ M with γx = αy. Let

δ = ∑_{i=0}^{p−2}(−1)^{i}α^{p−2−i}β^{i+1}γ^{i} = α^{p−2}β − α^{p−3}β^{2}γ + ⋯ − β^{p−1}γ^{p−2},

so that

αδ = (α + βγ)^{p−1}β, γδ = α^{p−1} − (α + βγ)^{p−1}.

Then we have

(α + βγ)^{p−1}x = (α^{p−1} − γδ)x = −αδy = −(α + βγ)^{p−1}βy,

so x + βy is in the kernel of (α + βγ)^{p−1}. Therefore there is an element z ∈ M with

(α + βγ)z = x + βy.

Then we have

(α + βγ)y = γx + βγy = γ(α + βγ)z

and therefore there is an element u ∈ M with

(α + βγ)^{p−1}u = y − γz.

Finally, we have

α(z − δu) = (x + βy − βγz) − (α + βγ)^{p−1}βu = (x + βy − βγz) − β(y − γz) = x,

and so x ∈ Im (α, M). □

Remark 1.9.10. Lemma 1.9.8 is a special case of Theorem 4.5.3. Proposition 1.9.9 is closely related to the discussion in Chapter 4.

The following summarises the basic properties of rank varieties that we shall be using. Proofs can be found in Carlson [74, 75] or in Benson [40].

Proposition 1.9.11. Let M and N be kE-modules.

(1) V_{E}^{rk}(M) is a closed homogeneous subvariety of V_{E}^{rk} = 𝔸^{r}(k).

(2) dim V_{E}^{rk}(M) is equal to the complexity of M.

(3) V_{E}^{rk}(M ⊕ N) = V_{E}^{rk}(M) ∪ V_{E}^{rk}(N).

(4) V_{E}^{rk}(M ⊗_{k} N) = V_{E}^{rk}(M) ∩ V_{E}^{rk}(N).

(5) If 0 → M_{1} → M_{2} → M_{3} → 0 is a short exact sequence of kE-modules then

V_{E}^{rk}(M_{2}) ⊆ V_{E}^{rk}(M_{1}) ∪ V_{E}^{rk}(M_{3}).

(6) (Carlson’s connectedness theorem) If V_{E}^{rk}(M) = V_{1} ∪ V_{2} where V_{1} and V_{2} are closed homogeneous subsets of V_{E}^{rk} with V_{1} ∩ V_{2} = {0} then M decomposes as M_{1} ⊕ M_{2} with V_{E}^{rk}(M_{1}) = V_{1} and V_{E}^{rk}(M_{2}) = V_{2}.

An exactly parallel discussion holds for exterior algebras.

Lemma 1.9.12 (Dade’s lemma for exterior algebras). Let Λ = Λ(X_{1}, …, X_{r}) be an exterior algebra over an algebraically closed field k. Then a finitely generated Λ-module M is projective if and only if it is projective on restriction to X_{α} for every 0 ≠ α ∈ 𝔸^{r}(k).

Proof. See Theorem 3.1 of Aramova, Avramov and Herzog [12]. □

Definition 1.9.13. Recall that V_{Λ}^{rk} denotes the affine space 𝔸^{r}(k). The rank variety V_{Λ}^{rk}(M) of a finitely generated Λ-module M is defined to be

V_{Λ}^{rk}(M) = {0} ∪ {0 ≠ α ∈ V_{Λ}^{rk} | M↓_{⟨X_{α}⟩} is not projective}.

Again, the test for projectivity on restriction to X_{α} is easy. The matrix for the action of X_{α} on M has rank at most half the dimension of M, with equality if and only if the restriction of M to X_{α} is projective.

Proposition 1.9.14. The rank varieties of Λ-modules satisfy the properties listed in Proposition 1.9.11 for rank varieties of kE-modules.

Proof. Again, see Theorem 3.1 of Aramova, Avramov and Herzog [12]. □

1.10. Carlson’s L_{ζ} Modules, and a Matrix Version

In this section, we describe Carlson’s L_{ζ} modules, and variations on them defined using matrices of elements of cohomology. This section will be crucial to the discussions in Chapter 12, where we also explore variations using relative cohomology.

Let 0 ≠ α = (λ_{1}, …, λ_{r}) ∈ 𝔸^{r}(k). Recall that X_{α} = ∑_{i=1}^{r} λ_{i}X_{i} ∈ kE and g_{α} = 1 + X_{α}, so that ⟨g_{α}⟩ ≅ ℤ/p. Let H^{∗}(⟨g_{α}⟩, k) = k[y_{α}] (p = 2) or k[y_{α}, x_{α}]/(y_{α}^{2}) (p odd), where y_{α} and x_{α} are the elements of degrees one and two described in Section 1.8.

The following theorem appears as Proposition 2.20 in Carlson [74].

Theorem 1.10.1. The restriction of y_{i} to ⟨g_{α}⟩ is equal to λ_{i}y_{α}. If p is odd, then the restriction of x_{i} to ⟨g_{α}⟩ is equal to λ_{i}^{p}x_{α}.

Proof. For the y_{i} this is easy, using the duality between H^{1}(E, k) and V_{E}^{rk}. If p is odd, the formula for the restriction of x_{i} comes from the fact that restriction commutes with the Bockstein map, together with the formula (1.8.1). □

Corollary 1.10.2. Let f be a polynomial in r variables. If p = 2 then the restriction of f(y_{1}, …, y_{r}) to ⟨g_{α}⟩ is equal to ∑ f(λ_{1}, …, λ_{r})y_{α}. If p is odd then the restriction of f(x_{1}, …, x_{r}) to ⟨g_{α}⟩ is equal to f(λ_{1}^{p}, …, λ_{r}^{p})x_{α}. □

This allows us to compute the rank varieties of Carlson’s L_{ζ} modules. Let ζ ∈ H^{n}(E, k) be represented by a cocycle ζ̂: Ω^{n}(k) → k. As long as ζ is non-zero, ζ̂ is surjective, and we write L_{ζ} for the kernel. Thus we have a short exact sequence

0 → L_{ζ} → Ω^{n}(k) ⟶^{ζ̂} k → 0.

If ζ = 0, we define L_{ζ} to be Ω^{n}(k) ⊕ Ω(k). The reasoning for this is that in both cases, we are really completing ζ̂ to a triangle in stmod(kE):

L_{ζ} → Ω^{n}(k) → k.

Since restriction to ⟨g_{α}⟩ gives a well defined functor from stmod(kE) to stmod(k⟨g_{α}⟩), we have the following.

Proposition 1.10.3. For p = 2, let ζ = f(y_{1}, …, y_{r}). Then V_{E}^{rk}(L_{ζ}) is the hypersurface defined by the equation f(Y_{1}, …, Y_{r}) = 0.

For p odd, let ζ = f(x_{1}, …, x_{r}). Then V_{E}^{rk}(L_{ζ}) is the hypersurface defined by the equation f^{(p)}(Y_{1}, …, Y_{r}) = 0, where f^{(p)} is obtained from f by raising all the coefficients to the pth power.

Proof. A point α is in V_{E}^{rk}(L_{ζ}) precisely when the restriction of ζ̂ is not an isomorphism in stmod(k⟨g_{α}⟩), namely when the restriction of ζ is zero. Now use Corollary 1.10.2. □

Corollary 1.10.4 (Carlson). Every closed homogeneous subvariety of V_{E}^{rk} is the rank variety of some finitely generated kE-module.

Proof. A closed homogeneous subvariety of affine space can always be expressed as an intersection of a finite set of closed homogeneous hypersurfaces. For such a hypersurface, we can apply Proposition 1.10.3. For the intersection, we tensor together these modules and use Proposition 1.9.11 (4). □

A variation on this is given by using matrices of cohomology elements. Let (f_{ij}) be an n × n matrix of homogeneous polynomials in r variables. Let us suppose that these satisfy the following condition.

Condition 1.10.5. There are integers a_{1}, …, a_{n} and b_{1}, …, b_{n} such that f_{ij}(y_{1}, …, y_{r}) (p = 2), respectively f_{ij}(x_{1}, …, x_{r}) (p odd), is an element of H^{a_{i}−b_{j}}(E, k).

Then we can represent it by a cocycle f̂_{ij}: Ω^{b_{j}}(k) → Ω^{a_{i}}(k). Thus the matrix (f_{ij}) gives us a map

(f̂_{ij}): ⨁_{j=1}^{n} Ω^{b_{j}}(k) → ⨁_{i=1}^{n} Ω^{a_{i}}(k).

Complete this to a triangle in stmod(kE):

⨁_{i=1}^{n} Ω^{a_{i}+1}(k) → L_{(f_{ij})} → ⨁_{j=1}^{n} Ω^{b_{j}}(k) → ⨁_{i=1}^{n} Ω^{a_{i}}(k).

Theorem 1.10.6. If p = 2 then V_{E}^{rk}(L_{(f_{ij})}) is the hypersurface defined by the equation det(f_{ij}) = 0. If p is odd, then V_{E}^{rk}(L_{(f_{ij})}) is the hypersurface defined by the equation det(f_{ij})^{(p)} = 0.

Proof. Again, a point α is in V_{E}^{rk}(L_{(f_{ij})}) precisely when the restriction of (f̂_{ij}) is not an isomorphism in stmod(⟨g_{α}⟩), namely when the determinant restricts to zero. Again use Corollary 1.10.2. □

1.11. Diagrams for Modules

We shall sometimes describe kE-modules by drawing diagrams. The meaning of these diagrams is that vertices represent basis elements and edges represent the action of the elements X_{i}. Not all modules have such diagrams, but when they do, the diagram is very convenient for understanding the module. More formal details of diagrams for modules can be found in Benson and Carlson [46]. We shall illustrate the ideas with some examples without going into formalities. It will be instructive to compare and contrast these diagrams with the ones occurring in Section 9.2.

For our first example, we let E = (ℤ/3)^{2}, and we draw the following diagram for the group algebra kE.

[figure: •^{1}; •; •; •; •; •; •; •; •]

Going downwards, single edges represent multiplication by X_{1} and double edges by X_{2}. If there is no edge of a particular kind leading downwards from a basis element, then the corresponding X_{i} acts as zero.

The top basis element, labelled “1”, represents the identity element of kE. So the two basis elements just below that are X_{1} and X_{2}, the three below that are X_{1}^{2}, X_{1}X_{2} and X_{2}^{2}, the two below that are X_{1}^{2}X_{2} and X_{1}X_{2}^{2}, and finally the bottom basis element is X_{1}^{2}X_{2}^{2}, which is annihilated by both X_{1} and X_{2}.

The augmentation map kE → k takes the top basis element to 1 and the remaining basis elements to zero, and the kernel of this map is Ω(k). Similarly, there is a map k → kE sending 1 to X_{1}^{2}X_{2}^{2}, whose cokernel is Ω^{−1}(k). It follows that the diagrams for these modules are as follows.

Ω(k) = [figure: •^{y_{1}}; •^{y_{2}}; •; •; •; •; •; •] Ω^{−1}(k) = [figure: •; •; •; •; •; •; •; •]

The top vertices of Ω(k) have been labelled with the exterior generators

y_{1}, y_{2} ∈ H^{1}(E, k) ≅ Ext_{kE}^{1}(k, k) ≅ Hom_{kE}(Ω(k), k).

The homomorphism ŷ_{1} : Ω(k) → k corresponding to y_{1} ∈ H^{1}(E, k), for example, sends the top left hand vertex to 1 and the remaining vertices to zero.

The projective cover of Ω(k) is two copies of kE, in such a way that the identity elements map to the two top vertices of Ω(k). The four vertices in the middle get hit twice, so there are diagonal elements in the kernel. It is not hard to see that this gives the following diagram for Ω^{2}(k).

Ω^{2}(k) = [figure: •^{y_{1}y_{2}}; •^{x_{1}}; •; •; •^{x_{2}}; •; •; •; •; •]

Similarly, the following is the diagram for Ω^{3}(k).

Ω^{3}(k) = [figure: •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •]

From the diagram for Ω^{2}(k), we can see immediately for example that there is a short exact sequence of the form

0 → Ω^{−1}(k) → Ω^{2}(k) → k ⊕ k → 0.

Under the isomorphism

H^{2}(E, k) ≅ Ext_{kE}^{2}(k, k) ≅ Hom_{kE}(Ω^{2}(k), k),

let x̂_{1} : Ω^{2}(k) → k correspond to x_{1} ∈ H^{2}(E, k). Then x̂_{1} is the homomorphism that takes the top left-hand vertex of the diagram for Ω^{2}(k) to 1 and the remaining basis elements to zero. So we have a Carlson module

L_{x_{1}} = Ker (x̂_{1}) = [figure: •; •; •; •; •; •; •; •; •]

Similarly we have

L_{x_{2}} = Ker (x̂_{2}) = [figure: •; •; •; •; •; •; •; •; •]

These are periodic modules with period two, i.e., Ω^{2}(L_{x_{i}}) ≅ L_{x_{i}}. We also have L_{x_{i}}^{∗} ≅ Ω(L_{x_{i}}). For p = 2 the syzygies of the trivial module are easier to compute, and are as follows:

Ω(k) = [figure: •; •; •] Ω^{2}(k) = [figure: •; •; •; •; •]

Ω^{−1}(k) = [figure: •; •; •] Ω^{−2}(k) = [figure: •; •; •; •; •]

For elementary abelian groups of rank three and higher, it is harder to draw diagrams except in very restricted circumstances, because the page is two dimensional. We will see just one rank four example in Section 12.2. But it is worth demonstrating how to draw diagrams for (ℤ/2)^{3}. For this purpose, we use a bird’s eye view, looking from the “top” of the module downwards. So for example Ω(k) and Ω^{−1}(k) are represented by a cube with the top, respectively bottom vertex removed:

[figure: Ω(k) =; Ω^{−1}(k) =]

To find Ω^{2}(k), we map three copies of kE onto the three top vertices of Ω(k), which in the diagram are top, bottom left and bottom right. The kernel then has the following diagram.

[figure: Ω^{2}(k) =]

The only thing to watch in this diagram is that the centre vertex really represents a vector space of dimension two, with three incoming maps and three outgoing maps corresponding to the three subspaces of dimension one defined over 𝔽_{2}. Thus Ω^{2}(k) has dimension 17.

It is easy to continue this way, so that we have

[figure: Ω^{3}(k) =]

There are now three vertices in this diagram that represent vector spaces of dimension two, while the rest are one dimensional. Thus Ω^{3}(k) is 31 dimensional.

Carlson modules can be described as before. So for example if we let x_{1} be the element of H^{1}(E, k) corresponding to the uppermost vertex in the diagram for Ω(k), then

[figure: L_{x_{1}} =; L_{x_{1}^{2}} =; L_{x_{1}^{3}} =]

1.12. Tensor Products

If G is any finite group, and M and N are kG-modules, we make M ⊗_{k} N into a kGmodule via

g(m ⊗ n) = gm ⊗ gn

for g ∈ G, m ∈ M and n ∈ N. Note, however, that for a linear combination of group elements the action is given by

(∑_{i} λ_{i}g_{i})(m ⊗ n) = ∑_{i} λ_{i}(g_{i}m ⊗ g_{i}n)

and not by

(∑_{i} λ_{i}g_{i})(m ⊗ n) = (∑_{i} λ_{i}g_{i}m) ⊗ (∑_{i} λ_{i}g_{i}n).

Indeed, the latter formula does not even define an action, because it is not linear. This can be seen in terms of the diagonal map Δ: kG → kG ⊗_{k} kG given by

Δ(∑_{i} λ_{i}g_{i}) = ∑_{i} λ_{i}(g_{i} ⊗ g_{i}).

Then any element x ∈ kG acts on M ⊗_{k} N by taking m ⊗ n to Δ(x)(m ⊗ n). The elements x ∈ kG satisfying Δ(x) = x ⊗ x are precisely the elements of G.

Warning 1.12.1. As a result of the way the action on tensor products is defined, if E is an elementary abelian p-group and M and N are kE-modules, then the restriction of M ⊗_{k} N to a cyclic shifted subgroup is not necessarily isomorphic to the tensor product of the restrictions.

Example 1.12.2. Let E = ⟨g_{1}, g_{2}⟩ ≅ ℤ/2 × ℤ/2 and let M and N both be the representation of dimension two given by

g_{1} ↦ (\begin{matrix}1 & 0 \\ 1 & 1 \end{matrix}) g_{2} ↦ (\begin{matrix}1 & 0 \\ λ & 1 \end{matrix})

with λ ∈ k \ {0, 1}. Let α = (λ, 1) ∈ 𝔸^{2}(k), so that

X_{α} = λ(g_{1} − 1) + (g_{2} − 1), g_{α} = 1 + X_{α}.

Then M and N each restricts to ⟨g_{α}⟩ to give a sum of two copies of the trivial module, while M ⊗_{k} N restricts to give a sum of two copies of the trivial module and an indecomposable module of dimension two.

There is a way to remedy this problem by changing the diagonal map. Setting

E = ⟨g_{1}, …, g_{r}⟩, X_{i} = g_{i} − 1 ∈ kE

as usual, the map Δ defined above sends J(kE) into

J(kE) ⊗ kE ⊕ kE ⊗ J(kE),

and

Δ(X_{i}) ≡ X_{i} ⊗ 1 + 1 ⊗ X_{i} (mod J(kE) ⊗ J(kE)).

If instead of Δ we use a different diagonal map Δ̃ given by

Δ̃(X_{i}) = X_{i} ⊗ 1 + 1 ⊗ X_{i}

then the tensor product defined using this new diagonal map does indeed commute with restriction to shifted subgroups. To distinguish between these two module structures on the tensor product, we write M ⊗_{k} N for the group theoretic tensor product defined using Δ and M ˜⊗_{k} N for the restricted Lie tensor product defined using Δ̃. This name comes from the fact that the diagonal map Δ̃ is the one obtained by viewing kE as the restricted enveloping algebra of the restricted Lie algebra given by the linear span of X_{1}, …, X_{r} with trivial Lie bracket and trivial pth power map. The reason why restriction to shifted subgroups commutes with this tensor product is that a shifted subgroup is the restricted enveloping algebra of a restricted Lie subalgebra.

We also remark that the difference between the two diagonal maps Δ and Δ̃ consists of “second order terms”. This observation will be useful in Chapter 4 when we want to show that the shift from one tensor product to the other does not alter the generic Jordan type.

Next, we discuss tensor products of modules for exterior algebras. Let Λ = Λ(X_{1}, …, X_{r}), and M and N be graded Λ-modules. Then M ⊗_{k} N is a Λ-module via

X_{i}(m ⊗ n) = X_{i}m ⊗ n + (−1)^{|m|}m ⊗ X_{i}n.

The corresponding diagonal map Δ: Λ → Λ ⊗_{k} Λ is given by

Δ(∑_{i} λ_{i}X_{i}) = ∑_{i} λ_{i}(X_{i} ⊗ 1 + 1 ⊗ X_{i}).

Warning 1.12.3. As long as we are not in characteristic two, the formula above for Δ is only an algebra homomorphism if we treat Λ as a graded algebra. This is because if we try to use this formula for the ungraded algebra then

Δ(X_{i})^{2} = 2(X_{i} ⊗ X_{i}) ≠ 0.

The signs introduced by the grading make sure that Δ(X_{i})^{2} = 0.

The consequence of this is that if we wish to form tensor products of Λ-modules then we must work in the graded context.

1.13. Duality

Just as with tensor products, there are two ways to define duality, namely the group theoretic dual and the restricted Lie dual, corresponding to the two different antipodes on kE. The antipode η: kE → kE for the group theoretic dual is the algebra (anti-)automorphism defined by η(g) = g^{−1} for g ∈ E. Thus

η(X_{i}) = −X_{i} + X_{i}^{2} − ⋯ + X_{i}^{p−1}.

The action of kE on the dual M^{∗} = Hom_{k}(M, k) of a kE-module M is given by (xf)(m) = f(η(x)m) for x ∈ kE, m ∈ M and f ∈ M^{∗}.

On the other hand, the antipode η̃: kE → kE for the restricted Lie dual is the algebra (anti-)automorphism defined by η̃(X_{i}) = −X_{i}. The action of kE on the restricted Lie dual M ˜∗ is given by (xf)(m) = f(η̃(x)m). Restriction to a shifted subgroup commutes with the restricted Lie dual, but not with the group theoretic dual.

The maps Δ and η make kE into a Hopf algebra, as do Δ̃ and η̃. The behaviours of these two different Hopf structures on kE are quite different.

Example 1.13.1. Let k be an algebraically closed field of characteristic p ≥ 3 and let E = ⟨g_{1}, g_{2}⟩ ≅ ℤ/p × ℤ/p. Let M_{λ,μ} be the kE-module

X_{1} ↦ (\begin{matrix}0 & 0 & 0 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{matrix}) X_{2} ↦ (\begin{matrix}0 & 0 & 0 \\ λ & 0 & 0 \\ μ & λ & 0 \end{matrix}) .

It is easy to check that for different pairs (λ, μ) these modules are non-isomorphic. The group theoretic dual M_{λ,μ}^{∗} is given by the matrices

X_{1} ↦ (\begin{matrix}0 & −1 & 1 \\ 0 & 0 & −1 \\ 0 & 0 & 0 \end{matrix} ) X_{2} ↦ (\begin{matrix}0 & −λ & λ^{2} − μ \\ 0 & 0 & −λ \\ 0 & 0 & 0 \end{matrix})

so that it is isomorphic to M_{λ,λ^{2}−λ−μ}, whereas the Lie theoretic dual M ˜∗ ; λ,μ is given by the matrices

X_{1} ↦ (\begin{matrix}0 & −1 & 0 \\ 0 & 0 & −1 \\ 0 & 0 & 0 \end{matrix} ) X_{2} ↦ (\begin{matrix}0 & −λ & −μ \\ 0 & 0 & −λ \\ 0 & 0 & 0 \end{matrix} )

so that it is isomorphic to M_{λ,−μ}.

Now let λ ∉ {0, 1} and M = M_{λ,0}. Set α = (λ, −1), so that X_{α} = λX_{1} − X_{2}. Then the restriction to X_{α} of M^{∗} has Jordan canonical form [2] [1] while the restriction of M ˜∗ is trivial, with Jordan type [1]^{3}. So for the group theoretic dual, the restriction of the dual to X_{α} can have a different Jordan type than the restriction of the module.

Furthermore, in this example the kernel of the map kE → End_{k}(M^{∗}) is not equal to the kernel of kE → End_{k}(M).

Example 1.13.2. Given a diagram for a module in the sense of Section 1.11, a diagram for the restricted Lie dual may be obtained by turning it upside down and negating the edges, in the following sense. Let m_{1}, …, m_{n} is a basis for M such that X_{i} takes m_{j} to λm_{j′}. Then with respect to the dual basis m_{1}^{∗}, …, m_{n}^{∗}, X_{i} takes m_{j′}^{∗} to −λm_{j}^{∗}. Often one can replace the elements of the dual basis by multiples in order to reduce the number of minus signs.

For example, if M is the module for ℤ/p × ℤ/p with diagram

[figure: •; •; •; •; •; •]

then M ˜∗ has diagram

[figure: •; •; •; •; •; •]

The sign problem is solved in this case by negating the basis elements in the middle row.

More care is needed with the group theoretic dual, and as an exercise the reader may wish to verify that in the example above, the group theoretic dual M^{∗} has the same diagram after a suitable change of basis.

Duality for exterior algebras works in a similar way. Let Λ = Λ(X_{1}, …, X_{r}). The antipode η: Λ → Λ is given by

η(X_{i}) = −X_{i}.

So the action of Λ on the dual M^{∗} = Hom_{k}(M, k) of a graded module M is given by X_{i}f(m) = −f(X_{i}m).

The maps Δ and η make Λ into a graded Hopf algebra.

1.14. Symmetric and Exterior Powers

We define the dth symmetric power S^{d}(M) of a module M to be the quotient of the d-fold tensor product M^{⊗d} by the linear span of the elements

m_{1} ⊗ ⋯ ⊗ m_{d} − m_{π(1)} ⊗ ⋯ ⊗ m_{π(d)}

with m_{1}, …, m_{d} ∈ M and π ∈ 𝔖_{d}, the symmetric group of degree d.^{1} We write m_{1} … m_{d} for the image of m_{1} ⊗ ⋯ ⊗ m_{d} in S^{d}(M). The following lemma is clear from the definition.

π(m_{1} ⊗ ⋯ ⊗ m_{d}) = m_{π^{−1}(1)} ⊗ ⋯ ⊗ m_{π^{−1}(d)}.

1Note that the natural action of 𝔖_{d} on M^{⊗d} is given by

Lemma 1.14.1. If M and N are modules then

S^{d}(M ⊕ N) ≅ ⨁_{i=0}^{d} S^{i}(M) ⊗ S^{d−i}(N).

Similarly, we define the dth exterior power Λ^{d}(M) as the quotient of M^{⊗d} by the elements

m_{1} ⊗ ⋯ ⊗ m_{d}

with m_{1}, …, m_{d} ∈ M, and such that there exist i ≠ j with m_{i} = m_{j}. As long as p > 2, this is the same as the quotient by the elements

m_{1} ⊗ ⋯ ⊗ m_{d} − sgn(π)m_{π(1)} ⊗ ⋯ ⊗ m_{π(d)}

with m_{1}, …, m_{d} ∈ M and π ∈ 𝔖_{d}. We write m_{1} ∧ ⋯ ∧ m_{d} for the image of m_{1} ⊗ … m_{d} in Λ^{d}(M).

Remark 1.14.2. The antisymmetrisation map

m_{1} ∧ ⋯ ∧ m_{d} ↦ ∑_{π∈𝔖_{d}} sgn(π)m_{π(1)} ⊗ ⋯ ⊗ m_{π(d)}

gives a natural embedding of Λ^{d}(M) in M^{⊗d}.

Beware, though, that S^{d}(M) can only be naturally embedded in M^{⊗d} when d! is invertible. The fixed points of 𝔖_{d} on M^{⊗d} give the d-fold divided power D^{d}(M), which is related to the d-fold symmetric power by D^{d}(M)^{∗} ≅ S^{d}(M^{∗}). The divided power D^{d}(M) is naturally isomorphic to S^{d}(M), provided d! is invertible; see Lemma 1.14.4.

Lemma 1.14.3. If M and N are modules then

Λ^{d}(M ⊕ N) ≅ ⨁_{i=0}^{d} Λ^{i}(M) ⊗ Λ^{d−i}(N).

Lemma 1.14.4. If d < p, then S^{d}(M) and Λ^{d}(M) are direct summands of M^{⊗d}.

Proof. The natural surjections M^{⊗d} → S^{d}(M) and M^{⊗d} → Λ^{d}(M) are split by the maps

m_{1} … m_{d} ↦ \frac{1}{d!} ∑_{π∈𝔖_{d}} m_{π(1)} ⊗ ⋯ ⊗ m_{π(d)},

respectively

m_{1} ∧ ⋯ ∧ m_{d} ↦ \frac{1}{d!} ∑_{π∈𝔖_{d}} sgn(π)m_{π(1)} ⊗ ⋯ ⊗ m_{π(d)}.  □

1.15. Schur Functions

We begin with some standard definitions concerning partitions. If

λ_{1} ≥ λ_{2} ≥ ⋯ ≥ λ_{t} ≥ 0

with ∑_{i=1}^{t} λ_{i} = d, we write [λ_{1}] … [λ_{t}] for the corresponding partition of d. We usually ignore trailing zeros, so that [λ_{1}] … [λ_{t}] represents the same partition as [λ_{1}] … [λ_{t}] [0] … [0]. This way we can assume that if we are working with several partitions, the values of t are equal. Repeated parts are denoted with indices, so that [2]^{3}[1] means [2] [2] [2] [1], for example.^{2}

Definition 1.15.1. If λ = [λ_{1}] … [λ_{t}] is a partition, then the conjugate partition λ′ = [λ′_{1}][λ′_{2}] … is defined by

λ′_{i} = |{j | λ_{j} ≥ i}|.

Lemma 1.15.2. For every partition λ we have λ′′ = λ. In other words, the conjugate of the conjugate is the original partition.

Definition 1.15.3. Let x_{1}, …, x_{n} be variables and let λ = [λ_{1}][λ_{2}] … be a partition of d. The Schur function s_{λ} is defined by

s_{λ} = s_{λ}(x_{1}, …, x_{n}) = \frac{det(x_{i}^{λ_{j}+n−j})_{1≤i,j≤n}}{det(x_{i}^{n−j})_{1≤i,j≤n}} .

The denominator is the Vandermonde determinant

∏_{i<j}(x_{i} − x_{j}),

and it divides the numerator because setting any two of the variables equal makes the numerator vanish. It follows that s_{λ} is a symmetric polynomial in x_{1}, …, x_{n} with integer coefficients.

For example if λ = [2] then

(1.15.1)  s_{λ} = \begin{matrix}| & \begin{matrix}x_{1}^{n+1} & ⋯ & x_{n}^{n+1} \\ x_{1}^{n−2} & & x_{n}^{n−2} \\ ⋮ & & ⋮ \\ 1 & ⋯ & 1 \end{matrix} & | \\ | & \begin{matrix}x_{1}^{n−1} & ⋯ & x_{n}^{n−1} \\ x_{1}^{n−2} & & x_{n}^{n−2} \\ ⋮ & & ⋮ \\ 1 & ⋯ & 1 \end{matrix} & | \end{matrix} = x_{1}^{2} + ⋯ + x_{n}^{2} + x_{1}x_{2} + x_{1}x_{3} + ⋯ + x_{n−1}x_{n}.

More generally, s_{[d]} is the complete symmetric function h_{d} of degree d.

The following formula, variously known as Jacobi–Trudi identity, the first Gambelli formula or the determinantal form, expresses s_{λ} in terms of the h_{j} (see for example Macdonald [189], Section I.3):

(1.15.2)  s_{λ} = det(h_{λ_{i}−i+j})_{1≤i,j≤r}

(r at least the number of parts in λ). For example we have

s_{[2][1]} = | \begin{matrix}h_{2} & h_{3} \\ h_{0} & h_{1} \end{matrix} | = h_{2}h_{1} − h_{3}.

The easiest way to write down this determinant is to put the h_{λ_{i}} down the diagonal and then fill in horizontally, decreasing the subscript to the left and increasing it to the right. Note that h_{0} = 1 and that if j < 0 then h_{j} = 0.

2It is more common to write [2^{3}, 1], but this notation is ambiguous, because for example it does not distinguish between the partitions [p^{2}] and [p]^{2} in our notation.

Let e_{d} be the elementary symmetric function of degree d, so that for example

e_{2} = x_{1}x_{2} + x_{1}x_{3} + ⋯ + x_{n−1}x_{n}.

Then e_{d} is equal to the Schur function s_{[1]^{d}}. The second Gambelli formula expresses s_{λ} in terms of the e_{j} (again see Macdonald [189], Section I.3):

s_{λ} = det(e_{λ′_{i}−i+j})_{1≤i,j≤r}

(r at least the number of parts of λ′, the conjugate of the partition λ).

For use in Section 2.4, we also record the obvious recurrence relations satisfied by the complete and elementary symmetric functions h_{d} and e_{d}.

Proposition 1.15.4. We have

(1) e_{d}(x_{1}, …, x_{n}) = e_{d}(x_{1}, …, x_{n−1}) + x_{n}e_{d−1}(x_{1}, …, x_{n−1}),

(2) h_{d}(x_{1}, …, x_{n}) = h_{d}(x_{1}, …, x_{n−1}) + x_{n}h_{d−1}(x_{1}, …, x_{n}). □

1.16. Schur Functors

If M is a module, we saw in Section 1.14 that there is a natural embedding of Λ^{d}(M) in M^{⊗d} and a natural surjection of M^{⊗d} onto S^{d}(M). If λ is a partition of d, with conjugate partition λ′, we look at the composite map

Λ^{λ′_{1}}(M) ⊗ Λ^{λ′_{2}}(M) ⊗ ⋯ → M^{⊗d} → S^{λ_{1}}(M) ⊗ S^{λ_{2}}(M) ⊗ · · · .

Here, the tensor factors of M^{⊗d} should be thought of as being indexed by the boxes in the partition λ. The exterior powers refer to the columns and the symmetric powers refer to the rows of the partition.

Definition 1.16.1. The Schur functor S^{λ} takes M to the image of the map described above.

Example 1.16.2. We have

S^{[d]}(M) = S^{d}(M), S^{[1]^{d}}(M) = Λ^{d}(M).

Example 1.16.3. Let λ = [3][1], λ′ = [2][1]^{2}:

\begin{matrix}m_{1} & m_{2} & m_{3} \\ m_{4} \end{matrix}

The maps

Λ^{2}(M) ⊗ M ⊗ M → M^{⊗4} → S^{3}(M) ⊗ M

are given by

(m_{1} ∧ m_{4}) ⊗ m_{2} ⊗ m_{3} → m_{1} ⊗ m_{2} ⊗ m_{3} ⊗ m_{4} − m_{4} ⊗ m_{2} ⊗ m_{3} ⊗ m_{1}
→ m_{1}m_{2}m_{3} ⊗ m_{4} − m_{2}m_{3}m_{4} ⊗ m_{1}.

This definition was introduced by Akin, Buchsbaum and Weyman [1] as a characteristic free definition of the Schur functor. Their definition associates this functor to the conjugate partition λ′ rather than λ. Our choice of notation is designed to fit better with the labelling of Schur functions in Section 1.15.

If we take for M the natural module of dimension n for the general linear group GL(n, k) then the modules S^{λ}(M) are the contragredient duals of the Weyl modules W^{λ}. In characteristic zero we have S^{λ}(M) ≅ W^{λ}, and these give a set of representatives for the isomorphism classes of simple polynomial representations of GL(n, k) as λ runs over the set of partitions with at most n parts. But in characteristic p they are not in general simple. In this case the simple module L^{λ}(M) is isomorphic to the unique simple submodule of S^{λ}(M) and the unique simple quotient of W^{λ}, and the modules L^{λ}(M) give a set of representatives for the isomorphism classes of simple polynomial representations as λ runs over the set of partitions with at most n parts.

If k has characteristic zero or characteristic p > d then there is another description of S^{λ}(M) as a direct summand of M^{⊗d}. In this case, the group algebra of the symmetric group k𝔖_{d} is semisimple, and its irreducible representations are indexed by partitions of d. If M is a representation of a group G then M^{⊗d} is a representation of G × 𝔖_{d}, where 𝔖_{d} permutes the tensor factors. So if e_{λ} ∈ k𝔖_{d} is a primitive idempotent corresponding to a partition λ of d then

S^{λ}(M) ≅ e_{λ}.M^{⊗d}.

All choices of e_{λ} are conjugate in k𝔖_{d}, so the right-hand side is independent of this choice up to isomorphism.

In this semisimple case, the Schur functors are simple functors. On the other hand, if p ≤ d then the Schur functor is often not simple. For example, even the pth symmetric power functor S^{p} has a subfunctor F^{p} (Frobenius pth power functor) where F^{p}(M) consists of the pth powers of elements of M. In general S^{λ} has a unique simple subfunctor. It is denoted L^{λ}. Every simple polynomial functor is isomorphic to a unique L^{λ}. The functors L^{λ} will appear again in Section 1.17.

The relationship between Schur functors and Schur functions is as follows. If we choose a basis v_{1}, …, v_{n} of M consisting of eigenvectors for a semisimple element g ∈ G with eigenvalues x_{1}, …, x_{n} then the eigenvalues of g on S^{λ}(M) are precisely the monomials appearing in s_{λ}(x_{1}, …, x_{n}). Thus the Schur function computes the character of the Schur functor.

Theorem 1.16.4. Suppose that λ is a partition of d and the characteristic of k is either zero or p > d. Then we have a determinantal form

S^{λ}(M) = det(S^{λ_{i}−i+j}(M))_{1≤i,j≤r}

(r at least the number of parts of λ). □

For example we have

S^{[2][1]}(M) = | \begin{matrix}S^{2}(M) & S^{3}(M) \\ S^{0}(M) & S^{1}(M) \end{matrix} | .

Since S^{0}(M) = k and S^{1}(M) = M, the meaning of this equation is that

S^{2}(M) ⊗ M ≅ S^{[2][1]}(M) ⊕ S^{3}(M).

1.17. Radical Layers of kE

In this section we describe the radical layers of kE in terms of Schur functors. Recall from Section 1.3 that kE is a truncated polynomial ring

kE = k[X_{1}, …, X_{r}]/(X_{1}^{p}, …, X_{r}^{p}).

The nth radical layer J^{n}(kE)/J^{n+1}(kE) is isomorphic to the image of the space of homogeneous polynomials of degree n in X_{1}, …, X_{r} in this quotient ring. It has a basis consisting of monomials in which each X_{i} appears to at most the (p − 1)st power. Thus we have

J^{(p−1)r+1}(kE) = 0.

We let GL(r, k) act on kE by linear substitutions on the linear span V of X_{1}, …, X_{r}. Hence GL(r, k) acts on the radical layers J^{n}(kE)/J^{n+1}(kE) for 0 ≤ n ≤ (p − 1)r, making it a quotient of the nth symmetric power S^{n}(V):

J^{n}(kE)/J^{n+1}(kE) ≅ S^{n}(V)/S^{n}(V) ∩ (X_{1}^{p}, …, X_{r}^{p}).

Thus for n ≥ p there is an exact sequence

F^{p}(V) ⊗ S^{n−p}(V) → S^{n}(V) → J^{n}(kE)/J^{n+1}(kE) → 0

where F^{p}(V) is the Frobenius pth power of V, spanned by the monomials X_{1}^{p}, …, X_{r}^{p}.

The submodule structure of S^{n}(V) was determined by Doty [105]. It follows from the calculations there that J^{n}(kE)/J^{n+1}(kE) is a simple GL(r, k)-module for 0 ≤ n ≤ (p − 1)r. The corresponding partition is given as follows. Let n = (p − 1)m + s with 0 ≤ s ≤ p − 1 and let L^{λ}(V) be the simple GL(r, k)-module corresponding to a partition λ. Thus L^{λ} is the simple functor discussed in Section 1.16. Then

J^{n}(kE)/J^{n+1}(kE) ≅ L^{[p−1]^{m}[s]}(V).

See Section 9.3 for how this structure is reflected in the vector bundles coming from the regular representation of kE.

For example, the radical layers of k(ℤ/3)^{3} are

k, V, S^{2}(V), L^{[2][1]}(V), L^{[2]^{2}}(V), L^{[2]^{2}[1]}(V), L^{[2]^{3}}(V).

of dimensions, 1, 3, 6, 7, 6, 3, 1. Note that for SL(3, k) we have L^{[2]^{2}}(V) ≅ S^{2}(V^{∗}), L^{[2]^{2}[1]}(V) ≅ V^{∗} and L^{[2]^{3}} is the trivial representation. For GL(3, k) one has to tensor with the determinant representation.

More generally for any p and r, we have

L^{[p−1]^{m}[s]}(V) ≅ L^{[p−1]^{r−m−1}[p−s]}(V^{∗}) ⊗ det .

Theorem 1.17.1. If n < p then the ideal of kE generated by the elements X_{α}^{n}, α ∈ 𝔸^{r}, is equal to J^{n}(kE).

Proof. We offer two proofs: one is at the level of elements and the other is at the level of representations.

The first proof goes as follows. Let I be the ideal generated by the X_{α}^{n}. Clearly I ⊆ J^{n}(kE). To prove that J^{n}(kE) ⊆ I, we will show that every monomial of degree n in X_{1}, …, X_{r} is in I.

We begin with the case r = 2, as this is easiest. For λ ∈ k we have

(X_{1} + λX_{2})^{n} = X_{1}^{n} + nλX_{1}^{n−1}X_{2} + \binom{n}{2}λ^{2}X_{1}^{n−2}X_{2}^{2} + ⋯ + λ^{n}X_{2}^{n} ∈ I.

Choose n + 1 distinct values a_{1}, …, a_{n+1} for λ. Since the matrix

A = (\begin{matrix}1 & na_{1} & \binom{n}{2}a_{1}^{2} & ⋯ & a_{1}^{n} \\ 1 & na_{2} & \binom{n}{2}a_{2}^{2} & ⋯ & a_{2}^{n} \\ ⋮ & & & ⋮ \\ 1 & na_{n+1} & \binom{n}{2}a_{n+1}^{2} & ⋯ & a_{n+1}^{n} \end{matrix})

is obtained by multiplying the columns of a Vandermonde matrix by binomial coefficients, which are non-zero because n < p, we have

det A = ∏_{i=0}^{n} \binom{n}{i}. ∏_{1≤i<j≤n+1}(a_{j} − a_{i}) ≠ 0.

It follows that A is invertible, and so each monomial X_{1}^{n}, X_{1}^{n−1}X_{2}, …, X_{2}^{n} is in I.

For larger values of r, the argument is similar except that the matrix is more complicated. For λ_{2}, …, λ_{r} ∈ k we have

(X_{1} + λ_{2}X_{2} + ⋯ + λ_{r}X_{r})^{n} = ∑_{\begin{matrix}0≤i_{2}≤n \\ ... \\ 0≤i_{r}≤n \end{matrix}} \binom{n}{i_{2} … i_{r}}λ_{2}^{i_{2}} … λ_{r}^{i_{r}}X_{1}^{n−i_{2}−···−i_{r}}X_{2}^{i_{2}} … X_{r}^{i_{r}}.

This sum has (n + 1)^{r−1} terms, but the ones with i_{2} + ⋯ + i_{r} > n are zero. Choose n + 1 distinct elements a_{1}, …, a_{n+1} ∈ k and consider this identity for each choice of values a_{1}, …, a_{n+1} for each of λ_{2}, …, λ_{r}, giving a total of (n + 1)^{r−1} such choices.

Now consider the matrix B obtained by taking a Kronecker tensor product of r −1 copies of the Vandermonde determinant

(\begin{matrix}1 & a_{1} & ⋯ & a_{1}^{n} \\ ⋮ & & & ⋮ \\ 1 & a_{n+1} & ⋯ & a_{n+1}^{n} \end{matrix}) .

Then

det B = ( ∏_{1≤i<j≤n+1}(a_{j} − a_{i}))^{r−1}

and so B is non-singular. Deleting the columns where the total degree is larger than n gives a rectangular matrix with full column rank. It follows that there exists a choice of rows giving a non-singular square matrix. Since n < p, multiplying the columns by the multinomial coefficients \binom{n}{i_{1}...i_{r}} again gives a nonsingular matrix, and so each monomial of degree n in X_{1}, …, X_{r} is in I.

The second proof using representation theory is much shorter. It consists of the observation that J^{n}(kE)/J^{n+1}(kE) is irreducible as a module for GL(r, k) (see above). The image in this space of the linear span of the X_{α}^{n} is a non-zero invariant subspace, and is hence the whole of J^{n}(kE)/J^{n+1}(kE). Now use Nakayama’s lemma. □

1.18. Twisted Versions of kE

Let p be an odd prime, and let k be a field of characteristic p. Consider the group G = (ℤ/p)^{2} ⋊ Q_{8} where Q_{8} = ⟨i, j, k | i^{2} = j^{2} = k^{2} = ijk⟩ is the quaternion group of order eight, and where the action of Q_{8} on (ℤ/p)^{2} is given by letting the central involution z = i^{2} = j^{2} = k^{2} act trivially, and the quotient Q_{8}/Z(Q_{8}) ≅ (ℤ/2)^{2} act by inverting the two generators of (ℤ/p)^{2}. Thus the quotient G/⟨z⟩ ≅ D_{2p} × D_{2p}, a direct product of two copies of the dihedral group of order 2p.

The principal block idempotent is e_{0} = \frac{1+z}{2}, and we have ekG ≅ k(G/⟨z⟩). The remaining block e_{1}kG, where e_{1} = 1 − e_{0} = \frac{1−z}{2} has just one simple module, which has dimension two. We have e_{1}kG ≅ Mat_{2}(C), where

C = k⟨X_{1}, X_{2}⟩/(X_{1}^{p}, X_{2}^{p}, X_{1}X_{2} + X_{2}X_{1}).

The algebra C can be regarded as a quantised version of k(ℤ/p)^{2}, and the module categories of C and e_{1}kG are equivalent.

Generalisations of the example above are studied in Benson and Green [50], Holloway and Kessar [158], Benson and Kessar [52]. The general setup is as follows. Let k be an algebraically closed field of characteristic p. Let q = (q_{ij}) ∈ Mat_{r}(k) be such that q_{ij}q_{ji} = 1 for 1 ≤ i, j ≤ r, and q_{ii} = 1 for 1 ≤ i ≤ r. Let n = (n_{1}, …, n_{r}) be an r-tuple of strictly positive integers. Let C_{q,n} (or just C_{q} if each n_{i} = 1) be the k-algebra with generators X_{1}, …, X_{r} and relations

X_{i}X_{j} = q_{ij}X_{j}X_{i} (1 ≤ i, j ≤ r), X_{i}^{p^{n_{i}}} = 0.

Such an algebra might reasonably be called a quantum complete intersection, cf. §2 of Avramov, Gasharov and Peeva [23]. The following statements are proved in [50, 158]:

(i) If q_{ij}^{p−1} = 1 for 1 ≤ i, j ≤ r then there exists a finite group G with a normal Sylow subgroup P ≅ ℤ/p^{n_{1}} × ⋯ × ℤ/p^{n_{r}}, a block idempotent e ∈ Z(kG) with defect group P, abelian inertial quotient and a single isomorphism class of simple modules, such that ekG ≅ Mat_{d}(C_{q,n}). Here, d is the dimension of the simple ekG-module.

(ii) If G is a finite group and e ∈ Z(kG) is a block idempotent with normal defect group isomorphic to ℤ/p^{n_{1}} × ⋯ × ℤ/p^{n_{r}}, abelian inertial quotient, and a single isomorphism class of simple modules, then there exists a matrix q = (q_{ij}) ∈ Mat_{r}(k) satisfying q_{ij}q_{ji} = 1 for 1 ≤ i, j ≤ r and q_{ii} = 1 for 1 ≤ i ≤ r such that ekG ≅ Mat_{d}(C_{q,n}). Again, d is the dimension of the simple ekG-module.

Notice that there is a gap between statements (i) and (ii): in statement (ii) it is not necessarily true that q_{ij}^{p−1} = 1.

The Ext algebra of the algebra C_{q,n} is also calculated in [158], under the assumption that each p^{n_{i}} > 2. The algebra Ext_{C_{q,n}}^{∗}(k, k) is the k-algebra with generators y_{1}, …, y_{r} in degree one and x_{1}, …, x_{r} in degree two^{3} satisfying the following relations:

x_{i}x_{j} = q_{ji}^{p^{n_{i}+n_{j}}} x_{j}x_{i}, y_{i}x_{j} = q_{ji}^{p^{n_{j}}} x_{j}y_{i}, y_{i}y_{j} = −q_{ji}y_{j}y_{i}, y_{i}^{2} = 0.

We shall be particularly interested in the case where all the n_{i} are equal to one, corresponding to the case where P is elementary abelian. This case is further analysed in [52], with the following conclusion. In case the algebra C_{q} arises from a block ekG as in (ii) above, with P elementary abelian, there exists a permutation σ of {1, …, r} such that for all 1 ≤ i, j ≤ p we have q_{σ(i),j} = q_{i,σ(j)} = q_{ij}^{p}. In particular, it follows that q_{σ(i),σ(j)} = q_{ij}^{p^{2}}. Since σ has to be of finite order, it follows that the q_{ij} lie in some finite subfield 𝔽_{p^{s}} of k.

3Beware that despite what is written in [158], the algebra Ext_{C_{q,n}}^{∗}(k, k) is not graded commutative.

Many theorems about representation theory of elementary abelian p-groups generalise naturally to this context, but there are some differences, and in general the representation theory of the algebras C_{q} discussed in this section is harder to analyse than that of elementary abelian p-groups.

Representations of C_{q} and of kE are both examples of nilvarieties of exponent p in the sense defined in Section 4.1. In particular, when we prove theorems about nilvarieties, we are also proving theorems about C_{q}-modules.

2 Cyclic Groups of Order p

2.1. Introduction

In this chapter, we discuss representations of ℤ/p in characteristic p. This will be applied when we examine tensor products, exterior and symmetric powers, and other Schur functors of modules of constant Jordan type. As a bonus, we also get information about the representations of SL(2, ℂ) in characteristic zero.

Tensor products, exterior powers and symmetric powers of representations of cyclic pgroups have been studied by many authors over the years, including Almkvist [2, 3, 4, 5], Almkvist and Fossum [6], Barry [30, 31], Bryant and Johnson [65], Carlson [71], Fossum [122, 123], Glasby, Praeger and Xia [135], Green [138], Hannula, Ralley and Reiner [143], Himstedt and Symonds [155], Hughes and Kemper [162], Kouwenhoven [179], McFall [192, 193], Nam and Oh [196, 197], Norman [199, 200], Ralley [211], Renaud [215, 216, 217, 218], Srinivasan [228], Symonds [233], Wehlau [244].

We restrict our attention to cyclic groups of order p. In Section 2.9 we give a recipe for computing Schur functors S^{λ} on indecomposable modules in this case, where λ is a partition of n and where n < p. The case of exterior and symmetric powers was addressed by Almkvist and Fossum [6], in terms of Gaussian polynomials g_{n,d}(q). The formula for a more general Schur functor S^{λ} generalises theirs, and involves generalised Gaussian polynomials g_{n,λ}(q). The computation of these polynomials is given by a hook formula. The same formula computes the Schur functors on the representations of SL(2, ℂ) in characteristic zero.

2.2. Modules for ℤ/p

If E ≅ ℤ/p = ⟨g⟩ is a cyclic group of order p, set X = g − 1 so that kE = k[X]/(X^{p}). If M is an indecomposable kE-module then the action of X on M is a single Jordan block of length between 1 and p with eigenvalue zero. We write J_{i} (1 ≤ i ≤ p) for the indecomposable module of dimension i, and by abuse of notation we also write J_{i} for a Jordan block of length i with eigenvalue zero. So a module is projective (or equivalently injective or flat) if and only if it is isomorphic to a direct sum of copies of J_{p}.

Let us begin by discussing subquotients of modules for a cyclic group of order p. Consider the forgetful functor from kE-modules to vector spaces. We can define functors F_{i,j} for 0 ≤ j < i ≤ p by taking the following subquotients of the forgetful functor:

F_{i,j}(M) = \frac{Ker (X^{j+1}) ∩ Im (X^{i−j−1})}{(Ker (X^{j+1}) ∩ Im (X^{i−j})) + (Ker (X^{j}) ∩ Im (X^{i−j−1}))}.

Using the second isomorphism theorem, we can see that M has a filtration where the F_{i,j}(M) are the filtered subquotients. In fact, each F_{i,j} is a simple functor (it has no non-trivial proper subfunctors) and the isomorphism types of these simple functors only depends on i and not on j. The effect of the functor F_{i,j} is to pick out the (j + 1)st socle layers of all the Jordan blocks of length i.

The lattice of subfunctors of the forgetful functor is distributive, and the poset of join irreducible elements in this lattice looks as follows, with each one labelled by the corresponding composition factor of the forgetful functor.

[figure: F_{p,p−1}; F_{p−1,p−2}; F_{p,p−2}; F_{p−2,p−3}; F_{p−1,p−3}; …; …; …; …; …; F_{1,0}; F_{p−1,1}; …; F_{p,1}; F_{p−2,0}; F_{p−1,0}; F_{p,0}]

In particular, the composition length of the forgetful functor is p(p + 1)/2. We shall see this diagram again when we describe certain vector bundles coming from modules of constant Jordan type in Section 8.2.

2.3. Tensor Products

In this section we shall compute the tensor products of the modules J_{i} for ℤ/p. We shall see that the group theoretic tensor product J_{i} ⊗_{k} J_{j} is always isomorphic to the restricted Lie tensor product J_{i} ˜⊗_{k} J_{j}.

Lemma 2.3.1. If 1 ≤ i < p then

J_{2} ⊗_{k} J_{i} ≅ J_{2} ˜⊗_{k} J_{i} ≅ J_{i−1} ⊕ J_{i+1}.

The case i = p is given by the formula

J_{2} ⊗_{k} J_{p} ≅ J_{2} ˜⊗_{k} J_{p} ≅ J_{p} ⊕ J_{p}.

Proof. This is straightforward, by computing the ranks of powers of X on these tensor

products. In the case of J_{2} ⊗_{k} J_{i} the matrix representing X is (\begin{matrix}J_{i} & I + J_{i} \\ 0 & J_{i} \end{matrix}), and so the

matrix representing X^{n} is (\begin{matrix}J_{i}^{n} & n(J_{i}^{n−1} + J_{i}^{n}) \\ 0 & J_{i}^{n} \end{matrix}). Thus we have

rank (X^{n}) = \begin{cases}2i − 2n & n < i \\ 1 & n = i < p \\ 0 & n = i = p. \end{cases}

In the case of J_{2} ˜⊗_{k} J_{i} the matrix representing X is (\begin{matrix}J_{i} & I \\ 0 & J_{i} \end{matrix}), and so the matrix representing

X^{n} is (\begin{matrix}J_{i}^{n} & nJ_{i}^{n−1} \\ 0 & J_{i}^{n} \end{matrix}). The rank of X^{n} is the same as before. □

It is possible to compute all tensor products of kE-modules inductively from the lemma, using associativity and the Krull–Schmidt theorem. For example, if we want to compute J_{3} ⊗_{k} J_{i} we use the identity

(J_{3} ⊗_{k} J_{i}) ⊕ J_{i} ≅ (J_{3} ⊕ J_{1}) ⊗_{k} J_{i} ≅ (J_{2} ⊗_{k} J_{2}) ⊗_{k} J_{i} ≅ J_{2} ⊗_{k} (J_{2} ⊗_{k} J_{i}).

The same holds for J_{3} ˜⊗_{k} J_{i}, and so we have J_{3} ⊗_{k} J_{i} ≅ J_{3} ˜⊗_{k} J_{i}. For example, provided i < p − 1 this gives the formula

J_{3} ⊗_{k} J_{i} ≅ J_{3} ˜⊗_{k} J_{i} ≅ J_{i−2} ⊕ J_{i} ⊕ J_{i+2}.

Continuing this way, an easy inductive argument gives the following theorem.

Theorem 2.3.2. If M and N are kE-modules with E cyclic of order p then

M ⊗_{k} N ≅ M ˜⊗_{k} N.

Proof. By the Krull–Schmidt theorem, it suffices to prove this with M and N indecomposable. In this case it follows inductively from Lemma 2.3.1 and the associativity of the tensor product J_{2} ⊗_{k} J_{i} ⊗_{k} J_{j}. □

In fact, it is not hard to write down an explicit formula for J_{i} ⊗_{k} J_{j}. The following formulas are special cases of Feit [119, Theorem VIII.2.7]. If i ≤ j and i + j ≤ p we have

J_{i} ⊗_{k} J_{j} ≅ J_{i} ˜⊗_{k} J_{j} ≅ ⨁_{s=1}^{i} J_{j−i+2s−1}

and

J_{p−i} ⊗_{k} J_{p−j} ≅ J_{p−i} ˜⊗_{k} J_{p−j} ≅ (p − i − j)J_{p} ⊕ (J_{i} ⊗_{k} J_{j}).

Using the commutativity of tensor product, all cases are covered by these two statements, since either i + j ≤ p or (p − i) + (p − j) ≤ p.

Note that for 1 ≤ ℓ ≤ p − 1 the number of copies of J_{ℓ} as a summand of a tensor product of Jordan blocks is at most one, and there is such a summand precisely in the cases

(2.3.1)  J_{i+j} ⊗_{k} J_{i+ℓ−1−j},  1 ≤ i ≤ p − ℓ,  0 ≤ j ≤ ℓ − 1.

Tables of tensor products of modules for ℤ/p can be found in Appendix A. For cyclic groups of order a power of a prime, Green [138] has described all tensor products of indecomposable modules, but the formulas are not quite so direct.

2.4. Gaussian Polynomials

In preparation for a discussion of symmetric and exterior powers of modules for ℤ/p, we introduce the Gaussian polynomials.

Definition 2.4.1. The (homogeneous) Gaussian polynomial G_{n,d}(X, Y) (n ≥ r ≥ 0) is defined to be

G_{n,d}(X, Y) = \frac{(X^{n} − Y^{n})(X^{n−1} − Y^{n−1}) … (X^{n−d+1} − Y^{n−d+1})}{(X^{d} − Y^{d})(X^{d−1} − Y^{d−1}) … (X − Y)}.

We shall see in Corollary 2.4.3 that this is indeed a polynomial in X and Y. The more usual (inhomogeneous) Gaussian polynomial is G_{n,d}(X, 1).

An easy calculation gives the following properties of the Gaussian polynomials. Proposition 2.4.2. We have

G_{n,d}(X, Y) = G_{n,d}(Y, X) = G_{n,n−d}(X, Y) = X^{n−d}G_{n−1,d−1}(X, Y) + Y^{d}G_{n−1,d}(X, Y). □

Corollary 2.4.3. G_{n,d}(X, Y) ∈ ℤ[X, Y] is a polynomial of degree d(n − d). Proof. This follows from the recurrence relation in Proposition 2.4.2 by induction. □ The relationship between the Gaussian polynomials and the complete and elementary

symmetric functions is given in the following proposition.

Proposition 2.4.4. We have

(1) e_{d}(X^{n−1}, X^{n−2}Y, …, XY^{n−2}, Y^{n−1}) = (XY)^{\binom{d}{2}}G_{n,d}(X, Y),

(2) h_{d}(X^{n}, X^{n−1}Y, …, Y^{n}) = G_{n+d,d}(X, Y). Proof. From Proposition 1.15.4 we have e_{d}(X^{n−1}, X^{n−2}Y, …, Y^{n−1}) = X^{d}e_{d}(X^{n−2}, X^{n−3}Y, …, Y^{n−2})

+ X^{d−1}Y^{n−1}e_{d−1}(X^{n−2}, X^{n−3}Y, …, Y^{n−2})

and h_{d}(X^{n}, X^{n−1}Y, …, Y^{n}) = X^{d}h_{d}(X^{n−1}, X^{n−2}Y, …, Y^{n−1}) + Y^{n}h_{d−1}(X^{n}, X^{n−1}Y, …, Y^{n}). The proposition now follows using the recurrence formula in Proposition 2.4.2 and induction on n. □

Definition 2.4.5. We define

g_{n,d}(q) = G_{n,d}(q, q^{−1}) = q^{−d(n−d)}G_{n,d}(q^{2}, 1) ∈ ℤ[q, q^{−1}].

Example 2.4.6. We have

(2.4.1)  g_{7,3}(q) = \frac{(q^{7} − q^{−7})(q^{6} − q^{−6})(q^{5} − q^{−5})}{(q^{3} − q^{−3})(q^{2} − q^{−2})(q − q^{−1})}
= q^{12} + q^{10} + 2q^{8} + 3q^{6} + 4q^{4} + 4q^{2} + 5
+ 4q^{−2} + 4q^{−4} + 3q^{−6} + 2q^{−8} + q^{−10} + q^{−12}.

Proposition 2.4.7. We have

(1) e_{d}(q^{n−1}, q^{n−3}, …, q^{−n+1}) = g_{n,d}(q),

(2) h_{d}(q^{n}, q^{n−2}, …, q^{−n}) = g_{n+d,d}(q). Proof. This follows directly from Proposition 2.4.4. □

2.5. Generalised Gaussian Polynomials and a Hook Formula

In this section we introduce the generalised Gaussian polynomials g_{n,λ}(q) and prove a hook formula for calculating them. This section is taken from Example 1 in Section I.3 of Macdonald [189], and will be needed in in Section 2.9.

Definition 2.5.1. Let λ be a partition of d. We define the generalised Gaussian polynomial for λ to be

g_{n,λ}(q) = s_{λ}(q^{n−1}, q^{n−3}, …, q^{−n+1})

where s_{λ} is the Schur function, see Definition 1.15.3.

Example 2.5.2. By Proposition 2.4.7, we have g_{n,[1]^{d}}(q) = g_{n,d}(q) and g_{n+1,[d]}(q) = g_{n+d,d}(q).

The following is the transpose of statement (1.7) in Chapter I of [189].

Lemma 2.5.3. Let λ be a partition with conjugate partition λ′, and let m ≥ λ_{1}, n ≥ λ′_{1}. Then the m + n numbers λ′_{i} + m − i (1 ≤ i ≤ m) and m − λ_{j} − 1 + j (1 ≤ j ≤ n) are a permutation of {0, 1, …, m + n − 1}.

Proof. The diagram for λ is contained in an n × m rectangle. Number the edges of the boundary between λ and its complement as in the diagram below, starting with zero at the top right and ending with m + n − 1 at the bottom left.

n = 6  m = 8  λ = [6] [3]^{2}[1] λ′ = [4] [3]^{2}[1]^{3} 0 1 2 3 4 5 6 7 8 9 10 11 12 13

The numbers on the horizontal edges are λ′_{i} + m − i (1 ≤ i ≤ m), while those on the vertical edges are (m + n − 1) − (λ_{j} + n − j) = m − λ_{j} − 1 + j (1 ≤ j ≤ n). □

Definition 2.5.4. The hook length of a partition λ at a box x = (i, j) ∈ λ is defined to be

h(x) = λ_{i} + λ′_{j} − i − j + 1.

This is the number of boxes directly below and directly to the right of x, including x itself.

Lemma 2.5.5. Let λ be a partition with at most n parts. Then we have

∏_{x∈λ}(q^{h(x)} − q^{−h(x)}) ∏_{1≤i<j≤n}(q^{λ_{i}−λ_{j}−i+j} − q^{−(λ_{i}−λ_{j}−i+j)}) = ∏_{i=1}^{n} ∏_{j=1}^{λ_{i}+n−i}(q^{j} − q^{−j}).

Proof. Let m = λ_{1}, and observe that n ≥ λ′_{1}. It follows from Lemma 2.5.3 that

∏_{i=1}^{m}(q^{λ′_{i}+m−i} − q^{−(λ′_{i}+m−i)}) ∏_{j=2}^{n}(q^{m−λ_{j}−1+j} − q^{−(m−λ_{j}−1+j)}) = ∏_{j=1}^{m+n−1}(q^{j} − q^{−j})

because the set of terms on the left is a permutation of the set of terms on the right. Note that we have omitted the term (q^{0} − q^{−0}) from both sides for obvious reasons, and that this explains why the second product on the left side omits the term j = 1 and the product on the right side omits the term j = 0.

The exponents λ′_{i} + m − i in the first product on the left are the hook lengths along the first row of λ. Writing down this identity for the partitions (λ_{i}, λ_{i+1}, …, λ_{n}) for 1 ≤ i ≤ n and multiplying, we obtain the identity in the statement of the lemma. □

Definition 2.5.6. The content of the box x = (i, j) is defined to be c(x) = j − i.

Proposition 2.5.7. Let λ be a partition with at most n parts. Then we have the following hook formula for g_{n,λ}(q):

g_{n,λ}(q) = ∏_{x∈λ} \frac{q^{n+c(x)} − q^{−(n+c(x))}}{q^{h(x)} − q^{−h(x)}} .

Remark 2.5.8. If λ has more than n parts then the formula still holds, but it says that zero is equal to zero.

Example 2.5.9. For the partition [3][2] we have the following values for n + c(x) and h(x):

\begin{matrix}n & n + 1 & n + 2 \\ n − 1 & n \end{matrix}  4 3 1 2 1

and so

g_{n,[3][2]}(q) = \frac{(q^{n+2} − q^{−n−2})(q^{n+1} − q^{−n−1})(q^{n} − q^{−n})^{2}(q^{n−1} − q^{−n+1})}{(q^{4} − q^{−4})(q^{3} − q^{−3})(q^{2} − q^{−2})(q − q^{−1})^{2}} ∈ ℤ[q, q^{−1}].

Proof of Proposition 2.5.7. By Definitions 1.15.3 and 2.5.1, we have

g_{n,λ}(q) = s_{λ}(q^{n−1}, q^{n−3}, …, q^{−n+1})

= \frac{det(q^{(n+1−2i)(λ_{j}+n−j)})_{1≤i,j≤n}}{det(q^{(n+1−2i)(n−j)})_{1≤i,j≤n}} .

The top of this fraction is some power of q times a Vandermonde determinant in the variables q^{2(λ_{j}+n−j)}, and the bottom is some power of q times the Vandermonde determinant in the variables q^{2(n−j)}. Using the usual formula for a Vandermonde determinant, we find that g_{n,λ}(q) is some power of q times

\frac{∏_{1≤i<j≤n}(q^{2(λ_{i}+n−i)} − q^{2(λ_{j}+n−j)})}{∏_{1≤i<j≤n}(q^{2(n−i)} − q^{2(n−j)})}.

We can either keep track directly of the power of q in front of this expression, or we can just use the fact that the result has to be invariant under swapping q and q^{−1}. Either way, we get

g_{n,λ}(q) = \frac{∏_{1≤i<j≤n}(q^{λ_{i}−λ_{j}−i+j} − q^{−(λ_{i}−λ_{j}−i+j)})}{∏_{1≤i<j≤n}(q^{j−i} − q^{−(j−i)})}.

The numerator on the right hand side of this equation is equal to the second product on the left hand side of Lemma 2.5.5, while the denominator is equal to ∏_{i=1}^{n} ∏_{j=1}^{n−i}(q^{j} − q^{−j}). Making these substitutions, we get

g_{n,λ}(q) = \frac{∏_{i=1}^{n} ∏_{j=1}^{λ_{i}+n−i}(q^{j} − q^{−j})}{∏_{i=1}^{n} ∏_{j=1}^{n−i}(q^{j} − q^{−j}) ∏_{x∈λ}(q^{h(x)} − q^{−h(x)})} = \frac{∏_{i=1}^{n} ∏_{j=n−i+1}^{λ_{i}+n−i}(q^{j} − q^{−j})}{∏_{x∈λ}(q^{h(x)} − q^{−h(x)})}.

Finally, we observe that the numbers n+c(x) for x in the ith row of λ are n−i+1, …, n−i+λ_{i}. So the numerator in the right hand side of the above expression for g_{n,λ}(q) is equal to

∏_{x∈λ}(q^{n+c(x)} − q^{−(n+c(x))}).  □

2.6. λ-Rings and Representations of SL(2, ℂ)

It is convenient to introduce the notion of a λ-ring in order to make further computations with symmetric and exterior powers. For more information on λ-rings, see Atiyah and Tall [18].

Definition 2.6.1. A λ-ring is a ring R together with operations λ^{n}: R → R (n ≥ 0) such that for all x, y ∈ R we have

(1) λ^{0}(x) = 1,

(2) λ^{1}(x) = x,

(3) λ^{n}(x + y) = ∑_{i+j=n} λ^{i}(x)λ^{j}(y).

Example 2.6.2. Lemma 1.14.3 shows that exterior powers on representation rings satisfy these properties.

In a λ-ring it is often convenient to introduce an indeterminate t and define

λ_{t}: R → R[[t]], λ_{t}(x) = ∑_{n≥0} λ^{n}(x)t^{n}.

In terms of this definition, axiom (3) states that

λ_{t}(x + y) = λ_{t}(x)λ_{t}(y).

Since we are working with representations in characteristic p, the exterior power operations Λ^{i} are better behaved for i < p than for i ≥ p. We therefore make the following definition.

Definition 2.6.3. A p-λ-ring is a ring R together with operations λ^{n} for 0 ≤ n < p satisfying axioms (1), (2) and (3).

In a p-λ-ring, we truncate λ_{t} in the obvious way:

λ_{t}: R → R[t]/(t^{p}), λ_{t}(x) = ∑_{0≤n<p} λ^{n}(x)t^{n}.

A special λ-ring is a λ-ring with further axioms detailing the behaviour of the λ-operations on products, and the behaviour of compositions of λ-operations. We shall need a modification of this concept to deal with the fact that we are working with representations in characteristic p.

We begin by defining certain polynomials involved in the definitions. Let x_{1}, …, x_{s} and y_{1}, …, y_{s} be indeterminates, and let e_{i} and e′_{i} be the elementary symmetric functions in x_{1}, …, x_{s}, respectively y_{1}, …, y_{s}. Let

P_{n}(e_{1}, …, e_{n}; e′_{1}, …, e′_{n})

be the coefficient of t^{n} in ∏_{j,k=1}^{s}(1 + x_{j}y_{k}t). As long as s ≥ n, these polynomials are independent of s and have integer coefficients. Similarly, we let

P_{m,n}(e_{1}, …, e_{mn})

be the coefficient of t^{m} in ∏_{i_{1}<···<i_{n}}(1 + x_{i_{1}} … x_{i_{n}}t). As long as s ≥ mn, these polynomials are independent of s and have integer coefficients.

Definition 2.6.4. A special λ-ring is a λ-ring satisfying the following further axioms:

(4) λ_{t}(xy) = P_{n}(λ^{1}(x), …, λ^{n}(x); λ_{1}(y), …, λ^{n}(y)),

(5) λ^{m}(λ^{n}(x)) = P_{m,n}(λ^{1}(x), …, λ^{mn}(x)). Note that the case n = 0 of (5) implies that λ_{t}(1) = 1 + t,

A special p-λ-ring is a p-λ-ring which satisfies axiom (4) for n < p and axiom (5) for mn < p.

Example 2.6.5. Using the exterior powers as the λ-operations, representation rings of characteristic zero representations are special λ-rings, while representation rings of characteristic p representations are special p-λ-rings.

Example 2.6.6. Let 𝕋 be the 1-dimensional torus, or circle group {z ∈ ℂ | |z| = 1}. Every finite dimensional continuous representation of 𝕋 is a direct sum of representations of dimension one. There is a representation of dimension one for each integer n, namely z ↦ z^{n}. Tensoring such representations adds the corresponding integer, and so the representation ring is the ring of Laurent polynomials ℤ[q, q^{−1}]. The λ-ring structure is given by λ_{t}(q^{n}) = 1+q^{n}t, and this makes ℤ[q, q^{−1}] into a special λ-ring.

Example 2.6.7. Next, consider the group SL(2, ℂ) of 2 × 2 complex matrices of determinant one. Every finite dimensional continuous representation is a direct sum of irreducible representations. There is one irreducible representation V_{n} for each integer n > 0, and V_{n} has dimension n. The restriction of V_{n} to

𝕋 = \{( \begin{matrix}z & 0 \\ 0 & z^{−1} \end{matrix} ) | |z| = 1\} ⊆ SL(2, ℂ)

corresponds to the Laurent polynomial q^{n−1} + q^{n−3} + ⋯ + q^{−n+1}. This identifies the representation ring of SL(2, ℂ) with the fixed points of ℤ[q, q^{−1}] under the involution which swaps q and q^{−1}. The special λ-ring structure is inherited from that of ℤ[q, q^{−1}].

Proposition 2.6.8. The representation Λ^{d}(V_{n}) of SL(2, ℂ) corresponds to the element g_{n,d}(q) ∈ ℤ[q, q^{−1}] while S^{d}(V_{n+1}) corresponds to the element g_{n+d,d}(q).

Proof. This follows from Proposition 2.4.4. □

Corollary 2.6.9. As representations of SL(2, ℂ) we have

Λ^{d}(V_{n+d}) ≅ S^{d}(V_{n+1}) ≅ S^{n}(V_{d+1}).  □

Definition 2.6.10. A λ-ideal in a λ-ring R is an ideal which is closed under the operations λ^{i}. A p-λ-ideal in a p-λ-ring is an ideal which is closed under the operations λ^{i} for i < p.

Lemma 2.6.11. If R is a special λ-ring (resp. special p-λ-ring) then an ideal I is a λ-ideal (resp. a p-λ-ideal) if and only if it is generated by a set of elements whose images under λ^{n} are in I for all n ≥ 0 (resp. for 0 ≤ n < p).

Proof. This is proved in Lemma 6.4 of [18] for special λ-rings, and the proof for special p-λ-rings is identical. □

The following lemma is a version of the splitting principle for λ-rings.

Lemma 2.6.12. Let x be an element of a special λ-ring (resp. special p-λ-ring) satisfying λ^{i}(x) = 0 for i > n (resp. for n < i < p). Then R[ξ]/(ξ^{n} − λ^{1}(x)ξ^{n−1} + ⋯ + (−1)^{n}λ^{n}(x)) with λ_{t}(ξ) = 1 + ξt is a special λ-ring (resp. special p-λ-ring).

Proof. This is proved in Lemma 6.5 of [18] for special λ-rings, and the proof for special p-λ-rings is identical. □

2.7. The Representation Ring of ℤ/p

Following Almkvist and Fossum [6], we examine the representation ring ℜ of the cyclic group of order p in characteristic p. This has a basis consisting of the representations J_{n} (1 ≤ n ≤ p), and the multiplication is determined by Lemma 2.3.1. Using the exterior powers as the λ-operations, ℜ is a special p-λ-ring.

Let us apply Lemma 2.6.12 to the element J_{2} of ℜ. Since λ_{t}(J_{2}) = 1 + J_{2}t + t^{2}, this gives us a ring

ℜ̂ = ℜ[q]/(q^{2} − J_{2}q + 1).

Rewriting the relation q^{2} − J_{2}q + 1 = 0, we obtain J_{2} = q + q^{−1}.

Lemma 2.7.1. In ℜ̂, J_{n} corresponds to the element

q^{n−1} + q^{n−3} + ⋯ + q^{−n+1}

for 1 ≤ n ≤ p.

Proof. By Lemma 2.3.1, if 1 ≤ n < p, we have the relation J_{2} ⊗_{k} J_{n} ≅ J_{n−1} + J_{n+1}. We also have

(q + q^{−1})(q^{n−1} + q^{n−3} + ⋯ + q^{−n+1}) = (q^{n} + q^{n−2} + ⋯ + q^{−n}) + (q^{n−2} + q^{n−4} + ⋯ + q^{−n+2}).

Combining these proves the lemma by induction on n. □

Proposition 2.7.2. We have ℜ̂ = ℤ[q, q^{−1}]/(q − 1)(q^{−1} − 1)(q^{p−1} + q^{p−3} + ⋯ + q^{−p+1}). This is a special p-λ-ring with λ-operations given by λ_{t}(q^{n}) = 1 + q^{n}t.

Proof. By Lemma 2.7.1, the representation J_{p} corresponds to the element q^{p−1} +q^{p−3} + ⋯ + q^{−p+1} of ℜ̂. By Lemma 2.3.1 we have

J_{2} ⊗ J_{p} ≅ J_{p} ⊕ J_{p}.

This gives the relation

(q + q^{−1})(q^{p−1} + q^{p−3} + ⋯ + q^{−p+1}) = 2(q^{p−1} + q^{p−3} + ⋯ + q^{−p+1})

or equivalently

(q − 1)(q^{−1} − 1)(q^{p−1} + q^{p−3} + ⋯ + q^{−p+1}) = 0.

The quotient of ℤ[q, q^{−1}] by this relation is a free ℤ-module of rank 2p with basis

q^{p}, q^{p−1}, …, q^{−p+1}.

Comparing ℤ-ranks, this quotient must equal ℜ̂.

By Lemma 2.6.12, we have λ_{t}(q) = 1 + qt. Repeatedly applying axiom (4) then gives λ_{t}(q^{n}) = 1 + q^{n}t. □

Remark 2.7.3. Proposition 2.7.2 allows us to identify ℜ with the fixed points of ℜ̂ under the automorphism interchanging q and q^{−1}.

The following corollary gives a close relationship between the representation theory of SL(2, ℂ) in characteristic zero and of ℤ/p in characteristic p.

Corollary 2.7.4. The map ℤ[q, q^{−1}] → ℜ̂ is a homomorphism of special p-λ-rings.

Proof. This follows by comparing Examples 2.6.6 and 2.6.7 with Proposition 2.7.2. □

Remark 2.7.5. Referring to Lemma 2.6.11, the corollary shows that the ideal in ℤ[q, q^{−1}] generated by the element

(q − 1)(q^{−1} − 1)(q^{p−1} + q^{p−3} + ⋯ + q^{−p+1})

is a p-λ-ideal. However, it is not a λ-ideal, as it is not closed under the operation λ^{p}. This explains why ℜ is not a special λ-ring.

Remark 2.7.6. It is often computationally easier to work in the further quotient

ℜ̄ = ℤ[q, q^{−1}]/(q^{p−1} + q^{p−3} + ⋯ + q^{−p+1}).

In this quotient, q^{p} is identified with q^{−p}. The 2p − 2 elements q^{p−1}, q^{p−2}, …, q^{−p+2} form a free ℤ-basis for ℜ̄. This may be regarded as the “stable” version of ℜ in which the projective module J_{p} has been set equal to zero.

The map from the representation ring of ℤ/p to ℜ̄ is not injective, because J_{p} goes to zero. However, the elements corresponding to J_{n} for 1 ≤ n ≤ p − 1 are linearly independent. If we combine with the dimension homomorphism to ℤ, we get an injective map from the representation ring of ℤ/p to ℤ×ℜ̄, sending J_{n} to the pair (n, q^{n−1}+q^{n−3}+⋯+q^{−n+1}) and in particular sending J_{p} to (p, 0). The image is not quite the fixed points of the automorphism of ℤ × ℜ̄ interchanging q and q^{−1}, but rather the elements of this of the form (n, f(q)) where f(1) ≡ n (mod p).

As another alternative, less aesthetically pleasing but somehow more convenient for calculation, we can work in the quotient

(2.7.1)  ℜ̃ = ℤ[q, q^{−1}]/(q − 1)(q^{p−1} + q^{p−3} + ⋯ + q^{−p+1}).

This quotient is a free ℤ-module of rank 2p − 1 with basis q^{p−1}, q^{p−2}, …, q^{−p+1}. Since

(q^{−1} + 1)(q − 1)(q^{p−1} + q^{p−3} + ⋯ + q^{−p+1}) = q^{p} − q^{−p},

in this quotient ring q^{p} is identified with q^{−p}. Furthermore, the elements

q^{p} + q^{p−2} + ⋯ + q^{−p+2}, q^{p−1} + q^{p−3} + ⋯ + q^{−p+1}, q^{p−2} + q^{p−4} + ⋯ + q^{−p}

are all identified, and these represent the element J_{p}.

In the ring ℜ̃, the elements representing J_{n} (1 ≤ n ≤ p) remain linearly independent, and the representation ring of ℤ/p is identified with the subring of ℜ̃ consisting of the fixed points of the automorphism interchanging q and q^{−1}.

Example 2.7.7. Let us compute the tensor product J_{4} ⊗ J_{5}. Using Lemma 2.7.1, this corresponds to the element

(q^{3} + q + q^{−1} + q^{−3})(q^{4} + q^{2} + 1 + q^{−2} + q^{−4}) = q^{7} + 2q^{5} + 3q^{3} + 4q + 4q^{−1} + 3q^{−3} + 2q^{−5} + q^{−7}.

If p ≥ 11 it follows that

J_{4} ⊗ J_{5} ≅ J_{8} ⊕ J_{6} ⊕ J_{4} ⊕ J_{2}.

If p = 7 the first and last term in the polynomial combine to give 2q^{7}, and so we get

J_{4} ⊗ J_{5} ≅ 2J_{7} ⊕ J_{4} ⊕ J_{2}.

Finally, if p = 5 then the term q^{7} is replaced by q^{−3}, q^{−7} is replaced by q^{3} and q^{−5} is replaced by q^{5} to get

4q^{5} + 4q^{3} + 4q + 4q^{−1} + 4q^{−3}

and so

J_{4} ⊗ J_{5} ≅ 4J_{5}.

For tensor products this is not so exciting, but for symmetric and exterior powers and other Schur functors it becomes a very good method of computation.

2.8. Symmetric and Exterior Powers of Jordan Blocks

Symmetric and exterior powers of Jordan blocks for a cyclic group of order p were investigated extensively in the work of Almkvist [2, 3, 4, 5], and Almkvist and Fossum [6]. We begin with the following observation, which appears as Proposition 2.10 from Chapter III of [6], and which will be needed in the proof of Theorem 5.12.1.

Theorem 2.8.1. As k(ℤ/p)-modules, S^{d}(J_{i}) is projective (i.e., a direct sum of copies of J_{p}) provided d < p and d + i > p.

Proof. Write m for a non-zero element of J_{i} spanning the fixed space of dimension one under ℤ/p. Then there is an injective homomorphism S^{d}(J_{i}) → S^{d+1}(J_{i}) which sends m_{1} … m_{d} to mm_{1} … m_{d}. The quotient module is isomorphic to S^{d+1}(J_{i−1}), where J_{i−1} is identified with J_{i}/(m). Thus there are exact sequences

0 → S^{d}(J_{i}) → S^{d+1}(J_{i}) → S^{d+1}(J_{i−1}) → 0.

If i = p and d < p − 1 then both S^{d}(J_{i}) and S^{d+1}(J_{i}) are projective (and hence injective), so S^{d+1}(J_{p−1}) is projective. Repeat the argument using downward induction on i. □

The following formulas are proved in Corollary III.1.4 and Proposition III.2.6 of Almkvist and Fossum [6]. See also Hughes and Kemper [162].

Theorem 2.8.2. As elements of the ring ℜ̂ described in Section 2.7, we have

Λ^{d}(J_{n}) = g_{n,d}(q), S^{d}(J_{n+1}) = g_{n+d,d}(q).

for d < p.

Proof. This follows from Proposition 2.6.8 and Corollary 2.7.4. □

Corollary 2.8.3. We have

S^{d}(J_{n+1}) ≅ S^{n}(J_{d+1})  d, n < p,
S^{d}(J_{n+1}) ≅ Λ^{d}(J_{n+d})  n + d ≤ p.  □

Remark 2.8.4. The isomorphisms in Corollary 2.8.3 were discovered by Almkvist and Fossum [6]. We describe a generalisation to Schur functors in Section 2.9. Another interesting formula proved in the same paper is

S^{mp+d}(J_{n+1}) ≅ S^{d}(J_{n+1}) ⊕ copies of J_{p} 0 ≤ d < p.

Example 2.8.5. Using equation (2.4.1) we calculate that for p ≥ 13 we have

S^{3}(J_{4}) ≅ J_{13} ⊕ J_{9} ⊕ J_{7} ⊕ J_{5} ⊕ J_{1}.

If p = 11 we replace q^{12} and q^{−12} by q^{−10} and q^{10} in (2.4.1) to get

S^{3}(J_{4}) ≅ J_{11} ⊕ J_{11} ⊕ J_{7} ⊕ J_{5} ⊕ J_{1}.

Using these methods, the following is an easy exercise.

Proposition 2.8.6. Let J_{a}, J_{b} be the indecomposable representations of dimensions a ≤ b ≤ p for a cyclic group of order p over k. Then

(1) J_{a} ⊗ J_{b} ≅ \begin{cases}J_{a+b−1} ⊕ J_{a+b−3} ⊕ J_{a+b−5} ⊕ ⋯ ⊕ J_{b−a+1} & p ≥ a + b − 1 \\ (a + b − p)J_{p} ⊕ J_{2p−a−b−1} \\ ⊕ J_{2p−a−b−3} ⊕ ⋯ ⊕ J_{b−a+1} & p ≤ a + b − 2 \end{cases}

(2) Λ^{2}(J_{a}) ≅ \begin{cases}J_{2a−3} ⊕ J_{2a−7} ⊕ J_{2a−11} ⊕ … & p ≥ 2a − 3 \\ (a − \frac{p+1}{2})J_{p} ⊕ J_{2p−2a−1} ⊕ J_{2p−2a−5} ⊕ … & p ≤ 2a − 5. \end{cases}

(3) S^{2}(J_{a}) ≅ Λ^{2}(J_{a+1}). □

Tables of symmetric and exterior powers of modules for ℤ/p can be found in Appendix A.

2.9. Schur Functors for SL(2, ℂ) and ℤ/p

In this section we describe how to compute Schur functors on representations of SL(2, ℂ) in characteristic zero and of ℤ/p in characteristic p. Recall from Example 2.6.7 that the representation ring of SL(2, ℂ) is isomorphic to ℤ[q, q^{−1}] as a λ-ring. To compute in the representation ring of ℤ/p we work in the quotient ring ℜ̂ of ℤ[q, q^{−1}] described in Section 2.7. The following theorem generalises Theorem 2.8.2.

Theorem 2.9.1. Let λ be a partition of d. Then we have the following.

(1) As a representation of SL(2, ℂ), the image of S^{λ}(V_{n}) in ℤ[q, q^{−1}] is equal to the generalised Gaussian polynomial g_{n,λ}(q).

(2) If d < p, as a representation of ℤ/p in characteristic p, the image of S^{λ}(J_{n}) (1 ≤ n ≤ p) in ℜ̂ is equal to g_{n,λ}(q).

Proof. From the determinantal form (1.15.2) we have

g_{n,λ}(q) = det(g_{n,λ_{i}−i+j}(q))_{1≤i,j≤r}

(r at least the number of parts of λ).

For the case of SL(2, ℂ), by Theorem 1.16.4 we have

S^{λ}(V_{n}) ≅ det(S^{λ_{i}−i+j}(V_{n}))_{1≤i,j≤r}.

Part (1) of the theorem therefore follows from the special case λ = [d], which is Proposition 2.6.8.

For the case of ℤ/p, because of the condition d < p, by Theorem 1.16.4 we have

S^{λ}(J_{n}) ≅ det(S^{λ_{i}−i+j}(J_{n}))_{1≤i,j≤r}.

Part (2) of the theorem therefore follows from the special case λ = [d], which is Theorem 2.8.2. □

Example 2.9.2. Let us calculate S^{[3][2]}(V_{4}) for SL(2, ℂ). At the same time we compute S^{[3][2]}(J_{4}) for ℤ/p with p ≥ 13. Using the hook formula of Proposition 2.5.7, we have

g_{4,[3][2]}(q) = \frac{(q^{6} − q^{−6})(q^{5} − q^{−5})(q^{4} − q^{−4})^{2}(q^{3} − q^{−3})}{(q^{4} − q^{−4})(q^{3} − q^{−3})(q^{2} − q^{−2})(q − q^{−1})^{2}}
= q^{11} + 2q^{9} + 4q^{7} + 6q^{5} + 8q^{3} + 9q + 9q^{−1} + 8q^{−3} + 6q^{−5} + 4q^{−7} + 2q^{−9} + q^{−11}

and so

S^{[3][2]}(V_{4}) ≅ V_{12} ⊕ V_{10} ⊕ 2V_{8} ⊕ 2V_{6} ⊕ 2V_{4} ⊕ V_{2} (for SL(2, ℂ))
S^{[3][2]}(J_{4}) ≅ J_{12} ⊕ J_{10} ⊕ 2J_{8} ⊕ 2J_{6} ⊕ 2J_{4} ⊕ J_{2} (for ℤ/p, p ≥ 13).

For smaller primes p > 5, we identify q^{p} with q^{−p} as usual, and we get

S^{[3][2]}(J_{4}) ≅ \begin{cases}2J_{11} ⊕ 2J_{8} ⊕ 2J_{6} ⊕ 2J_{4} ⊕ J_{2} & p = 11 \\ 8J_{7} ⊕ J_{4} & p = 7. \end{cases}

Further results of such calculations may be found in Appendix A.

The following corollary generalises the isomorphism S^{d}(J_{n+1}) ≅ Λ^{d}(J_{n+d}) of Almkvist and Fossum (see Corollary 2.8.3).

Corollary 2.9.3. Suppose that λ be a partition of d and that the characteristic of k is either zero or p > d. Suppose further that there exists an integer e such that for each diagonal box x = (i, i) of λ the quantity λ_{i} − λ′_{i} is equal to e.

(1) As representations of SL(2, ℂ) we have

S^{λ′}(V_{n+e}) ≅ S^{λ}(V_{n}).

(2) Provided that d < p, n ≤ p and n+e ≤ p, as representations of ℤ/p in characteristic p we have

S^{λ′}(J_{n+e}) ≅ S^{λ}(J_{n}).  □

Example 2.9.4. One obvious class of partitions with the property described in the corollary is the hook partitions [j][1]^{d−j} with e = 2j − d + 1. So we have

S^{[d−j+1][1]^{j−1}}(V_{n+2j−d+1}) ≅ S^{[j][1]^{d−j}}(V_{n}) (for SL(2, ℂ))
S^{[d−j+1][1]^{j−1}}(J_{n+2j−d+1}) ≅ S^{[j][1]^{d−j}}(J_{n}) (for ℤ/p, p > d).

Another example is [4][3][1] with e = 2, giving

S^{[3][2]^{2}[1]}(V_{n+2}) ≅ S^{[4][3][1]}(V_{n}) (for SL(2, ℂ))
S^{[3][2]^{2}[1]}(J_{n+2}) ≅ S^{[4][3][1]}(J_{n}) (for ℤ/p, p ≥ 11).

3 Background from Algebraic Geometry

Many readers will wish to skip over this chapter, which is designed to provide background material from algebraic geometry in the form needed for our considerations. We make no attempt to be either general or comprehensive, but limit ourselves to the task in hand. More details can be found in EGA I and EGA II (Grothendieck [141, 142]), FAC (Serre [224]), and Chapters I–III of Hartshorne [147].

The first four sections of this chapter are all that is needed in order to read Chapter 4. The rest of this chapter consists of prerequisites for Chapter 6.

3.1. Affine Space and Affine Varieties

Let k be an algebraically closed field. We write 𝔸^{r} = 𝔸^{r}(k) for affine r-space over k, consisting of r-tuples (a_{1}, …, a_{r}) with each a_{i} ∈ k. Let X_{i} be the r-tuple (0, …, 1, …, 0) with a 1 in the ith place and 0 in all the other places. Thus X_{1}, …, X_{r} is the standard basis for 𝔸^{r}.

Let Y_{1}, …, Y_{r} be the linear functions on 𝔸^{r} defined by Y_{i}(X_{j}) = δ_{i,j} (Kronecker delta). Thus the coordinate ring of 𝔸^{r} is the ring k[Y_{1}, …, Y_{r}] of polynomial functions on 𝔸^{r}.

If I is an ideal in k[Y_{1}, …, Y_{r}] we write

V (I) = {a ∈ 𝔸^{r} | ∀f ∈ I, f(a) = 0},

the set of simultaneous zeros of I. Hilbert’s Nullstellensatz states that if f ∈ k[Y_{1}, …, Y_{r}] vanishes at every point in V (I) then some power of f is in I. It follows that V (I) = V (I′) if and only if \sqrt{I} = \sqrt{I′}, where

\sqrt{I} = \{f ∈ k[Y_{1}, …, Y_{r}] | ∃ n, f^{n} ∈ I\}.

An ideal I is said to be a radical ideal if I = \sqrt{I}.

We endow 𝔸^{r} with the Zariski topology in which the closed sets are the sets of the form V (I) as I runs over the ideals in k[Y_{1}, …, Y_{r}], together with the empty set. Thus the nonempty closed sets are in one-to-one correspondence with the radical ideals. We refer to the closed subsets as closed subvarieties. Each point (a_{1}, …, a_{r}) forms a closed set corresponding to the maximal ideal 𝔪 = (Y_{1} − a_{1}, …, Y_{r} − a_{r}).

An ideal is radical if and only if it is a finite intersection of prime ideals. Since V (I ∩I′) = V (I) ∪ V (I′), the irreducible closed sets, namely those which cannot be written as a union of two proper closed subsets, are precisely the closed sets corresponding to the prime ideals. Thus the closed irreducible subvarieties are those of the form V (𝔭) with 𝔭 prime.

Definition 3.1.1. An affine variety over k is a closed subvariety V of 𝔸^{r}(k) for some r. The Zariski topology on V is inherited from that on 𝔸^{r} via the subspace topology.

Example 3.1.2. GL(n, k) is the closed subvariety of 𝔸^{n^{2}+1} defined by n^{2} matrix entries a_{i,j} together with another coordinate a_{0} and the single equation Y_{0}. det(Y_{i,j}) = 1. The same device can be used to make any open set in 𝔸^{r} defined by the non-vanishing of a single polynomial into a closed subvariety of 𝔸^{r+1}.

3.2. Generic Points and Closed Points

If V = V (𝔭) is a closed irreducible subvariety of 𝔸^{r} corresponding to a prime ideal 𝔭 ⊆ k[Y_{1}, …, Y_{r}], we write k[V] for the integral domain k[Y_{1}, …, Y_{r}]/𝔭. This is the ring of polynomial functions on V, and is called the coordinate ring of V. We write k(V) for the field of fractions of k[V]. This is the field of rational functions on V, and is called the function field of V.

Let Ȳ_{1}, …, Ȳ_{r} be the images of Y_{1}, …, Y_{r} in k[V] ⊆ k(V). Then the point

Ȳ_{1}X_{1} + ⋯ + Ȳ_{r}X_{r} ∈ 𝔸^{r}(k(V))

is called the generic point of V. It is a canonically defined point in 𝔸^{r}(k(V)). It is independent of the choice of basis X_{1}, …, X_{r} of 𝔸^{r}(k). In fact it lies in the subvariety V_{k(V)} obtained from V by extending scalars from k to k(V), since the coordinates satisfy the equations defining V.

Let K be algebraically closed and have transcendence degree at least r over k. Then there is a k-embedding of k(V) into K. Let t_{1}, …, t_{r} be the images of Ȳ_{1}, …, Ȳ_{r} under such an embedding. Then the point

(3.2.1)  t_{1}X_{1} + ⋯ + t_{r}X_{r}

is called a generic point of V, and it is an element of V_{K}.

Conversely, given any extension field K of k and any point α in 𝔸^{r}(K), we write it in the form (3.2.1) with t_{1}, …, t_{r} ∈ K. Letting 𝔭 ⊆ k[Y_{1}, …, Y_{r}] be the ideal of polynomial relations satisfied by t_{1}, …, t_{r} over k, we see that 𝔭 is a prime ideal and so there is a corresponding closed irreducible subvariety V ⊆ 𝔸^{r}(k) with k[V] = k[Y_{1}, …, Y_{r}]/𝔭. This gives a uniquely determined field homomorphism k(V) → K such that the induced map 𝔸^{r}(k(V)) → 𝔸^{r}(K) sends the generic point of V to α, namely Ȳ_{i} ↦ t_{i}. We then say that α is a generic point for V. In this way, every point of 𝔸^{r}(K) is a generic point for some closed irreducible subvariety of 𝔸^{r}(k). If the irreducible subvariety V is a single point defined over k then we say the generic point is a closed point.

Lemma 3.2.1. Suppose that we are given polynomials p_{1}, …, p_{n} ∈ k[Y_{1}, …, Y_{r}] and a closed irreducible subvariety V of 𝔸^{r}(k). Then the following are equivalent:

• p_{1}, …, p_{n} vanish at the generic point of V in 𝔸^{r}(k(V))

• p_{1}, …, p_{n} vanish at some generic point of V defined over some extension field of k • p_{1}, …, p_{n} vanish on every point of V.

If these conditions fail, then the points of V where p_{1}, …, p_{n} do not vanish form a dense open subset of V in the Zariski topology. □

Because of the correspondence between generic points and prime ideals, it is customary to throw in abstract generic points to make the spectrum of k[Y_{1}, …, Y_{r}]. More generally, we have the following definition.

Definition 3.2.2. If R is a commutative ring, we write Spec R for the set of prime ideals of R. We endow this with the Zariski topology in which the closed sets are the sets of the form

V (I) = {𝔭 ∈ Spec R | I ⊆ 𝔭},

where I is an ideal in R. If r ∈ R, we write V (r) for the closed set determined by the principal ideal (r), and

D(r) = {𝔭 | r ∉ 𝔭}

for its open complement. Every open set is a union of open sets of the form D(r), so they form a base for the Zariski topology.

If R → R′ is a ring homomorphism then f^{∗}: Spec R′ → Spec R takes a prime ideal in R′ to its inverse image in R.

So if we examine Spec k[Y_{1}, …, Y_{r}] we find that the closed points are the maximal ideals (Y_{1} − a_{1}, …, Y_{r} − a_{r}) corresponding to points (a_{1}, …, a_{r}) ∈ 𝔸^{r}(k). The remaining points are generic points for closed irreducible subvarieties of 𝔸^{r}(k). From now on, we shall write 𝔸^{r} or 𝔸^{r}(k) for Spec k[Y_{1}, …, Y_{r}] and talk of closed points when we wish to refer to elements of the form (a_{1}, …, a_{r}).

3.3. Projective Space and Projective Varieties

Points in projective space correspond to lines through the origin in affine space. More precisely, we put an equivalence relation on the closed points of 𝔸^{r}(k) \ {0} by deeming that (a_{1}, …, a_{r}) and (a′_{1}, …, a′_{r}) are equivalent if and only if there exists 0 ≠ λ ∈ k such that for i = 1, …, r we have a′_{i} = λa_{i}. The set of equivalence classes is denoted ℙ^{r−1}(k) or just ℙ^{r−1}, and called projective (r − 1)-space. The point in ℙ^{r−1} defined by a representative (a_{1}, …, a_{r}) ∈ 𝔸^{r} \ {0} is denoted (a_{1} : …: a_{r}).

The corresponding ring theoretic construction is to put a grading on k[Y_{1}, …, Y_{r}] in such a way that each Y_{i} has degree one.

Definition 3.3.1. We say that a polynomial f(Y_{1}, …, Y_{r}) ∈ k[Y_{1}, …, Y_{r}] is homogeneous of degree n if every monomial appearing with non-zero coefficient has degree exactly n. This is equivalent to the statement that for λ ∈ k we have

f(λY_{1}, …, λY_{r}) = λ^{n}f(Y_{1}, …, Y_{r}).

For the grading on k[Y_{1}, …, Y_{r}], a homogeneous element of degree n is a homogeneous polynomial of degree n in Y_{1}, …, Y_{r}. An ideal I in k[Y_{1}, …, Y_{r}] is said to be homogeneous if it is generated by homogeneous elements.

The point of this is that a closed subvariety of 𝔸^{r} is a union of lines through the origin if and only if it is the set of zeros of a homogeneous ideal. So we define a closed subvariety of ℙ^{r−1} to be the set of simultaneous zeros of a homogeneous ideal I ⊆ k[Y_{1}, …, Y_{r}]. These form the closed sets in the Zariski topology on ℙ^{r−1}. Thus there is a one-to-one correspondence between the closed subsets of ℙ^{r−1} and the homogeneous radical ideals of k[Y_{1}, …, Y_{r}]. Under this correspondence, the closed irreducible subsets correspond to the homogeneous prime ideals.

The ideal 𝔪 = (Y_{1}, …, Y_{r}) is the unique maximal homogeneous ideal, and corresponds to the empty subset of ℙ^{r−1}. This is called the irrelevant ideal, because it corresponds to the origin in 𝔸^{r}, which was removed in order to construct ℙ^{r−1}.

The points in ℙ^{r−1} correspond to the maximal homogeneous ideals among those properly contained in the irrelevant ideal. Namely, the point (a_{1} : …: a_{r}) ∈ ℙ^{r−1} corresponds to the homogeneous ideal generated by the elements a_{i}Y_{j} − a_{j}Y_{i} for i ≠ j.

Again, we can throw in abstract generic points for closed irreducible subvarieties of ℙ^{r−1} to form the projective spectrum of the graded ring k[Y_{1}, …, Y_{r}]. More generally, we have the following definition, see EGA II [142, §2.3].

Definition 3.3.2. Let R = ⨁_{i≥0} R_{i} be a commutative graded ring. We write Proj R for the set of homogeneous prime ideals not containing the irrelevant ideal ⨁_{i>0} R_{i}.

We endow this with the Zariski topology, in which the closed sets are the sets of the form

V (I) = {𝔭 ∈ Proj R | I ⊆ 𝔭}

where I is a homogeneous ideal in R.

Thus Proj k[Y_{1}, …, Y_{r}] has as its closed points the ideals (a_{i}Y_{j}−a_{j}Y_{i}, i ≠ j) corresponding to the points (a_{1} : …: a_{r}) ∈ ℙ^{r−1}(k). The remaining points are generic points for other closed irreducible subvarieties of ℙ^{r−1}(k). As in the affine case, we shall write ℙ^{r−1} or ℙ^{r−1}(k) for Proj k[Y_{1}, …, Y_{r}], and refer to closed points when we wish to refer only to elements of the form (a_{1} : …: a_{r}).

Projective space ℙ^{r−1} has an open cover by r copies of 𝔸^{r−1}. At the level of closed points, the ith open set U_{i} is the set of (a_{1} : …: a_{r}) such that a_{i} ≠ 0. Such a point has a unique representative with a_{i} = 1, and the remaining coordinates give the correspondence with points in 𝔸^{r−1}. At the level of Proj k[Y_{1}, …, Y_{r}], U_{i} is the set of homogeneous primes

not containing Y_{i}. This open set may be thought of as Spec k[Y_{1}/Y_{i}, … ↑^{i} …, Y_{r}/Y_{i}]. These copies of 𝔸^{r−1} covering ℙ^{r−1} are called the affine patches.

Definition 3.3.3. A projective variety over k is a closed subvariety of ℙ^{r−1}(k) for some r. If V is a projective variety then V = Proj R where R = k[Y_{1}, …, Y_{r}]/I for some homogeneous ideal I ⊆ k[Y_{1}, …, Y_{r}]. A projective variety V has an open cover consisting of the

V ∩ U_{i} = Spec R[Y_{i}^{−1}]_{0}

for which Y_{i} is not nilpotent in R.

Definition 3.3.4. A quasi-projective variety over k is a locally closed subset of ℙ^{r−1}, i.e., the intersection of an open subset and a closed subset, for some r.

Thus affine varieties and projective varieties are examples of quasi-projective varieties. But, for example, 𝔸^{2}(k) \ {0} is a quasi-projective variety which is neither an affine variety nor a projective variety. On the other hand, every quasi-projective variety has a finite open cover by affine varieties.

Definition 3.3.5. A regular map of quasi-projective varieties is one which is locally given by polynomials at every point. A rational map, on the other hand, is locally given by polynomials on some dense open subset, and not defined on the complement. Two rational maps are equal if and only if they agree where they are both defined.

Thus for example the regular maps 𝔸^{r} → 𝔸^{1} (or equivalently 𝔸^{1} → k) are the elements of k[Y_{1}, …, Y_{r}] while the rational maps are the elements of k(Y_{1}, …, Y_{r}). Regular or rational maps to k are given the structure of a (commutative) k-algebra via the addition and multiplication in k.

The only regular maps ℙ^{r−1} → 𝔸^{1} are the constant maps, whereas the rational functions are the elements of k(Y_{1}/Y_{r}, …, Y_{r−1}/Y_{r}), namely the rational functions of Y_{1}, …, Y_{r} whose numerator and denominator have the same degree.

3.4. Tangent Spaces

Let V be a quasi-projective variety over k and a a (closed) point of V. We wish to define the tangent space to V at a. This is supposed to be a local concept in the sense that if V′ is an open subset of V containing a then the tangent space to V and to V′ at a are naturally isomorphic. So by intersecting with an affine patch and taking the closure, we may assume that V is a closed subset of 𝔸^{r} for some r.

The linear space L_{V,a} of V at a = (a_{1}, …, a_{r}) is defined to be the set of b = (b_{1}, …, b_{r}) ∈ 𝔸^{r} with the property that for all ϕ ∈ k[Y_{1}, …, Y_{r}] that vanish on V we have

(3.4.1)  ∑_{i=1}^{r} (\frac{∂ϕ}{∂Y_{i}} ) (a)(b_{i} − a_{i}) = 0.

See for example Shafarevich [227, §II.1.3]. The notation is a little confusing here, so let me spell it out. What we are doing is evaluating the function \frac{∂ϕ}{∂Y_{i}} at a, multiplying by b_{i} − a_{i} and then summing. This amounts to evaluating the element of the dual vector space of 𝔸^{r} with coordinates (\frac{∂ϕ}{∂Y_{i}} ) (a) on the vector b − a ∈ 𝔸^{r}.

The tangent space T_{V,a} is defined to be the vector space whose elements are the vectors b − a with b ∈ L_{V,a}. The linear space L_{V,a} is the coset a + T_{V,a}. Thus the tangent space is the subspace defined by the equations

(3.4.2)  ∑_{i=1}^{r} (\frac{∂ϕ}{∂Y_{i}} ) (a)b_{i} = 0.

For f ∈ k[Y_{1}, …, Y_{r}] we write d_{a}(f)(b) for ∑_{i=1}^{r} (\frac{∂f}{∂Y_{i}} ) (a)b_{i}. So d_{a}(f) is a linear form, and (3.4.2) says that T_{V,a} is defined by the equations d_{a}(ϕ)(b) = 0 for ϕ ∈ I. We also have

d_{a}(f_{1} + f_{2}) = d_{a}(f_{1}) + d_{a}(f_{2})
d_{a}(λf) = λd_{a}(f) (λ ∈ k)
(3.4.3)  d_{a}(f_{1}f_{2}) = f_{1}(a)d_{a}(f_{2}) + d_{a}(f_{1})f_{2}(a).

Now d_{a}(f)(b) is well defined for f ∈ k[V] = k[Y_{1}, …, Y_{r}]/I and b ∈ T_{V,a}. It vanishes for f a constant function, so we may as well restrict to 𝔪_{a}/I, the maximal ideal in k[V] corresponding to a, which we shall denote 𝔪_{V,a}. Furthermore, by (3.4.3), d_{a}(f_{1}f_{2})(b) = 0 for f_{1}, f_{2} ∈ 𝔪_{V,a}. So we obtain a well defined bilinear evaluation map

(3.4.4)  𝔪_{V,a}/𝔪_{V,a}^{2} × T_{V,a} → k.

Proposition 3.4.1. The pairing (3.4.4) induces an isomorphism

T_{V,a} ≅ (𝔪_{V,a}/𝔪_{V,a}^{2})^{∗}.

Proof. It suffices to show that the pairing (3.4.4) is perfect. Since every linear form in T_{V,a} is the restriction to T_{V,a} of a linear polynomial in 𝔪_{a}, it remains to show that if f ∈ 𝔪_{a} satisfies d_{a}(f)(b) = 0 for all b ∈ T_{V,a} then f ∈ 𝔪_{a}^{2} + I. By definition of T_{V,a}, given such an f there exists ϕ ∈ I such that d_{a}(f) = d_{a}(ϕ). Examining the Taylor expansion of f − ϕ about a, we see that f − ϕ ∈ 𝔪_{a}^{2} as desired. □

We shall need the following lemma.

Lemma 3.4.2. Let V be a locally closed subset of 𝔸^{r} and let ρ: 𝔸^{1} → V be a rational map of the form

ρ(t) = a + t^{i}b + t^{i+1}c(t)

where c(t) is defined at t = 0. Then b ∈ T_{V,a}.

Proof. We may as well assume that V is closed. If f ∈ k[Y_{1}, …, Y_{r}] vanishes at a then

(f ∘ ρ)(t) = f(a + t^{i}b + t^{i+1}c(t)).

Consider the Taylor expansion of f around a:

f(x) = d_{a}(f)(x − a) + ∑_{j≤k} λ_{j,k}(x_{j} − a_{j})(x_{k} − a_{k}) + · · · .

Then

(f ∘ ρ)(t) = d_{a}(f)(b)t^{i} + d_{a}(f)(c(t))t^{i+1} + ∑_{j≤k} λ_{j,k}(t^{i}b_{j} + t^{i+1}c_{j}(t))(t^{i}b_{k} + t^{i+1}c_{k}(t)) + · · · .

If f is in the ideal defining V then we must have (f ∘ ρ)(t) = 0. Comparing coefficients of t^{i}, this implies that d_{a}(f)(b) = 0. Thus b ∈ T_{V,a}. □

3.5. Presheaves and Sheaves

We want to keep track of regular functions on open subsets of a variety, and we do this by means of sheaves.

Definition 3.5.1. Let X be a topological space. A presheaf ℱ of abelian groups, or rings, or modules, or more generally of objects in a category 𝒞, assigns to each open subset U ⊆ X an object ℱ(U) in 𝒞 and to each inclusion U ⊆ U′ ⊆ X a restriction map ℱ(U′) → ℱ(U). This must send the inclusion U ⊆ U to the identity map on ℱ(U), and a composite inclusion U ⊆ U′ ⊆ U′′ to the composite ℱ(U′′) → ℱ(U′) → ℱ(U). In other words, ℱ is a contravariant functor on the category of open sets and inclusions. A morphism of presheaves is a natural transformation of functors.

For a presheaf to be a sheaf, the category 𝒞 is required to have pullbacks. A presheaf ℱ on X is then defined to be a sheaf if the following gluing condition is satisfied. Given an open cover {U_{α}} of an open set U, ℱ(U) is the pullback of the restriction maps U_{α} → U_{α} ∩ U_{β}. If 𝒞 is a category of abelian groups, rings, modules, etc., we can express this as a condition on elements. Given elements x_{α} ∈ ℱ(U_{α}) such that the restrictions of x_{α} and x_{β} to ℱ(U_{α} ∩ U_{β}) agree for all α and β, there exists a unique x ∈ ℱ(U) that restricts to x_{α} on each U_{α}.

Example 3.5.2. Let V be a quasi-projective variety. The structure sheaf 𝒪_{V} of V assigns to each open subset U ⊆ V the k-algebra of regular functions U → k. If U ⊆ U′ ⊆ V, the restriction map 𝒪_{V} (U′) → 𝒪_{V} (U) is just given by restriction of functions. The gluing condition is easy to verify in this case.

Definition 3.5.3. A ringed space (X, 𝒪_{X}) is a topological space X together with a sheaf of commutative rings 𝒪_{X}.

Example 3.5.4. If V is a quasi-projective variety and 𝒪_{V} is its structure sheaf then (V, 𝒪_{V}) is a ringed space.

Definition 3.5.5. If ℱ is a sheaf on a space X, we write Γ(X, ℱ) for ℱ(X), and we refer to its elements as global sections of ℱ.

Definition 3.5.6. If U is an open subset of X and ℱ is a sheaf on X, we write ℱ|_{U} for the restriction of ℱ to U. This is a sheaf on U in an obvious way.

The following lemma says that sheaves may be constructed locally. This is not true of presheaves.

Lemma 3.5.7. Suppose we are given an open cover {U_{α}} of a space X, sheaves ℱ_{α} on U_{α}, and isomorphisms

ϕ_{α,β}: ℱ_{α}|_{U_{α}∩U_{β}} ⟶^{≅} ℱ_{β}|_{U_{α}∩U_{β}},

such that ϕ_{α,α} is the identity, and such that the diagram

\xymatrix{ℱ_{α}|_{U_{α}∩U_{β}∩U_{γ}} \ar[rr]^{ϕ_{α,γ}} \ar[dr]_{ϕ_{α,β}} &  & ℱ_{γ}|_{U_{α}∩U_{β}∩U_{γ}} \\  & ℱ_{β}|_{U_{α}∩U_{β}∩U_{γ}} \ar@{-->}[ur]_{ϕ_{β,γ}}}

commutes for all α, β, γ. Then there exists a sheaf ℱ on X and isomorphisms ϕ_{α}: ℱ|_{U_{α}} → ℱ_{α} such that the diagram

\xymatrix{ℱ|_{U_{α}∩U_{β}} \ar[rr]^{ϕ_{β}} \ar[dr]_{ϕ_{α}} &  & ℱ_{β}|_{U_{α}∩U_{β}} \\  & ℱ_{α}|_{U_{α}∩U_{β}} \ar@{-->}[ur]_{ϕ_{α,β}}}

commutes for all α, β.

The sheaf ℱ is unique up to natural isomorphism.

Proof. By the gluing condition, an element of ℱ(U) is uniquely determined by the choice of an element x_{α} ∈ ℱ_{α}(U ∩ U_{α}) for each α such that ϕ_{α,β}(x_{α}|_{U∩U_{α}∩U_{β}}) = x_{β}|_{U∩U_{α}∩U_{β}} for all α, β. One checks that this uniquely determines a sheaf ℱ satisfying the given conditions. □

3.6. Stalks and Sheafification

Definition 3.6.1. The stalk of a presheaf ℱ at a point x ∈ X is defined to be

ℱ_{x} = lim_{\overrightarrow{x∈U}} ℱ(U).

Example 3.6.2. In the case of the structure sheaf 𝒪_{V} on a quasi-projective variety V the stalk 𝒪_{V,x} at x ∈ V is the ring of germs of regular functions at x. It is a local ring, whose maximal ideal 𝔪_{x} consists of the germs of functions vanishing at x. We write k(x) for the field 𝒪_{V,x}/𝔪_{x}.

Definition 3.6.3. The inclusion of the category of sheaves on X into the category of presheaves on X has a left adjoint called sheafification. If ℱ is a presheaf and U ⊆ X is open, then the sheafification ℱ^{+} of ℱ assigns to U the functions s from U to the union of the stalks ⋃_{x∈U} ℱ_{x} such that

(1) each s(x) ∈ ℱ_{x}, and

(2) for each x ∈ U there is an open neighbourhood x ∈ U′ ⊆ U and f ∈ ℱ(U) whose image in each stalk ℱ_{y} is equal to s(y) for all y ∈ U′.

There is an obvious map ℱ → ℱ^{+} which is the unit of the adjunction. If ℱ is already a sheaf, then ℱ → ℱ^{+} is an isomorphism.

The sheafification functor is particularly useful because many constructions on sheaves naturally produce presheaves, which then need to be sheafified.

For example, if ℱ and ℱ′ are two sheaves of abelian groups on X, then the presheaf whose value on an open set U is the tensor product ℱ(U) ⊗ ℱ′(U) is usually not a sheaf. We write ℱ ⊗ ℱ′ for its sheafification.

As another example, given a morphism ϕ: ℱ → ℱ′ of sheaves of abelian groups on X, the kernel is already a sheaf. The presheaf cokernel, assigning to each open set U ⊆ X the abelian group ℱ′(U)/ϕ(ℱ(U)), is not necessarily a sheaf. We write ℱ′/ϕ(ℱ) for the sheafification of this presheaf. With this definition of quotient sheaf, sheaves of abelian groups on X form an abelian category.

Remark 3.6.4. Closely related to the fact that quotients need sheafification is the following. If 0 → ℱ → ℱ′ → ℱ′′ → 0 is a short exact sequence of sheaves on X then

0 → Γ(X, ℱ) → Γ(X, ℱ′) → Γ(X, ℱ′′)

is exact. In other words, the global section functor from sheaves to abelian groups is left exact, but not necessarily right exact. We shall examine its right derived functors in Section 3.10.

3.7. The Language of Schemes

We shall sometimes use the language of schemes rather than varieties where this is the appropriate generality. In this section, we gather the necessary definitions. For the purpose of this work, the reader who does not wish to think about schemes can usually just think of varieties, with the proviso that these must include generic points as described earlier in this chapter.

We saw in Section 3.2 that Spec R, for R a commutative ring, is a generalisation of affine variety which emphasises the role of generic points for closed irreducible subvarieties. The analogue of the sheaf of regular functions is the following.

Definition 3.7.1. Let X = Spec R. We define a sheaf of commutative rings 𝒪_{X} as follows. For 𝔭 ∈ X, let R_{𝔭} be the localisation of R at 𝔭. If U ⊆ X is open then 𝒪_{X}(U) is the set of functions s from U to ⋃_{𝔭∈U} R_{𝔭}, such that

(1) each s(𝔭) ∈ R_{𝔭}, and

(2) for each 𝔭 ∈ U there is an open neighbourhood 𝔭 ∈ U′ ⊆ U and elements x, y ∈ R such that for each 𝔮 ∈ U′ we have y ∉ 𝔮 and s(𝔮) = x/y ∈ R_{𝔮}.

We call 𝒪_{X} the structure sheaf on X. With the obvious ring operations and restriction maps, (X, 𝒪_{X}) is a ringed space, called the spectrum of R.

Definition 3.7.2. An affine scheme is a ringed space (X, 𝒪_{X}) which is isomorphic to the spectrum of a commutative ring.

A scheme is a ringed space (X, 𝒪_{X}) which is locally affine. In other words, each x ∈ X has an open neighbourhood x ∈ U ⊆ X such that (U, 𝒪_{X}|_{U}) is an affine scheme.

A scheme (X, 𝒪_{X}) is reduced if each 𝒪_{X}(U) is a reduced ring; in other words, 𝒪_{X}(U) has no non-zero nilpotent elements.

A scheme is Noetherian if it has a finite cover by open affine subsets of the form Spec R with R Noetherian.

Example 3.7.3. An affine variety V = Spec k[Y_{1}, …, Y_{r}]/I ⊆ 𝔸^{r} together with the sheaf of regular functions 𝒪_{V} is an example of an affine scheme. It is reduced and Noetherian.

Example 3.7.4. If V = Proj k[Y_{1}, …, Y_{r}]/I ⊆ ℙ^{r−1} is a projective variety, with its sheaf of regular functions 𝒪_{V}, then (V, 𝒪_{V}) is a reduced Noetherian scheme. This is because if U_{i} is the ith affine patch of ℙ^{r−1} then {(V ∩ U_{i}, 𝒪_{V} |_{V ∩U_{i}})} is an open cover of V by reduced Noetherian affine schemes.

Definition 3.7.5. If (X, 𝒪_{X}) is a scheme, then the stalk 𝒪_{X,x} of 𝒪_{X} at x ∈ X is a local ring. A ringed space with this property is called a locally ringed space.

If (X, 𝒪_{X}) is a locally ringed space and x ∈ X, we write k(x) for the quotient of 𝒪_{X,x} by its maximal ideal 𝔪_{x}.

3.8. Sheaves of Modules

The material in this section is mostly taken from §2 of FAC [224]; see also §3.2–3.4 of EGA II [142] and §II.5 of Hartshorne [147].

If (X, 𝒪_{X}) is a ringed space, an 𝒪_{X}-module ℱ is a sheaf of abelian groups on X together with an 𝒪_{X}(U)-module structure on each ℱ(U), compatible with the restriction maps on 𝒪_{X} and on ℱ.

For a morphism of 𝒪_{X}-modules ℱ → ℱ′, it is required that each ℱ(U) → ℱ′(U) is an 𝒪_{X}(U)-module homomorphism, as well as requiring that ℱ → ℱ′ is a morphism of sheaves of abelian groups. With these definitions, the 𝒪_{X}-modules form an abelian category.

If ℱ and ℱ′ are sheaves of 𝒪_{X}-modules, we write ℱ ⊗_{𝒪_{X}} ℱ′ for the sheafification of the presheaf assigning to an open set U ⊆ X the 𝒪_{X}(U)-module ℱ(U)⊗_{𝒪_{X}(U)}ℱ′(U). The presheaf assigning to U ⊆ X the 𝒪_{X}(U)-module Hom_{𝒪_{X}(U)}(ℱ(U), ℱ′(U)) is already a sheaf, denoted ℋom_{𝒪_{X}}(ℱ, ℱ′), with global sections Γ(X, ℋom_{𝒪_{X}}(ℱ, ℱ′)) = Hom_{𝒪_{X}}(ℱ, ℱ′). Thus ℱ ⊗_{𝒪_{X}} ℱ′ and ℋom_{𝒪_{X}}(ℱ, ℱ′) are 𝒪_{X}-modules, and we have a natural isomorphism

ℋom_{𝒪_{X}}(ℱ ⊗_{𝒪_{X}} ℱ′, ℱ′′) ≅ ℋom_{𝒪_{X}}(ℱ, ℋom_{𝒪_{X}}(ℱ′, ℱ′′)).

Definition 3.8.1. Let (X, 𝒪_{X}) be a locally ringed space. If ℱ is an 𝒪_{X}-module, the fibre of ℱ at a point x ∈ X is defined to be the k(x)-vector space ℱ_{x} ⊗_{𝒪_{X,x}} k(x).

For an affine scheme (V, 𝒪_{V}) with V = Spec R, the global section functor for 𝒪_{V} -modules has a left adjoint, constructed as follows.

Definition 3.8.2. Let V = Spec R be an affine scheme. If M is an R-module, we write M^{∼} for the 𝒪_{V} -module defined as follows. If U ⊆ V is open, then M^{∼}(U) is the set of functions s from U to the union of the localisations ⋃_{𝔭∈U} M_{𝔭}, such that

(1) each s(𝔭) ∈ M_{𝔭}, and

(2) for each 𝔭 ∈ U there is an open neighbourhood 𝔭 ∈ U′ ⊆ U and elements m ∈ M, f ∈ R such that for each 𝔮 ∈ U′ we have f ∉ 𝔮 and s(𝔮) = m/f ∈ M_{𝔮}.

With the obvious restriction maps, this is an 𝒪_{V} -module.

Lemma 3.8.3. Let V = Spec R be an affine scheme. If M is an R-module and ℱ is an 𝒪_{V} -module, we have natural isomorphisms

Hom_{𝒪_{V}} (M^{∼}, ℱ) ≅ Hom_{R}(M, Γ(V, ℱ))

giving an adjunction between the functors ^{∼} and Γ(V, −).

Definition 3.8.4. Let (X, 𝒪_{X}) be a scheme. An 𝒪_{X}-module ℱ is quasi-coherent if X can be covered by affine open subsets U_{α} = Spec R_{α} such that the restriction of ℱ to each U_{α} is isomorphic to M_{α}^{∼} for some R_{α}-module M_{α}. If the M_{α} can be taken to be finitely generated then ℱ is coherent.

Quasi-coherence is equivalent to the statement that the restriction of ℱ to every affine open subset U = Spec R ⊆ V is of the form M^{∼} for some R-module M. Given suitable Noetherian hypotheses, coherence is equivalent to the same condition with M finitely generated. See Proposition II.5.4 of Hartshorne [147].

In particular, a quasi-coherent 𝒪_{V} -module for V = Spec R affine is precisely one of the form M^{∼} for M a R-module.

Theorem 3.8.5. Let V = Spec R be an affine scheme. The functors ^{∼} and Γ(V, −) give an equivalence of categories between R-modules and quasi-coherent 𝒪_{V} -modules. This restricts to an equivalence between finitely generated R-modules and coherent 𝒪_{V} -modules provided R is Noetherian.

Proof. See Corollary II.5.5 of Hartshorne [147]. □

3.9. Coherent Sheaves on Projective Varieties

There is a similar but more complicated equivalence of categories in the projective case, using graded modules.

Definition 3.9.1. Let R = k[Y_{1}, …, Y_{r}]/I with deg(Y_{i}) = 1 and I a homogeneous ideal, and let V = Proj R. Then V has an affine open cover V ∩ U_{i} = Spec R[Y_{i}^{−1}]_{0}, i = 1, …, r. If M is a graded R-module then M[Y_{i}^{−1}]_{0} is an R[Y_{i}^{−1}]_{0}-module. Let ℱ_{i} be the 𝒪_{V ∩U_{i}}-module M[Y_{i}^{−1}]_{0}^{∼}. Then the hypotheses of Lemma 3.5.7 are satisfied, and the ℱ_{i} glue together to give an 𝒪_{V} -module M^{∼}.

Example 3.9.2. If M is a graded R-module and j ∈ ℤ, we define the jth shift of M, denoted M(j), via M(j)_{i} = M_{i+j}. In particular, regarding R as an R-module, this gives us an R-module R(j). We write 𝒪_{V} (j) for the 𝒪_{V} -module R(j)^{∼}. The sheaf 𝒪_{V} (1) is called the twisting sheaf associated with the chosen embedding V ↪ ℙ^{r−1}. For any 𝒪_{V} -module ℱ we write ℱ(j) for 𝒪_{V} (j) ⊗_{𝒪_{V}} ℱ. It is easy enough to check that 𝒪_{V} (j) ⊗_{𝒪_{V}} 𝒪_{V} (j′) ≅ 𝒪_{V} (j + j′), and so we have ℱ(j)(j′) = ℱ(j + j′). We call ℱ(j) the jth twist of ℱ. For example, M(j)^{∼} is naturally isomorphic to M^{∼}(j).

Definition 3.9.3. In the other direction, we have a functor Γ_{∗} from 𝒪_{V} -modules to graded R-modules given as follows. If ℱ is an 𝒪_{V} -module, we make a graded vector space Γ_{∗}(ℱ) whose degree j elements are given by the global sections of the jth twist,

Γ_{j}(ℱ) = Γ(V, ℱ(j)), j ∈ ℤ.

We give Γ_{∗}(ℱ) a graded R-module structure as follows. If r ∈ R_{i} then it determines a global section of 𝒪_{V} (i). If x is a global section of ℱ(j) then rx ∈ Γ_{i+j}(ℱ) defined by the map

Γ(V, 𝒪_{V} (i)) ⊗_{k} Γ(V, ℱ(j)) → Γ(V, ℱ(i + j))

induced by the natural isomorphism

𝒪_{V} (i) ⊗_{𝒪_{V}} ℱ(j) ⟶^{≅} ℱ(i + j).

Lemma 3.9.4. (i) There are natural isomorphisms

Hom_{𝒪_{V}} (M^{∼}, ℱ) ≅ Hom_{R}(M, Γ_{∗}(ℱ))

giving an adjunction between the functors ^{∼} and Γ_{∗}.

(ii) If ℱ is a quasi-coherent 𝒪_{V} -module then the counit of the adjunction is a natural isomorphism Γ_{∗}(ℱ)^{∼} → ℱ.

Proof. In FAC [224, §59], see also EGA II [142, §3.3], natural maps α_{M}: M → Γ_{∗}(M^{∼}) and β_{ℱ}: Γ_{∗}(ℱ)^{∼} → ℱ are constructed, and in FAC Propositions 59.7, 59.8, EGA II (3.3.5.2) it is shown that the composites

M^{∼} ⟶^{α_{M}^{∼}} Γ_{∗}(M^{∼})^{∼} ⟶^{β_{M^{∼}}} M^{∼}
Γ_{∗}(ℱ) ⟶^{α_{Γ∗(ℱ)}} Γ_{∗}(Γ_{∗}(ℱ)^{∼}) ⟶^{Γ_{∗}(β_{ℱ} )} Γ_{∗}(ℱ)

are isomorphisms. By Theorem IV.2 of Mac Lane [190], this implies part (i) of the lemma. Part (ii) is proved in FAC §65, see also EGA II [142, (3.4.4)], Hartshorne [147, Prop. 5.15]. □

Warning 3.9.5. Even if M is finitely generated, Γ_{∗}(M^{∼}) need not be, because there may be infinitely many negative degrees in which it is non-zero.

Definition 3.9.6. If M is a graded R-module and d ∈ ℤ, we write M_{≥d} for the graded R-module ⨁_{j≥d} M_{j} obtained by replacing the terms in M of degree strictly less than d by zero.

Lemma 3.9.7. If M is a finitely generated graded R-module then the unit of the adjunction M → Γ_{∗}(M^{∼}) is an isomorphism in large enough degrees, in the sense that there exists d ∈ ℤ such that M_{≥d} → Γ_{∗}(M^{∼})_{≥d} is an isomorphism.

Proof. An element of Γ_{j}(M^{∼}) may be described as follows. For each affine open set U_{i} ⊆ ℙ^{r−1} defined by Y_{i}, we must name a fraction m_{i}/Y_{i}^{n_{i}} with deg(m_{i}) = j + n_{i}, n_{i} ≥ 0, in such a way that for i ≠ j we have Y_{j}^{n_{j}}m_{i} = Y_{i}^{n_{i}}m_{j}. Two such elements are equivalent, (m_{1}/Y_{1}^{n_{1}}, …, m_{r}/Y_{r}^{n_{r}}) ∼ (m′_{1}/Y_{1}^{n′_{1}}, …, m′_{r}/Y_{r}^{n′_{r}}), if each Y_{i}^{n′_{i}}m_{i} − Y_{i}^{n_{i}}m′_{i} is annihilated by some power of Y_{i}. The map β_{M} takes m ∈ M_{j} to (m/1, …, m/1).

The kernel of β_{M} is the set of m ∈ M annihilated by some power of each Y_{i}, and hence by every element of large enough degree. Since M is finitely generated, this submodule is also finitely generated, and hence non-zero only in a finite number of degrees.

Similarly, for any (m_{1}/Y_{1}^{n_{1}}, …, m_{r}/Y_{r}^{n_{r}}) in Γ_{j}(M), multiplication by Y_{i}^{n_{i}} takes it to an element equivalent to (m_{i}/1, …, m_{i}/1). So every element of the cokernel is annihilated by some power of each Y_{i}, and hence by every element of large enough degree. But the cokernel is not necessarily finitely generated, so a further argument is still necessary.

We note that since M is finitely generated, it has a finite filtration with filtered quotients of the form R/𝔭(n), where 𝔭 is a homogeneous prime ideal and n ∈ ℤ. A short exact sequence of modules 0 → M → M′ → M′′ → 0 gives rise to a diagram

\xymatrix{0 \ar[r] & M \ar[r] \ar[d] & M′ \ar[r] \ar[d] & M′′ \ar[r] \ar[d] & 0 \\ 0 \ar[r] & Γ_{∗}(M^{∼}) \ar[r] & Γ_{∗}(M′ ^{∼}) \ar[r] & Γ_{∗}(M′′ ^{∼})}

and so it suffices to prove the theorem with M = R/𝔭(n), or equivalently with M = R/𝔭. Replacing R by R/𝔭, we may assume that 𝔭 = 0, R is a graded integral domain and M = R, so that M^{∼} = 𝒪_{V}. In this case, α: R → Γ_{∗}(𝒪_{V}) is an embedding of R as a subring of ∩_{i=1}^{r}R[Y_{i}^{−1}], where the intersection takes place in the graded field of fractions of R.

Let x = (x_{1}/Y_{1}^{n_{1}}, …, x_{r}/Y_{r}^{n_{r}}) ∈ Γ_{j}(𝒪_{V}) with j ≥ 0. Then for some n ≥ 0, every element of R_{≥n} sends x into R, and since j ≥ 0, it sends it into R_{≥n}. So by induction on q, for every q ≥ 1 we have Y_{1}^{n}x^{q} ∈ R_{≥n}. Thus for all q ≥ 1 we have x^{q} ∈ Y_{1}^{−n}R, which is a finitely generated R-module. So x is in the integral closure of R in its homogeneous field of fractions, which is a finitely generated R-module. Thus Γ_{≥0}(𝒪_{V}) is a finitely generated R-module. So the part of the cokernel in non-negative degrees is also finitely generated, and is annihilated by R_{≥n}, and is hence zero in all large enough degrees. □

Definition 3.9.8. We say that a graded R-module M is quasi-finitely generated if for some d ∈ ℤ, M_{≥d} is finitely generated.^{1}

We say that M has finite length if the total dimension of M as a k-vector space is finite, and that M has quasi-finite length if for some d ∈ ℤ, M_{≥d} is the zero module.

We write mod(R), qmod(R), fl(R), qfl(R) for the categories of R-modules which are finitely generated, respectively quasi-finitely generated, finite length, quasi-finite length.

Proposition 3.9.9. If ℱ is coherent then Γ_{∗}(ℱ) is quasi-finitely generated. Furthermore, ℱ ≅ M^{∼} for a suitable finitely generated R-module M.

Proof. By a theorem of Serre (see Hartshorne [147, Theorem II.5.17]) ℱ(n) is generated by a finite number of global sections for some large enough n ∈ ℤ. Let M be the submodule of Γ_{∗}(ℱ) generated by these global sections. Then M^{∼} → Γ_{∗}(ℱ)^{∼} is an isomorphism. By Lemma 3.9.4 (ii), Γ_{∗}(ℱ)^{∼} → ℱ is also an isomorphism, so ℱ ≅ M^{∼}, and M is finitely generated. It now follows from Lemma 3.9.7 that there exists d ∈ ℤ such that M_{≥d} → Γ_{∗}(ℱ)_{≥d} is an isomorphism, and so Γ_{∗}(ℱ) is quasi-finitely generated. □

Definition 3.9.10. We write mod(R)/fl(R) for the category whose objects are the same as those in mod(R), but whose morphisms are given by

Hom_{mod(R)/fl(R)}(M, M′) = lim_{d→∞} Hom_{R}(M_{≥d}, M′_{≥d}).

In other words, a morphism from M to M′ in mod(R)/fl(R) is given by a homomorphism M_{≥d} → M′_{≥d} for some d ∈ ℤ, and two such morphisms are equal if they agree for some d′ ≥ d.

1In FAC [224, §56], this is referred to as “M vérifie la condition (TF).”

Similarly we form the category qmod(R)/qfl(R) where the objects are the same as those in qmod(R) and the morphisms are defined in the same way as above.

Another way of saying this is that fl(R), respectively qfl(R), is a Serre subcategory of mod(R), respectively qmod(R), with Serre quotients mod(R)/fl(R) and qmod(R)/qfl(R).

Lemma 3.9.11. The inclusion mod(R) → qmod(R) induces an equivalence of categories

mod(R)/fl(R) ⟶^{≃} qmod(R)/qfl(R).

Proof. This follows from the following easily verified facts.

(1) Every quasi-finitely generated module M is isomorphic in qmod(R)/qfl(R) to the finitely generated module M_{≥d} for d large enough.

(2) A finitely generated module is isomorphic to the zero module in qmod(R)/qfl(R) if and only if it has finite length. □

Theorem 3.9.12. The functors ^{∼} and Γ_{∗} induce an equivalence of categories between qmod(R)/qfl(R) and the category of coherent sheaves on V = Proj R.

Proof. This follows from Lemma 3.9.4, Lemma 3.9.7 and Proposition 3.9.9. □

Corollary 3.9.13. The functors ^{∼} and Γ_{≥0} induce an equivalence of categories between mod(R)/fl(R) and coherent sheaves on V = Proj R.

Proof. This follows from Theorem 3.9.12 and Lemma 3.9.11. □

The case of greatest interest to us is the following.

Corollary 3.9.14. The functors ^{∼} and Γ_{≥0} induce an equivalence of categories between mod(k[Y_{1}, …, Y_{r}])/fl(k[Y_{1}, …, Y_{r}]) and coherent sheaves on ℙ^{r−1}. □

3.10. Cohomology of Sheaves

Let 𝒪 be a sheaf of rings on a space X, and consider the category of 𝒪-modules. We saw in Remark 3.6.4 that a short exact sequence of 𝒪-modules 0 → ℱ → ℱ′ → ℱ′′ → 0 gives rise to an exact sequence

0 → Γ(X, ℱ) → Γ(X, ℱ′) → Γ(X, ℱ′′).

In other words, Γ(X, −) is left exact, but not right exact. So it has right derived functors H^{i}(X, −), which we now describe.

Lemma 3.10.1. The category of 𝒪-modules has enough injectives.

Proof. See Hartshorne [147, Proposition III.2.2]. □

Example 3.10.2. If V = Spec R is an affine variety and I is an injective R-module then I^{∼} is an injective 𝒪_{V} -module.

It follows that every 𝒪-module ℱ has an injective resolution, namely a sequence

0 → ℐ_{0} → ℐ_{1} → ℐ_{2} → ⋯

which is exact except in degree zero, where the kernel is isomorphic to ℱ. Applying the global section functor, we obtain a complex

0 → Γ(X, ℐ_{0}) → Γ(X, ℐ_{1}) → Γ(X, ℐ_{2}) → · · · .

The cohomology of this complex in degree i is denoted

H^{i}(X, ℱ).

A morphism of sheaves ℱ → ℱ′ extends to a morphism of resolutions, unique up to homotopy,

\xymatrix{0 \ar[r] & ℐ_{0} \ar[r] \ar[d] & ℐ_{1} \ar[r] \ar[d] & ℐ_{2} \ar[r] \ar[d] & ⋯ \\ 0 \ar[r] & ℐ′_{0} \ar[r] & ℐ′_{1} \ar[r] & ℐ′_{2} \ar[r] & ⋯}

and hence there is an induced map

H^{i}(X, ℱ) → H^{i}(X, ℱ′).

In particular, given two resolutions of the same sheaf, extending the identity map induces a natural isomorphism from H^{i}(X, ℱ) defined using one resolution to that defined using the other, so that this definition of cohomology of sheaves is independent of the choice of resolution up to natural isomorphism.

Since Γ is left exact, we have a natural isomorphism

H^{0}(X, ℱ) ≅ Γ(X, ℱ).

Given a short exact sequence of sheaves 0 → ℱ → ℱ′ → ℱ′′ → 0, the horseshoe lemma constructs a short exact sequence of resolutions, and hence a long exact sequence in cohomology

0 → Γ(X, ℱ) → Γ(X, ℱ′) → Γ(X, ℱ′′)
→ H^{1}(X, ℱ) → H^{1}(X, ℱ′) → H^{1}(X, ℱ′′)
→ H^{2}(X, ℱ) → H^{2}(X, ℱ′) → H^{2}(X, ℱ′′) → · · · .

Example 3.10.3. Let V = Spec R be an affine variety and let M be an R-module. If

0 → I_{0} → I_{1} → I_{2} → ⋯

is an injective resolution of M as an R-module, then

0 → I_{0}^{∼} → I_{1}^{∼} → I_{2}^{∼} → ⋯

is an injective resolution of M^{∼} because ^{∼} is exact. Taking global sections gives us back the injective resolution of M. It follows that for i > 0 we have

H^{i}(V, M^{∼}) = 0.

Thus quasi-coherent 𝒪_{V} -modules with V affine only have cohomology in degree zero.

This gives rise to a way of computing cohomology using a Čech complex. Let 𝒪 be a sheaf of rings on a space X. Given a finite open cover 𝔘 = {U_{1}, …, U_{r}} of a space X, and a sheaf ℱ on X, we set

Č^{n}(𝔘, ℱ) = ⨁_{1≤i_{0}<···<i_{n}≤r} ℱ(U_{i_{0}} ∩ ⋯ ∩ U_{i_{n}}),

(so Č^{n}(𝔘, ℱ) = 0 for n ≥ r).^{2} We form a complex

0 → Č^{0}(𝔘, ℱ) ⟶^{d} Č^{1}(𝔘, ℱ) ⟶^{d} Č^{2}(𝔘, ℱ) ⟶^{d} ⋯

via the alternating sum of the restriction maps

(dx)_{i_{0},...,i_{n+1}} = ∑_{j=0}^{n+1}(−1)^{j}xi_{0},...↑...,i_{n+1}^{j}|_{U_{i_{0}}∩···∩U_{i_{n+1}}}.

We write Ȟ^{i}(𝔘, ℱ) for the cohomology of this Čech complex.

Theorem 3.10.4. Let 𝔘 = {U_{1}, …, U_{r}} be an affine open cover of a quasi-projective variety V and let ℱ be a quasi-coherent 𝒪_{V} -module. Then there are natural isomorphisms

H^{i}(V, ℱ) ≅ Ȟ^{i}(𝔘, ℱ).

In particular, H^{i}(V, ℱ) = 0 for i ≥ r.

Proof. See Theorem III.4.5 of Hartshorne [147]. □ Corollary 3.10.5. Let V ⊆ ℙ^{r−1} be a projective variety and ℱ a quasi-coherent 𝒪_{V} -

module. Then H^{i}(V, ℱ) = 0 for i ≥ r.

Proof. Let U_{i} be the affine open subset of V given by intersecting with the ith affine patch of ℙ^{r−1}, 1 ≤ i ≤ r. Apply Theorem 3.10.4 to this open cover. □

2These definitions extend in an obvious way to infinite totally ordered open covers, but we shall not need this.

4 Jordan Type

In this chapter, following Friedlander, Pevtsova and Suslin [128] we introduce maximal Jordan type, generic Jordan type and modules of constant Jordan type, in the context of the representation theory of elementary abelian p-groups.

4.1. Nilvarieties

Suppose we are given a matrix A ∈ Mat_{n×m}(k[V]). Then given an integer s with 0 ≤ s ≤ n, for a point v ∈ V, the condition that

rank (A(v)) ≤ s

is given by the vanishing of the set of (s + 1) × (s + 1) minors of A (one for each subset of size s + 1 of the rows and subset of size s + 1 of the columns). Using the discussion in Section 3.2, we obtain the following.

Theorem 4.1.1. Given A = A(Ȳ_{1}, …, Ȳ_{r}) ∈ Mat_{n×m}(k[V]), the following are equivalent:

(1) rank (A) = s, where A is regarded as an element of Mat_{n×m}(k(V)).

(2) rank (A(t_{1}, …, t_{r})) = s, where t_{1}, …, t_{r} are elements of some field extension K of k such that the ideal of polynomial relations satisfied by t_{1}, …, t_{r} is the prime ideal 𝔭 defining V.

(3) rank (A(v)) ≤ s for all points v ∈ V, and there exists v ∈ V with rank (A(v)) = s.

(4) rank (A(v)) = s for all v in a dense open subset of V.

Proof. This follows from Lemma 3.2.1. □

We shall need the following consequence of Theorem 4.1.1 when we study generic kernels in Section 5.7 and generic images in Section 5.10.

Proposition 4.1.2. Let A_{1}, A_{2} ∈ Mat_{n×m}(k), regarded as maps from k^{m} to k^{n}.

(1) Suppose that for all λ, μ ∈ k, not both zero, λA_{1} + μA_{2} is injective. Then

⋂_{(λ,μ)≠(0,0)} Im (λA_{1} + μA_{2}) = 0.

(2) Suppose that for all λ, μ ∈ k, not both zero, λA_{1} + μA_{2} is surjective. Then

∑_{(λ,μ)≠(0,0)} Ker (λA_{1} + μA_{2}) = k^{m}.

Proof. Since these statements are dual, it suffices to prove (1). Suppose that 0 ≠ v ∈ k^{n} is in the image of λA_{1} + μA_{2} for all (λ, μ) ≠ (0, 0). This condition is equivalent to the statement that the augmented matrix (λA_{1} + μA_{2}|v) has rank m for all (λ, μ) ≠ (0, 0), and so by Theorem 4.1.1 it remains true over any extension field of k. In particular, it holds over the field of Laurent series k[[t]][t^{−1}]. So there exist u_{i} ∈ k^{m} (i ∈ ℤ) such that (A_{1} + tA_{2}) ∑_{i} u_{i}t^{i} = v, with u_{i} = 0 for i sufficiently large negative. Comparing coefficients of powers of t, we see that u_{i} = 0 for i < 0 and that A_{1}u_{0} = v, A_{2}u_{i} = A_{1}u_{i+1} (i ≥ 0). Since k^{m} is finite dimensional, there exists a non-trivial linear relation between the u_{i} with i ≥ 0. Choose s such that u_{s} ∈ ⟨u_{0}, …, u_{s−1}⟩. Then

A_{2}u_{s} ∈ ⟨A_{2}u_{0}, …, A_{2}u_{s−1}⟩ = ⟨A_{1}u_{1}, …, A_{1}u_{s}⟩ ⊆ ⟨A_{1}u_{0}, …, A_{1}u_{s−1}⟩.

Then for all (λ, μ) ≠ (0, 0) the map

λA_{1} + μA_{2} : ⟨u_{0}, …, u_{s−1}⟩ → ⟨A_{1}u_{0}, …, A_{1}u_{s−1}⟩

is injective, and hence an isomorphism. This is impossible, because it implies that A_{1}^{−1}A_{2} has no eigenvectors on ⟨u_{0}, …, u_{s−1}⟩. □

Another consequence of Theorem 4.1.1 of more immediate use to us is the following.

Corollary 4.1.3. Let A_{1}, …, A_{r} ∈ Mat_{n}(k). Then the following are equivalent:

(1) For all λ_{1}, …, λ_{r} ∈ k the matrix λ_{1}A_{1} + ⋯ + λ_{r}A_{r} is nilpotent.

(2) For all field extensions K of k and for all t_{1}, …, t_{r} ∈ K, the matrix t_{1}A_{1} +⋯+t_{r}A_{r} is nilpotent.

(3) The matrix t_{1}A_{1} + ⋯ + t_{r}A_{r} ∈ Mat_{n}(k(t_{1}, …, t_{r})) is nilpotent, where t_{r}, …, t_{r} are independent transcendentals over k.

Proof. This follows by applying Theorem 4.1.1 to the powers of λ_{1}A_{1} + ⋯ + λ_{r}A_{r} with V = 𝔸^{r}(k). □

Definition 4.1.4. We say that A_{1}, …, A_{r} span a nilvariety of rank r if the equivalent conditions of Corollary 4.1.3 are satisfied.

Example 4.1.5. If A_{1}, …, A_{r} are commuting nilpotent matrices then they span a nilvariety. If, in addition, k has characteristic p and the pth powers of A_{1}, …, A_{r} are zero, then this nilvariety corresponds to the representation of the elementary abelian p-group ⟨g_{1}, …, g_{r}⟩ ≅ (ℤ/p)^{r} given by g_{i} ↦ I + A_{i}.

Example 4.1.6. The matrices

A_{1} = (\begin{matrix}0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{matrix}) and A_{2} = (\begin{matrix}0 & 0 & 0 \\ 1 & 0 & 0 \\ 0 & −1 & 0 \end{matrix})

span a nilvariety, but do not commute. We have

(Y_{1}A_{1} + Y_{2}A_{2})^{3} = 0

for all values of Y_{1} and Y_{2}. In fact, for all values of Y_{1} and Y_{2}, not both zero, the matrix Y_{1}A_{1} + Y_{2}A_{2} has a single Jordan block with eigenvalue zero. We shall see this example again in Section 10.6.

Example 4.1.7. Similar examples of larger dimension may also be constructed. The following examples of dimension five and seven with a single Jordan block are taken from a paper of Causa, Re and Teodorescu [85]. In each case we give the matrix Y_{1}A_{1} + Y_{2}A_{2}:

(\begin{matrix}0 & Y_{1} & 0 & 0 & 0 \\ 2Y_{2} & 0 & Y_{1} & 0 & 0 \\ 0 & −Y_{2} & 0 & Y_{1} & 0 \\ 0 & 0 & Y_{2} & 0 & Y_{1} \\ 0 & 0 & 0 & −2Y_{2} & 0 \end{matrix}) ,  (\begin{matrix}0 & 0 & −Y_{2} & 0 & 0 \\ 0 & 0 & Y_{1} & 0 & −Y_{2} \\ 0 & 0 & 0 & 0 & Y_{1} \\ Y_{1} & Y_{2} & 0 & 0 & 0 \\ Y_{2} & 0 & 0 & −Y_{1} & 0 \end{matrix}) ,

(\begin{matrix}0 & Y_{1} & 0 & 0 & 0 & 0 & 0 \\ −4Y_{2} & 0 & Y_{1} & 0 & 0 & 0 & 0 \\ 0 & 2Y_{2} & 0 & Y_{1} & 0 & 0 & 0 \\ 0 & 0 & −Y_{2} & 0 & Y_{1} & 0 & 0 \\ 0 & 0 & 0 & Y_{2} & 0 & Y_{1} & 0 \\ 0 & 0 & 0 & 0 & −2Y_{2} & 0 & Y_{1} \\ 0 & 0 & 0 & 0 & 0 & 4Y_{2} & 0 \end{matrix}) .

It is shown in [85] that in each odd dimension n the variety of matrices looking like the first and third of these examples but with possibly different coefficients below the diagonal is a complete intersection of dimension (n − 3)/2, and that there are no even dimensional examples. See Theorems 10.10.1 and 10.10.2.

Corollary 4.1.8. For A, B_{1}, …, B_{r} ∈ Mat_{n}(k), the following are equivalent.

(1) rank (A) = rank (A + t_{1}B_{1} + ⋯ + t_{r}B_{r}), where t_{1}, …, t_{r} are independent transcendentals over k.

(2) rank (A) ≥ rank (A + λ_{1}B_{1} + ⋯ + λ_{r}B_{r}) for every (λ_{1}, …, λ_{r}) ∈ 𝔸^{r}(K), for every extension field K of k.

(3) rank (A) ≥ rank (A + λ_{1}B_{1} + ⋯ + λ_{r}B_{r}) for every (λ_{1}, …, λ_{r}) ∈ 𝔸^{r}(k).

(4) rank (A) = rank (A + λ_{1}B_{1} + ⋯ + λ_{r}B_{r}) for (λ_{1}, …, λ_{r}) in a dense open subset of 𝔸^{r}(k).

Proof. This also follows from Theorem 4.1.1; the open subset where the ranks are equal is non-empty because it contains the origin, and is therefore dense. □

Definition 4.1.9. Let A, B_{1}, …, B_{r} ∈ Mat_{n}(k). We say that A is rank maximal with respect to B_{1}, …, B_{r} if the equivalent conditions of Corollary 4.1.8 are satisfied.

Remark 4.1.10. It follows from Corollary 4.1.8 that if A is rank maximal with respect to B_{1}, …, B_{r} and λ_{1}, …, λ_{r} are such that

rank (A) = rank (A + λ_{1}B_{1} + ⋯ + λ_{r}B_{r})

then A + λ_{1}B_{1} + ⋯ + λ_{1}B_{r} is rank maximal with respect to A, B_{1}, …, B_{r}.

4.2. Matrices and Tangent Spaces

In preparation for the proof of Gerstenhaber’s Theorem 4.3.1, in this section we examine the tangent space of a conjugacy class in Mat_{n}(k) at a matrix A.

Definition 4.2.1. Let A ∈ Mat_{n}(k). We write 𝒞cℓ(A) for the locally closed subvariety of Mat_{n}(k) consisting of matrices conjugate to A. In other words, it is the set of matrices of the form BAB^{−1} with B ∈ GL_{n}(k).

Proposition 4.2.2. The tangent space T_{A,𝒞cℓ(A)} consists of the matrices of the form W A − AW with W ∈ Mat_{n}(k).

Proof. Consider the morphism of algebraic varieties 𝔸^{1} → 𝒞cℓ(A) given by

t ↦ (I + tW)A(I + tW)^{−1} = A + t(W A − AW) + t^{2}C(t)

where C(t) ∈ Mat_{n}(k(t)) is defined at t = 0. It follows from Lemma 3.4.2 that W A − AW ∈ T_{𝒞cℓ(A),A}.

The morphism GL_{n}(k) → 𝒞cℓ(A) given by B ↦ BAB^{−1} induces a surjective map T_{GL_{n}(k),I} → T_{𝒞cℓ(A),A}, and so all elements of T_{𝒞cℓ(A),A} are of this form. □

Proposition 4.2.3. Let U(t) ∈ GL_{n}(k(t)) be a matrix of rational functions, and suppose that A ∈ Mat_{n}(k) satisfies

U(t)AU(t)^{−1} = A + t^{j}B + t^{j+1}C(t)

where B ∈ Mat_{n}(k) and C(t) ∈ Mat_{n}(k(t)) such that C(0) is defined. Then B ∈ T_{𝒞cℓ(A),A}, and there is a matrix W such that B = W A − AW.

Proof. We begin by remarking that U(0) does not have to be defined, and even if it is defined, it does not have to be invertible. However, det U(t) ∈ k(t) has only a finite number of zeros and poles, so there is a dense open subset of 𝔸^{1}(k) consisting of those t such that U(t)AU(t)^{−1} is defined. We can also see from the right hand side of the equation that this definition extends to a regular map in an open neighbourhood of t = 0 that takes the value A at t = 0. The function

f(t) = A + t^{j}B + t^{j+1}C(t)

is therefore a rational map from 𝔸^{1} to 𝒞cℓ(A) ⊆ 𝔸^{n^{2}} that is defined at t = 0. It follows by Lemma 3.4.2 that we have B ∈ T_{𝒞cℓ(A),A}. The existence of W then follows from Proposition 4.2.2. □

4.3. A Theorem of Gerstenhaber

Theorem 4.3.1 (Gerstenhaber [131, Prop. 1.9]). Let A, B ∈ Mat_{n}(k). Then the following are equivalent:

(1) If t is transcendental over k then A ∼ A + tB as elements of Mat_{n}(k(t)).

(2) There exists a finite sequence of matrices W_{1}, W_{2}, …, W_{s} such that the following equations hold:

W_{1}A − AW_{1} = B, W_{i}A − AW_{i} = BW_{i−1} for i = 2, …, s, and BW_{s} = 0.

(3) There exists an infinite sequence of matrices W_{1}, W_{2}, … such that the following equations hold:

W_{1}A − AW_{1} = B, W_{i}A − AW_{i} = BW_{i−1} for i ≥ 2.

Proof. Since there are missing details in [131], we have chosen to write out the proof in detail. If (1) holds, then there is a matrix W (t) ∈ Mat_{n}(k(t)) such that

W (t)AW (t)^{−1} = A + tB.

Rewriting this as

(4.3.1)  W (t)A = (A + tB)W (t)

and comparing coefficients, we see that (3) is the statement that W (t) may be chosen to be a power series in t of the form

(4.3.2)  I + W_{0}t + W_{1}t^{2} + …

while (2) is the statement that W (t) may be chosen to be a polynomial in t of this form. It is clear that (2) implies (1), so we shall show that (1) implies (3), which in turn implies (2).

We begin by showing that (1) implies (3). Given U(t) ∈ Mat_{n}(k(t)) conjugating A to A + tB, we shall inductively construct a power series W (t) ∈ Mat_{n}(k[[t]]) of the form (4.3.2) which also conjugates A to A + tB.

Now examine the equation U(t)AU(t)^{−1} = A + tB. Applying Proposition 4.2.3 to this equation, we see that there exists a matrix W_{1} such that W_{1}A − AW_{1} = B.

Next, examine the expression

(I + tW_{1})^{−1}(A + tB)(I + tW_{1}) = A + t^{2}BW_{1} + ⋯

as elements of Mat_{n}(k[[t]]). Since this matrix is also conjugate to A in Mat_{n}(k(t)), it follows that BW_{1} ∈ T_{𝒞cℓ(A),A} and again applying Proposition 4.2.3, there exists a matrix W_{2} such that W_{2}A − AW_{2} = B.

Proceeding by induction, we obtain matrices W_{1}, W_{2}, … such that for all i ≥ 1 we have

(I + tW_{1} + ⋯ + t^{i−1}W_{i−1})^{−1}(A + tB)(I + tW_{1} + ⋯ + t^{i−1}W_{i−1}) = A + t^{i}BW_{i−1} + ⋯

Therefore BW_{i−1} ∈ T_{A}, and this allows us to find a matrix W_{i} such that W_{i}A−AW_{i} = BW_{i−1}.

To see that (3) implies (2), we note that given any sequence W′_{1}, W′_{2}, … satisfying (3) and a sequence K_{1}, K_{2}, … of matrices that commute with A, the sequence W_{1}, W_{2}, … defined by

W_{i} = W′_{i} + W′_{i−1}K_{1} + ⋯ + W′_{1}K_{i−1} + K_{i}

also satisfies (3).

For s sufficiently large, the subspace of Mat_{n}(k) spanned by I, W′_{1}, W′_{2}, … contains W′_{s}, and so we may find scalar multiples of the identity K_{1}, …, K_{s} such that

W_{s} = W′_{s} + W′_{s−1}K_{1} + ⋯ + W′_{1}K_{s−1} + K_{s} = 0.

Then we can take W_{i} = 0 for i ≥ s, and we see that (2) is satisfied. □

Corollary 4.3.2. Let A, B ∈ Mat_{n}(k), and let C ∈ Mat_{n}(k) be a matrix commuting with A and B. If A ∼ A + tB in Mat_{n}(k(t)) then also A ∼ A + tBC.

Proof. The matrices C^{i}W_{i} satisfy:

(CW_{1})A − A(CW_{1}) = BC, (C^{i}W_{i})A − A(C^{i}W_{i}) = (BC)(C^{i−1}W_{i−1}) for i = 2, …, s,

and (BC)(C^{s}W_{s}) = 0.  □

The version in several variables works the same way, but with more notation.

Theorem 4.3.3. Let A, B_{1}, …, B_{r} ∈ Mat_{n}(k). Then the following are equivalent:

(1) If t_{1}, …, t_{r} are independent transcendentals over k then A ∼ A + t_{1}B_{1} + ⋯ + t_{r}B_{r}

as elements of Mat_{n}(k(t_{1}, …, t_{r})).

(2) There exist matrices W_{i_{1},...,i_{r}}, one for each sequence (i_{1}, …, i_{r}) of integers, such that

W_{i_{1},...,i_{r}}A − AW_{i_{1},...,i_{r}} = B_{1}W_{i_{1}−1,...,i_{r}} + ⋯ + B_{r}W_{i_{1},...,i_{r}−1},

and furthermore, W_{i_{1},...,i_{r}} = 0 if and only of the indices are negative, W_{0,...,0} = I, and:

W_{i_{1},...,i_{r}} = 0 for all but finitely many values of (i_{1}, …, i_{r}).

(3) There exist matrices W_{i_{1},...,i_{r}} as in (2) but without the last finiteness condition.

Proof. This proof follows the same lines as the proof of Theorem 4.4.10. This time, the conjugating matrix is given by

W (t_{1}, …, t_{r}) = ∑_{(i_{1},...,i_{r})} W_{i_{1},...,i_{r}}t_{1}^{i_{1}} … t_{r}^{i_{r}}

and the conjugation equation is given by

{□ \atop (4.3.3)} W (t_{1}, …, t_{r})A = (A + t_{1}B_{1} + ⋯ + t_{r}B_{r})W (t_{1}, …, t_{r}).

4.4. Dominance Order and Nilpotent Jordan Types

Definition 4.4.1. The dominance order is the partial order on partitions of n defined as follows: [a_{1}] … [a_{t}] ⪰ [b_{1}] … [b_{t}] if and only if for all i with 1 ≤ i ≤ t

∑_{j=1}^{i} a_{j} ≥ ∑_{j=1}^{i} b_{j}.

Example 4.4.2. It is easy to check that [3] [1] ⪰ [2] [2]. More generally, if a_{i−1} > a_{i} and a_{j} > a_{j+1} (with i < j) then

[a_{1}] … [a_{i} + 1] … [a_{j} − 1] … [a_{t}] ⪰ [a_{1}] … [a_{i}] … [a_{j}] … [a_{t}].

It is not difficult to verify that [a_{1}] … [a_{t}] ⪰ [b_{1}] … [b_{t}] if and only if we can move from one to the other through a sequence of such steps (see for example James and Kerber [171, §1.4]).

Lemma 4.4.3. Conjugation of partitions reverses the dominance ordering: a ⪰ b if and only if b′ ⪰ a′. The latter condition may be written as follows:

∀ s > 0 ∑_{j} min(a_{j}, s) ≤ ∑_{j} min(b_{j}, s).

Proof. See James and Kerber [171, §1.4]. □

If A is a nilpotent matrix then all its eigenvalues are zero, so its Jordan canonical form is a diagonal sum of matrices J_{a}, where J_{a} is a Jordan block of length a with eigenvalue zero, namely the a × a matrix

(\begin{matrix}0 & 0 & ⋯ & 0 & 0 \\ 1 & 0 & & 0 & 0 \\ 0 & 1 & ⋯ & 0 & 0 \\ ⋮ & & ... & & ⋮ \\ 0 & 0 & ⋯ & 1 & 0 \end{matrix}) .

We write J_{[a_{1}]...[a_{t}]} for an n × n matrix which is a diagonal sum of Jordan blocks of lengths a_{1}, …, a_{t} with a_{1} ≥ ⋯ ≥ a_{t}. So letting GL(n, k) act on Mat_{n}(k) by conjugation, this puts the GL(n, k)-orbits of nilpotent matrices in one to one correspondence with partitions of n.

Definition 4.4.4. We put a partial order on nilpotent n × n matrices, which we also call the dominance order as follows. If A, B ∈ Mat_{n}(k) are nilpotent, we write A ⪰ B if for all s > 0

rank (A^{s}) ≥ rank (B^{s}).

This depends only on the Jordan canonical forms of A and B, and so we can regard it as a partial order on nilpotent Jordan types.

Lemma 4.4.5. The dominance order on nilpotent matrices corresponds to the dominance order on partitions, in the sense that J_{[a_{1}]...[a_{t}]} ⪰ J_{[b_{1}]...[b_{t}]} if and only if [a_{1}] … [a_{t}] ⪰ [b_{1}] … [b_{t}].

Proof. Let A = J_{[a_{1}]...[a_{t}]}, B = J_{[b_{1}]...[b_{t}]}. Then using Lemma 4.4.3 we have

A ⪰ B ⟺ ∀s > 0 nullity(A^{s}) ≤ nullity(B^{s})

⟺ ∀s > 0 ∑_{j} min(a_{j}, s) ≤ ∑_{j} min(b_{j}, s)

⟺ [a_{1}] … [a_{t}] ≥ [b_{1}] … [b_{t}].  □

Definition 4.4.6. We write A ∼ B to denote that n×n matrices A and B are conjugate; namely if there exists an invertible matrix S such that S^{−1}AS = B.

Lemma 4.4.7. For nilpotent n × n matrices A and B, we have A ⪰ B and B ⪰ A if and only if A ∼ B.

Proof. This follows from the fact that A ∼ B if and only if A and B have the same Jordan canonical form. □

The following proposition is due to Gerstenhaber [131, Prop. 1.6]. See also Gerstenhaber [133], Hesselink [153], O’Halloran [201].

Proposition 4.4.8. If A and B are nilpotent n × n matrices then the following are equivalent:

(1) A ⪰ B.

(2) The GL_{n}(k)-orbit of B is in the closure of the GL_{n}(k)-orbit of A.

Proof. Since rank (A) ≤ s is a closed condition (see §3.2), (2) imples (1). To prove the converse, it suffices by the observation of Example 4.4.2 to prove that if a_{1} ≥ a_{2} > 0 then J_{[a_{1}] [a_{2}]} is in the closure of the orbit of J_{[a_{1}+1] [a_{2}−1]}. To see this, write down a one parameter family of matrices of the form

(\begin{matrix}J_{a_{1}} & 0 \\ U & J_{a_{2}} \end{matrix})

where

U = (\begin{matrix}0 & … & 0 \\ ⋮ & & ⋮ \\ 0 & … & t \end{matrix})

is a matrix with t in the bottom right hand corner and all other entries equal to zero. When t = 0 this is J_{[a_{1}] [a_{2}]}, while if t ≠ 0, computing ranks of powers shows that it is conjugate to J_{[a_{1}+1] [a_{2}−1]}. □

Example 4.4.9. Consider the one parameter family of matrices

(\begin{matrix}0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & t & 1 & 0 \end{matrix}) .

When t = 0 this has Jordan type [2]^{2}, whereas when t ≠ 0 it has Jordan type [3] [1]. This shows that J_{[2]^{2}} is in the closure of the orbit of J_{[3] [1]}.

Theorem 4.4.10. Let A, B ∈ Mat_{n}(k) be matrices spanning a nilvariety. Then the following are equivalent:

(1) For all λ ∈ k, A ⪰ A + λB.

(2) For all extension fields K of k and for all t ∈ K, A ⪰ A + tB.

(3) If t is transcendental over k then A ∼ A + tB as elements of Mat_{n}(k(t)).

(4) There exists a finite sequence of matrices W_{1}, W_{2}, …, W_{s} such that the following equations hold:

W_{1}A − AW_{1} = B, W_{i}A − AW_{i} = BW_{i−1} for i = 2, …, s, and BW_{s} = 0.

(5) There exists an infinite sequence of matrices W_{1}, W_{2}, … such that the following equations hold:

W_{1}A − AW_{1} = B, W_{i}A − AW_{i} = BW_{i−1} for i ≥ 2.

Proof. This follows from Theorem 4.3.1 and Corollary 4.1.8. □

Definition 4.4.11. Let A, B ∈ Mat_{n}(k) span a nilvariety. We say that A is maximal with respect to B if the equivalent conditions of the theorem above are satisfied.

Theorem 4.4.12. Let A, B_{1}, …, B_{r} ∈ Mat_{n}(k) be matrices spanning a nilvariety. Then the following are equivalent:

(1) For all λ_{1}, …, λ_{r} ∈ k, A ⪰ A + λ_{1}B_{1} + ⋯ + λ_{r}B_{r}.

(2) For all extension fields K of k and for all t_{1}, …, t_{r} ∈ K, A ⪰ A + t_{1}B_{1} + ⋯ + t_{r}B_{r}.

(3) If t_{1}, …, t_{r} are independent transcendentals over k then A ∼ A + t_{1}B_{1} + ⋯ + t_{r}B_{r} as elements of Mat_{n}(k(t_{1}, …, t_{r})).

(4) There exist matrices W_{i_{1},...,i_{r}}, one for each sequence (i_{1}, …, i_{r}) of integers, such that

W_{i_{1},...,i_{r}}A − AW_{i_{1},...,i_{r}} = B_{1}W_{i_{1}−1,...,i_{r}} + ⋯ + B_{r}W_{i_{1},...,i_{r}−1},

and furthermore, W_{i_{1},...,i_{r}} = 0 if and only of the indices are negative, W_{0,...,0} = I, and:

W_{i_{1},...,i_{r}} = 0 for all but finitely many values of (i_{1}, …, i_{r}).

(5) There exist matrices W_{i_{1},...,i_{r}} as in (4) but without the last finiteness condition.

Proof. This follows from Theorem 4.3.3 and Corollary 4.1.8. □

Definition 4.4.13. Let A, B_{1}, …, B_{r} ∈ Mat_{n}(k) span a nilvariety. We say that A is maximal with respect to B_{1}, …, B_{r} if the equivalent conditions of the theorem above are satisfied.

Remark 4.4.14. Just as in Remark 4.1.10, it follows from Theorem 4.4.12 that if A is maximal with respect to B_{1}, …, B_{r} and λ_{1}, …, λ_{r} are such that

A ∼ A + λ_{1}B_{1} + ⋯ + λ_{r}B_{r}

then A + λ_{1}B_{1} + ⋯ + λ_{1}B_{r} is maximal with respect to A, B_{1}, …, B_{r}.

Theorem 4.4.15. Let A, B_{1}, …, B_{r} ∈ Mat_{n}(k) span a nilvariety, and let C_{1}, …, C_{r} ∈ Mat_{n}(k) be matrices commuting with each other and with A, B_{1}, …, B_{r}. If A is maximal with respect to B_{1}, …, B_{r} then it is also maximal with respect to B_{1}, …, B_{r}, B_{1}C_{1}, …, B_{r}C_{r}.

Proof. Look at the equation

(4.4.1) W̃ (t_{1}, …, t_{r}, s_{1}, …, s_{r})A =

(A + t_{1}B_{1} + ⋯ + t_{r}B_{r} + s_{1}B_{1}C_{1} + ⋯ + s_{r}B_{r}C_{r})W̃ (t_{1}, …, t_{r}, s_{1}, …, s_{r}).

Setting

W̃ (t_{1}, …, t_{r}, s_{1}, …, s_{r}) = ∑ W̃_{i_{1},...,i_{r},j_{1},...,j_{r}}t_{1}^{i_{1}} … t_{r}^{i_{r}}s_{1}^{j_{1}} … s_{r}^{j_{r}},

we see by comparing coefficients that we can satisfy this equation by setting

W̃_{i_{1},...,i_{r},j_{1},...,j_{r}} = C_{1}^{j_{1}} … C_{r}^{j_{r}}W_{i_{1}+j_{1},...,i_{r}+j_{r}}.

Namely, if W (t_{1}, …, t_{r}) satisfies Equation (4.3.3) then W̃ (t_{1}, …, t_{r}, s_{1}, …, s_{r}) satisfies Equation (4.4.1). □

Remark 4.4.16. A different, inductive proof of the above theorem is given in [128].

The crucial theorem of Friedlander, Pevtsova and Suslin [128, Theorem 1.9], which appears below as Theorem 4.4.18, is a strengthening of Theorem 4.4.15 in the case where C is nilpotent. We begin with the single variable case.

Proposition 4.4.17. Suppose that A, B ∈ Mat_{n}(k) span a nilvariety and C ∈ Mat_{n}(k) is a nilpotent matrix commuting with A and B.

(1) If A is rank maximal with respect to B then rank (A) = rank (A + BC).

(2) If A is maximal with respect to B then A ∼ A + BC.

Proof. (1) Let W = ∑_{μ∈k^{×}} Ker (A + μB), the span of the kernels of A + μB as B runs over the non-zero elements of k. On each Ker (A + μB), and hence on W, the matrices A and B have the same image. So for all λ ∈ k we have (A + λB)(W) ⊆ A(W), with equality for all but finitely many values of λ ∈ k.

Since A is rank maximal with respect to B, we have

dim Ker (A + λB) = dim Ker A

for all but finitely many values of λ ∈ k.

Combining these, for all but finitely many values of λ ∈ k^{×} we have

dim Ker A|_{W} = dim Ker (A + λB)|_{W} = dim Ker (A + λB) = dim Ker A,

where the second equality follows from the definition of W. It follows that Ker (A) ⊆ W.

Now C commutes with A and B, and hence C(Ker (A + μB)) ⊆ Ker (A + μB). On Ker (A + μB), A acts in the same way as −μB and so A + BC acts in the same way as A(I − μ^{−1}C). Since C is nilpotent, I − μ^{−1}C is invertible and so A and A + BC have the same image. It follows that A(W) = (A + BC)(W).

Next, we show that Ker (A + BC) ⊆ W. To do this, we prove by downward induction on i that if v ∈ Ker (A + BC) then C^{i}v ∈ W, this being trivially true for large i because C is nilpotent: if C^{i+1}v ∈ W then AC^{i}v = −BC^{i+1}v ∈ B(W) = A(W). So there exists w ∈ W such that AC^{i}v = Aw. Then C^{i}v = (C^{i}v − w) + w ∈ Ker (A) + W = W.

We now have Ker (A) ⊆ W, Ker (A + BC) ⊆ W, and A(W) = (A + BC)(W). So Ker (A) and Ker (A + BC) have the same codimension in W and hence also in k^{n}, and so rank (A) = rank (A + BC).

(2) This follows by applying (1) to the powers of A and of A + BC. □

Theorem 4.4.18 (Friedlander, Pevtsova and Suslin [128]). Suppose that A, B_{1}, …, B_{r} span a nilvariety and C_{1}, …, C_{r} ∈ Mat_{r}(k) are nilpotent matrices commuting with each other and with A, B_{1}, …, B_{r}. If A is maximal with respect to B_{1}, …, B_{r} then

A ∼ A + B_{1}C_{1} + ⋯ + B_{r}C_{r}.

Proof. By Theorem 4.4.15, A is maximal with respect to B_{1}, …, B_{r}, B_{1}C_{1}, …, B_{r}C_{r}. By Proposition 4.4.17 we have A ∼ A + B_{1}C_{1} and so by Remark 4.4.14, it follows that A + B_{1}C_{1} is also maximal with respect to B_{1}, …, B_{r}, B_{1}C_{1}, …, B_{r}C_{r}. Continuing by induction, we obtain

A ∼ A + B_{1}C_{1} ∼ A + B_{1}C_{1} + B_{2}C_{2} ∼ ⋯ ∼ A + B_{1}C_{1} + ⋯ + B_{r}C_{r}.  □

4.5. Generic and Maximal Jordan Type

We now return to the representation theory of an elementary abelian p-group

E = ⟨g_{1}, …, g_{r}⟩ ≅ (ℤ/p)^{r}

or an exterior algebra Λ = Λ(X_{1}, …, X_{r}). Our notation is the same as in Section 1.1. In particular, in the case of an elementary abelian group we set X_{i} = g_{i} − 1 ∈ J(kE). In either case, if

α = (λ_{1}, …, λ_{r}) ∈ 𝔸^{r}(k),

we write X_{α} = ∑_{i} λ_{i}X_{i} for the corresponding element of J = J(kE) or J(Λ), g_{α} = 1 + X_{α}, and if α ≠ 0, we write E_{α} = ⟨g_{α}⟩ for the corresponding cyclic shifted subgroup. We also recall the convention that when we talk of a kE-module or a Λ-module, we mean a finitely generated module.

Let M be a kE-module or a Λ-module. Then for each x ∈ J we can look at the Jordan type of the nilpotent matrix x_{M} representing the action of x on M with respect to some given basis. In general, if x, x′ ∈ J with x − x′ ∈ J^{2} then it can happen that x_{M} and x′_{M} have different Jordan types. However, we’ll show that generically this does not happen.

Example 4.5.1. Let p = 2 and r = 3, and let M be the four dimensional kE-module given by

g_{1} ↦ (\begin{matrix}1 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 1 & 1 \end{matrix})  g_{2} ↦ (\begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \end{matrix})  g_{3} ↦ (\begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 1 \end{matrix}) .

Then X_{3} has Jordan type [2] [1]^{2} while X_{3} + X_{1}X_{2} has Jordan type [1]^{4}. Note that X_{3} ≡ X_{3} + X_{1}X_{2} (mod J^{2}(kE)).

Definition 4.5.2. We say that x ∈ J has maximal Jordan type on M if the Jordan type of x_{M} is maximal in the dominance order (Definition 4.4.4) among the matrices of elements of J.

Theorem 4.5.3 (Friedlander, Pevtsova and Suslin). Let M be a kE-module or a Λmodule, on which J does not act as zero.

(1) The elements of J^{2} are not of maximal Jordan type on M.

(2) All elements of maximal Jordan type on M have the same Jordan type, and this is the same as the Jordan type of the generic element of J on M.

(3) The set of elements of J of maximal Jordan type on M is a union of cosets of J^{2} in J, and is the inverse image in J of a homogeneous dense open subset of (J/J^{2})\{0}.

Proof. Choose 0 ≠ β ∈ 𝔸^{r}(k) such that the Jordan type of (X_{β})_{M} is maximal among all (X_{α})_{M}. Then (X_{β})_{M} is maximal with respect to (X_{1})_{M}, …, (X_{r})_{M} (Definition 4.4.11). It follows from Corollary 4.1.8 that there is a dense open subset U of 𝔸^{r}(k) such that the Jordan type of (X_{α})_{M} for α ∈ U is the same as the Jordan type of (X_{β})_{M}. Since J does not act as zero on M, this set U does not contain the origin. Furthermore, there is a unique Jordan type for which this happens, because any two dense open subsets of 𝔸^{r}(k) have non-empty intersection.

Any element y ∈ J^{2} can be written in the form y = X_{1}C_{1}+⋯+X_{r}C_{r} with C_{1}, …, C_{r} ∈ J. It follows from Theorem 4.4.15 that (X_{β})_{M} is maximal with respect to

(X_{1})_{M}, …, (X_{r})_{M}, (X_{1}C_{1})_{M}, …, (X_{r}C_{r})_{M}

so that for all α ∈ 𝔸^{r}(k) we have (X_{β})_{M} ⪰ (X_{α} + y)_{M}. Since every element of J can be written as X_{α} + y for suitable α and y, it follows that X_{β} has maximal Jordan type on M.

It follows from Theorem 4.4.18 that (X_{β})_{M} ∼ (X_{β} +y)_{M}. On the other hand, if (X_{β})_{M} ∼ (X_{α} +y)_{M} then (X_{α} +y)_{M} is maximal with respect to (X_{1})_{M}, …, (X_{r})_{M}. Applying the same argument, we find that (X_{α} + y)_{M} ∼ (X_{α})_{M} and so α ∈ U. In particular it follows that α ≠ 0. Thus we have shown that the elements of maximal Jordan type form the inverse image in J of a dense open subset of (J/J^{2})\{0} and all have the same Jordan type. Finally, it follows from Theorem 4.4.12 with A = (X_{β})_{M} that this Jordan type is the same as that of the generic element of J. □

Definition 4.5.4. The generic Jordan type of a kE-module or Λ-module M is defined to be the Jordan type of the generic element of J on M. Theorem 4.5.3 shows that this is the same as the that of the elements of maximal Jordan type on M, and these elements are given by the inverse image in J of a dense open subset

U_{max}(M) ⊆ (J/J^{2}) \ {0}.

We refer to this set U_{max}(M) as the maximal Jordan set of M, and we regard it as a subset of 𝔸^{r}(k). It is the complement of a closed homogeneous subset of affine space, so it defines an open subvariety Ū_{max}(M) ⊆ ℙ^{r−1}(k).

Remark 4.5.5. We can similarly talk about the generic rank of a kE-module or Λ-module M, which is the rank of the generic element of J on M. This is equal to the maximum rank of an element of J on M, and also equal to the maximal rank of X_{α} on M for α ∈ 𝔸^{r}(k). We say that such an element of J has maximal rank on M.

The following theorem shows that U_{max}(M) is well behaved under field extensions.

Theorem 4.5.6. Let K be a field containing k. If M is a kE-module or a Λ-module and

α = (λ_{1}, …, λ_{r}) ∈ 𝔸^{r}(K)

then the Jordan canonical form of X_{α} ∈ J(KE) or J(K ⊗_{k} Λ) on K ⊗_{k} M is equal to the generic Jordan type of M if and only if there exists f ∈ k[Y_{1}, …, Y_{r}] such that f(α) = 0 but f(β) = 0 for all β ∈ U_{max}(M). Otherwise its Jordan canonical form is dominated by the generic Jordan type of M.

Proof. This follows from Lemma 3.2.1 and the fact that Jordan canonical form of X_{α} is determined by the vanishing of polynomials given by minors of the action on M. □

Corollary 4.5.7. Let M be a kE-module or a Λ-module and K be an algebraically closed extension field of k. Then the generic Jordan types of M and K ⊗_{k} M are the same.

Let I be the ideal of k[Y_{1}, …, Y_{r}] of functions vanishing on 𝔸^{r}(k)\U_{max}(M). Then K ⊗_{k} I is the ideal of K[Y_{1}, …, Y_{r}] of functions vanishing on 𝔸^{r}(K) \ U_{max}(K ⊗_{k} M). □

Proposition 4.5.8. The generic Jordan type of M ⊕ N is the direct sum of the generic Jordan types of M and N. Furthermore,

U_{max}(M ⊕ N) = U_{max}(M) ∩ U_{max}(N).

Proof. This is an easy consequence of the definition and Theorem 4.5.3. □

Proposition 4.5.9. If the generic Jordan type of M is

[p]^{n_{p}} [p − 1]^{n_{p−1}} … [i]^{n_{i}} … [1]^{n_{1}}

then that of Ω(M) is

[p]^{m_{p}} [p − 1]^{n_{1}} … [p − i]^{n_{i}} … [1]^{n_{p−1}}

for some value of m_{p}. Furthermore, we have U_{max}(Ω(M)) = U_{max}(M).

Proof. This follows easily from the fact that Ω and restriction to a cyclic shifted subgroup commute up to projective summands. □

Corollary 4.5.10. The generic Jordan type of Ω^{2}(M) is the same as that of M apart from possibly the number of Jordan blocks of length p. □

Remark 4.5.11. We shall see in Section 8.5 that if M has constant Jordan type (see Section 5.1) then the modules Ω^{2}(M) and M are distinguished by the vector bundles on projective space associated to them.

Theorem 4.5.12. Let n < p. For a kE-module M, the following are equivalent:

(1) The generic Jordan type of M has all Jordan blocks of length at most n.

(2) For all α ∈ 𝔸^{r}(k), the action of X_{α} on M has all Jordan blocks of length at most n.

(3) For all extension fields K of k and all α ∈ 𝔸^{r}(K), the action of X_{α} on K ⊗_{k} M has all Jordan blocks of length at most n.

(4) Rad^{n}(M) = 0.

Proof. This follows from Theorem 1.17.1 and Theorem 4.5.6. □

4.6. Tensor Products

The tensor product of modules for a cyclic group of order p were discussed in Section 2.3. In particular, it was shown there that the isomorphism type of the tensor product does not depend on whether we use the usual group theoretic tensor product M ⊗_{k} N or the restricted Lie tensor product M ˜⊗_{k} N.

Theorem 4.6.1 (Friedlander, Pevtsova and Suslin). If M and N are two kE-modules then

(1) The generic Jordan type of M ⊗_{k} N is the tensor product of the generic Jordan types of M and N.

(2) U_{max}(M ⊗_{k} N) ⊇ U_{max}(M) ∩ U_{max}(N). The same statements hold for the restricted Lie tensor product M ˜⊗_{k} N.

Proof. We begin by proving the theorem for M ˜⊗_{k} N. Let α ∈ U_{max}(M) ∩ U_{max}(N). We have

(M ˜⊗_{k} N)↓_{E_{α}} ≅ M↓_{E_{α}} ˜⊗_{k} N↓_{E_{α}}

and so the isomorphism type of this restriction does not depend on which such α we use. It follows from Theorem 4.5.3 that this dense open subset is contained in U_{max}(M ˜⊗_{k} N), and that the generic Jordan type of M ˜⊗_{k} N is the tensor product of those of M and N.

To prove the theorem for M ⊗_{k} N, we argue as follows. Let α ∈ U_{max}(M) ∩ U_{max}(N). Then (X_{α})_{M} is maximal with respect to (X_{1})_{M}, …, (X_{r})_{M}, and so by Theorem 4.4.12 if t_{1}, …, t_{r} are independent transcendentals we have

(4.6.1)  (X_{α})_{M} ∼ (X_{α})_{M} + t_{1}(X_{1})_{M} + ⋯ + t_{r}(X_{r})_{M}.

Similarly, if t′_{1}, …, t′_{r} are further independent transcendentals we have

(4.6.2)  (X_{α})_{N} ∼ (X_{α})_{N} + t′_{1}(X_{1})_{N} + ⋯ + t′_{r}(X_{r})_{N}.

So we have

(X_{α})_{M ˜⊗_{k}N} = (X_{α})_{M} ⊗ I_{N} + I_{M} ⊗ (X_{α})_{N}

∼ (X_{α})_{M} ⊗ I_{N} + I_{M} ⊗ (X_{α})_{N} (4.6.3)

+ t_{1}(X_{1})_{M} ⊗ I_{N} + ⋯ + t_{r}(X_{r})_{M} ⊗ I_{N}
+ I_{M} ⊗ t′_{1}(X_{1})_{N} + ⋯ + I_{M} ⊗ t′_{r}(X_{r})_{N}.

Indeed, if a matrix A effects the conjugation (4.6.1) and B effects (4.6.2) then A ⊗ B effects the conjugation (4.6.3).

Now if X_{α} = ∑_{i} λ_{i}X_{i} then (X_{α})_{M⊗_{k}N} is equal to (X_{α})_{M ˜⊗_{k}N} + ∑_{i} λ_{i}(X_{i})_{M} ⊗ (X_{i})_{N} plus a sum of terms of the form Y_{M} ⊗ Z_{N} where Y and Z are in J(kE). Such a term can be written as (Y_{M} ⊗I_{N})(I_{M} ⊗Z_{N}), and is therefore a sum of products of nilpotent matrices with matrices appearing in (4.6.3). All the matrices commute, so by Theorem 4.4.15 we have

(X_{α})_{M ˜⊗_{k}N} ∼ (X_{α})_{M⊗_{k}N}.

The version of the theorem for the group theoretic tensor product now follows. □

Corollary 4.6.2. If λ is a partition of d with d < p and S^{λ} is the Schur functor corresponding to λ (see Section 1.16) then the generic Jordan type of S^{λ}(M) is S^{λ} of the generic Jordan type of M, and U_{max}(S^{λ}(M)) ⊇ U_{max}(M).

Proof. It follows from the discussion in Section 1.16 that if d < p then S^{λ}(M) is a direct summand of M^{⊗d}. It follows from Theorem 4.6.1 that M^{⊗d} has generic Jordan type equal to the tensor dth power of the generic Jordan type of M, with U_{max}(M^{⊗d}) ⊇ U_{max}(M). The corollary now follows by applying Proposition 4.5.8. □

If d ≥ p the statement in the corollary is false. The following examples are the results of discussions with Kay Jin Lim.

Let P be the regular representation of kE, of dimension p^{r}. Then looking at the permutation basis e_{1}, …, e_{p^{r}} for kE, we see that the exterior pth power Λ^{p}(P) is a direct sum in which most of the summands are free. The non-free summands correspond to wedge products e_{i_{1}} ∧⋯∧e_{i_{p}} where the corresponding group elements form a coset of a cyclic subgroup. These summands are isomorphic to the permutation modules on cyclic subgroups. In particular, if r > 1 then Λ^{p}(P) is generically free. So its generic Jordan type is [p]^{\frac{1}{p}}^{\binom{p^{r}}{p}}. On the other hand, P has generic Jordan type [p]^{p^{r−1}}, and applying Λ^{p} to this gives [p]^{\frac{1}{p} \binom{p^{r}}{p}−p^{r−2}}[1]^{p^{r−1}}.

Now look at the exact sequence

0 → Ω(k) → P → k → 0.

Taking the symmetric pth power of the chain complex P → k (in degrees one and zero), we obtain a complex whose homology is Λ^{p}(Ω(k)) in degree p. This gives us an exact sequence

0 → Λ^{p}(Ω(k)) → Λ^{p}(P) → Λ^{p−1}(P) → ⋯ → P → k → 0.

Since Λ^{p−1}(P), …, P are projective, this gives us a short exact sequence

0 → Λ^{p}(Ω(k)) → Λ^{p}(P) → Ω^{p−1}(k) → 0.

The generic Jordan type of Ω^{p−1}(k) is of the form [p]^{?}[1], while the generic Jordan type of Λ^{p}(P) is of the form [p]^{?}. It follows that the generic Jordan type of Λ^{p}(Ω(k)) is of the form [p]^{?}[p − 1] while Λ^{p} of the generic Jordan type of Ω(k) is of the form [p]^{?}[1]^{p^{r−1}−1}. The set U_{max}(Λ^{p}(P)) is the complement of the subvariety corresponding to the union of the (p^{r} − 1)/(p − 1) cyclic subgroups of E.

For an explicit example, let p = 3 and E = (ℤ/3)^{2}. Then the generic Jordan type of Ω(k) is [3]^{2}[2], and the generic Jordan type of Λ^{3}(Ω(k)) is [3]^{18}[2], whereas Λ^{3} of the generic Jordan type of Ω(k) is [3]^{18}[1]^{2}. The set U_{max}(Λ^{3}(Ω(k))) is the complement of the subvariety corresponding to the union of the four cyclic subgroups of E.

5 Modules of Constant Jordan Type

5.1. Introduction and Definitions

Modules of constant Jordan type were introduced by Carlson, Friedlander and Pevtsova [81]. The definition is as follows.

Definition 5.1.1. We say that a finite dimensional kE-module M has constant Jordan type [a_{1}] … [a_{t}] if the restriction of M to every cyclic shifted subgroup has Jordan canonical form given by blocks of length a_{1}, …, a_{t}. The notation here is the same as in Section 4.4.

Remark 5.1.2. This definition appears to depend on a choice of basis for the elementary abelian group E. In fact the notion is independent of choice of basis, as we shall see in Lemma 5.1.7.

Definition 5.1.3. Similarly, we say that a nilvariety of rank r spanned by matrices A_{1}, …, A_{r} has constant Jordan type [a_{1}] … [a_{t}] if for all λ_{1}, …, λ_{r} ∈ k, not all zero, the matrix

λ_{1}A_{1} + ⋯ + λ_{r}A_{r}

has Jordan canonical form given by blocks of length a_{1}, …, a_{t}.

Example 5.1.4. Suppose that A_{1}, …, A_{r} span a linear variety of m × n matrices of constant rank s. Then the (n + m) × (n + m) matrices

B_{i} = (\begin{matrix}0 & 0 \\ A_{i} & 0 \end{matrix}) , 1 ≤ i ≤ r

commute, and every linear combination squares to zero. The kE-module M given by

X_{i} ↦ B_{i}, g_{i} ↦ I + B_{i} (1 ≤ i ≤ r)

satisfies Rad^{2}(M) = 0, and has constant Jordan type [2]^{s}[1]^{n+m−2s}.

Examples 4.1.6 and 4.1.7 give two dimensional spaces of matrices of constant rank. Here are some larger dimensional examples. Consider the space spanned by the 5 × 5 matrices

Y_{1}A_{1} + Y_{2}A_{2} + Y_{3}A_{3} = ( \begin{matrix}0 & 0 & 0 & Y_{1} & Y_{2} \\ 0 & 0 & Y_{1} & Y_{2} & Y_{3} \\ 0 & −Y_{1} & 0 & Y_{3} & 0 \\ −Y_{1} & −Y_{2} & −Y_{3} & 0 & 0 \\ −Y_{2} & −Y_{3} & 0 & 0 & 0 \end{matrix} ) .

It is easy to check that every non-zero linear combination of A_{1}, A_{2} and A_{3} has rank four. This gives a k(ℤ/p)^{3}-module of dimension 10 and constant Jordan type [2]^{4}[1]^{2}. The diagram for this module is

[figure: •; •; •; •; •; •; •; •; •; •]

In this diagram, we have to keep in mind that the edges on the left hand side have a minus sign associated with them.

A larger example (Westwick [248]) is given by the 10 × 10 matrices of constant rank 8

Y_{1}A_{1} + Y_{2}A_{2} + Y_{3}A_{3} + Y_{4}A_{4} = (\begin{matrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & Y_{1} & Y_{2} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & Y_{1} & Y_{2} & 0 & Y_{3} \\ 0 & 0 & 0 & 0 & 0 & −Y_{1} & Y_{2} & 0 & Y_{3} & Y_{4} \\ 0 & 0 & 0 & 0 & Y_{1} & Y_{2} & 0 & Y_{3} & Y_{4} & 0 \\ 0 & 0 & 0 & −Y_{1} & 0 & 0 & Y_{3} & −Y_{4} & 0 & 0 \\ 0 & 0 & Y_{1} & −Y_{2} & 0 & 0 & Y_{4} & 0 & 0 & 0 \\ 0 & −Y_{1} & −Y_{2} & 0 & −Y_{2} & −Y_{4} & 0 & 0 & 0 & 0 \\ −Y_{1} & −Y_{2} & 0 & −Y_{3} & Y_{4} & 0 & 0 & 0 & 0 & 0 \\ −Y_{2} & 0 & −Y_{3} & −Y_{4} & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & −Y_{3} & −Y_{4} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{matrix} ) ,

giving a k(ℤ/p)^{4}-module of dimension 20 and constant Jordan type [2]^{8}[1]^{4}. The diagram for this module, with the same caveat about minus signs, and with the direction of an edge indicating which X_{i} is involved, is

[figure: •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •]

Further examples of modules of constant Jordan type will appear throughout this chapter as well as Chapter 9.

Remark 5.1.5. Papers on modules of constant Jordan type include Baland [25, 26], Benson [41, 42, 44, 43], Benson and Pevtsova [53], Carlson and Friedlander [80], Carlson, Friedlander and Pevtsova [81], Carlson, Friedlander and Suslin [82], Friedlander and Pevtsova [127].

Papers on linear varieties of matrices of constant rank and on nilvarieties of matrices of constant rank include Atkinson and Westwick [19], Beasley [37], Boralevi, Faenzi and Mezzetti [60], Fania and Mezzetti [117], Gerstenhaber [130, 131, 132, 133, 134], Ilic and Landsberg [166], Manivel and Mezzetti [191], Sylvester [231], Westwick [245, 246, 247, 248].

In Section 10.1 we use the theory of Chern numbers to prove theorems of Sylvester and Westwick about matrices of constant rank, and apply this to modules of constant Jordan type.

The following proposition shows that it is not always obvious whether or not a module has constant Jordan type.

Proposition 5.1.6. Let E = (ℤ/p)^{2} with p ≥ 3, and let M_{n} (n ≥ 2) be the module defined by the diagram

[figure: •^{u_{1}}; •^{u_{2}}; •^{u_{3}}; …; •^{u_{n−1}}; v_{1} •; •; •; •; …; •; • v_{n}; •_{w_{1}}; •_{w_{2}}; •_{w_{3}}; …; •_{w_{n−1}}]

More explicitly, M_{n} has basis elements u_{1}, …, u_{n−1}, v_{1}, …, v_{n}, w_{1}, …, w_{n−1} with

X_{1}(u_{i}) = v_{i}  X_{2}(u_{i}) = v_{i+1}  X_{1}(v_{i+1}) = X_{2}(v_{i}) = w_{i}  (1 ≤ i ≤ n − 1)

and all other basis elements sent to zero by X_{1} and X_{2}. Then M_{n} has constant Jordan type if and only if n is divisible by p. The Jordan type is [3]^{n−2}[2]^{2}.

Proof. Let 0 ≠ (λ_{1}, λ_{2}) ∈ 𝔸^{2}(k). The rank of λ_{1}X_{1} + λ_{2}X_{2} is equal to 2n − 2. The rank of (λ_{1}X_{1} + λ_{2}X_{2})^{2} is equal to n − 1 if either λ_{1} or λ_{2} is zero. If they are both non-zero, the rank is either n or n − 1. The latter happens if and only if some linear combination of the u_{i} is in the kernel. This happens if and only if n is divisible by p, and then the kernel is the linear span of the element

λ_{1}^{n−2}u_{1} − 2λ_{1}^{n−3}λ_{2}u_{2} + 3λ_{1}^{n−4}λ_{2}^{2}u_{3} − ⋯ − (n − 1)λ_{2}^{n−2}u_{n−1}.  □

Lemma 5.1.7. The following are equivalent.

(1) M has constant Jordan type.

(2) The maximal Jordan set U_{max}(M) is equal to 𝔸^{r}(k) \ {0}.

(3) The restriction of M to every element of J(kE) \ J^{2}(kE) has the same Jordan canonical form.

Proof. This follows immediately from the definitions together with Theorem 4.5.3. □

Theorem 5.1.8. Let M be a kE-module of constant Jordan type and let K be an algebraically closed field containing k. Then M ⊗_{k} K is a KE-module of the same constant Jordan type.

Proof. This follows immediately from Corollary 4.5.7. □

Theorem 5.1.9. Let M and N be kE-modules.

(1) M and N both have constant Jordan type if and only if M ⊕ N has constant Jordan type.

(2) If M and N have constant Jordan type then so does M ⊗_{k} N.

(3) If M has constant Jordan type and λ is a partition of d with d < p then S^{λ}(M) (where S^{λ} is the Schur functor corresponding to λ as in Section 1.16) has constant Jordan type.

The Jordan type of the sum, tensor product and S^{λ}(M) is obtained by applying the same operation to the Jordan type of M (and N).

Proof. (1) It is obvious that if M and N have constant Jordan type then so does M ⊕N. The converse follows from Proposition 4.5.8.

(2) This follows from Theorem 4.6.1.

(3) This follows from Corollary 4.6.2. □

Warning 5.1.10. It follows from part (3) of the theorem that if M has constant Jordan type then the exterior powers Λ^{d}(M) and the symmetric powers S^{d}(M) have constant Jordan type when d < p. This does NOT remain true for d ≥ p. The examples at the end of Section 4.6 show that if a module M has constant Jordan type then Λ^{p}(M) does not necessarily have constant Jordan type.

Definition 5.1.11 (Carlson, Friedlander and Pevtsova [81].). The stable constant Jordan type of M is obtained by omitting the Jordan blocks of length p. So M has stable constant Jordan type [a_{1}] … [a_{t}] if the restriction of M to every cyclic shifted subgroup has Jordan canonical form given by blocks of length a_{1}, …, a_{t} together with some number (possibly zero) of blocks of length p. In the case where t = 0, we write ∅ for the empty partition, to denote this stable constant Jordan type.

Proposition 5.1.12. If M has constant Jordan type then so do Ω(M) and Ω^{−1}(M). If the stable Jordan type of M is [a_{1}] … [a_{t}] then the stable Jordan type of Ω(M) and Ω^{−1}(M) are both [p − a_{1}] … [p − a_{t}].

Proof. This follows from the fact that for 0 ≠ α ∈ 𝔸^{r}(k) the module Ω(M)↓_{X_{α}} is stably isomorphic to Ω(M↓_{X_{α}}) and Ω^{−1}(M)↓_{X_{α}} is stably isomorphic to Ω^{−1}(M↓_{X_{α}}). □

Lemma 5.1.13. Given modules of constant Jordan type, the stable constant Jordan type of a direct sum or tensor product is the same operation on the stable constant Jordan types.

The same is true for symmetric and exterior dth powers and more generally for Schur functors S^{λ} with λ a partition of d, provided d < p.

Proof. The first part follows from Theorem 5.1.9 together with the observation that the tensor product of a Jordan block of length p with anything is a sum of Jordan blocks of length p. The second part then follows from Corollary 4.6.2. □

In the case where all Jordan blocks have length p, in other words the stable Jordan type is the empty partition ∅, we have the following.

Proposition 5.1.14. If M has stable constant Jordan type ∅ then M is projective.

Proof. This is a restatement of Dade’s Lemma 1.9.5. □

Corollary 5.1.15. If M has stable constant Jordan type ∅ then the dimension of M is divisible by p^{r}.

Remark 5.1.16. We shall see in Section 10.6 that for p = 2, a nilvariety of rank r of stable constant Jordan type ∅ has to have dimension divisible by 2^{r}. Example 4.1.6 shows that the same is not true if p is odd.

At the other extreme, when there are no Jordan blocks of length p, we have the following.

Proposition 5.1.17. If M has constant Jordan type with all Jordan blocks of length at most n, with n < p, then Rad^{n}(M) = 0.

Proof. This follows from Theorem 4.5.12. □

Theorem 5.1.18. Suppose that r ≥ 2. If M has constant Jordan type with all blocks of the same length then either M is projective or E acts trivially on M.

Proof. If all the blocks have length p then we can apply Dade’s Lemma 1.9.5 to deduce that M is projective. So we need to show that if all blocks are of equal length less than p then E acts trivially on M.

Let n be the lengths of the Jordan blocks, with n < p. Then by Theorem 4.5.12 we have Rad^{n}(M) = 0. If n ≥ 2, it follows that all non-zero linear combinations of X_{1} and X_{2} induce an isomorphism from M/Rad(M) to Rad(M)/Rad^{2}(M). Using X_{1} to identify these two spaces, it follows that X_{2} has no eigenvalues on this space. So M/Rad(M) = 0, and hence M = 0.

So we are left with the case n = 1, in which case E acts trivially on M. □

5.2. Homogeneous Modules

Our first examples of modules of constant Jordan type are the modules Ω^{n}(k), n ∈ ℤ. The reason why they are of constant Jordan type is that they are homogeneous. The Jordan type is [1] if n is even and [p − 1] if n is odd. Note that if p = 2 then [1] = [p − 1] and there is no dependence on the parity of n.

Definition 5.2.1. A kE-module M is said to be homogenous if given any maximal shifted subgroup E′ with choice of basis, the corresponding automorphism ρ: kE → kE (see Definition 1.3.1) satisfies ρ^{∗}(M) ≅ M. Recall that the group of all such automorphisms is isomorphic to GL(r, k).

We shall see in Chapter 8 that there are many modules of constant Jordan type that are not homogeneous. An example is also given later in this section.

Proposition 5.2.2. The class of homogeneous kE-modules is closed under direct sums and summands.

Proof. Closure under direct sums is clear, so we prove closure under direct summands. If M is homogeneous and M = M_{1} ⊕ ⋯ ⊕ M_{t} with the M_{i} indecomposable, then for ρ ∈ GL(r, k), ρ^{∗}(M) ≅ M and so by the Krull–Schmidt theorem, ρ^{∗}(M_{1}) is isomorphic to some M_{i}. Let H be the subgroup of GL(r, k) consisting of those ρ such that ρ^{∗}(M_{1}) ≅ M_{1}. Then for ρ, ρ′ in different left cosets of H in GL(r, k), the isomorphism type of ρ^{∗}(M_{1}) and (ρ′)^{∗}(M_{1}) are not the same. Therefore H is a closed subgroup of finite index in GL(r, k). Since GL(r, k) is connected, this implies that H = GL(r, k) and the proposition is proved. □

Proposition 5.2.3. If M is a homogeneous kE-module and j > 0 then Rad^{j}(M), M/Rad^{j}(M), Soc^{j}(M), M/Soc^{j}(M), Ω(M) and Ω^{−1}(M) are all homogeneous. □

Proposition 5.2.4. If M is homogeneous then M has constant Jordan type.

Proof. The automorphisms ρ in the definition of homogeneity are transitive on cyclic shifted subgroups. □

Corollary 5.2.5. The modules Ω^{n}(k) have constant Jordan type for n ∈ ℤ. □

It is easy to construct a large supply of homogeneous modules. For example we can start with any homogeneous module, such as k, and allow ourselves to apply the operations Ω^{n} for n ∈ ℤ, Rad^{i} and Soc^{i}, and more generally any quotient of one submodule of the form Rad^{i} or Soc^{i} by another contained in it.

Example 5.2.6. The modules kE/J^{2}(kE) and Soc^{2}(kE) are both homogeneous modules of constant Jordan type [2] [1]^{r−1}.

More generally, we have the following lemma, which will be used in the proof of Proposition 5.3.11.

Lemma 5.2.7. For j ≤ p, the modules kE/J^{j}(kE) and Soc^{j}(kE) are homogeneous modules of constant Jordan type

[j] [j − 1]^{r−1}[j − 2]^{\binom{r}{2}}[j − 3]^{\binom{r+1}{3}} … [1]^{\binom{r+j−3}{j−1}}.

Proof. This is easy to see using the explicit basis for kE/J^{j}(kE) consisting of monomials in X_{1}, …, X_{r} of degree at most j − 1 and looking at the action of X_{1}. Using the obvious Jordan decomposition, each Jordan block has a unique monomial not involving X_{1}. So the number of Jordan blocks of length j −i is equal to the number of monomials in X_{2}, …, X_{r} of degree i, namely \binom{r−2+i}{i}. The module Soc^{j}(kE) is dual, so it has the same Jordan type. □

Example 5.2.8. If p = 3 and E = (ℤ/3)^{2} then Ω(k) was described in Section 1.11. This has constant Jordan type [3]^{2}[2]. Taking the quotient by the socle gives us a module with the following diagram:

[figure: •; •; •; •; •; •; •]

This is a module of constant Jordan type [3] [2]^{2}.

As another example, Soc^{3}(Ω^{2}(k)) has the following diagram:

[figure: •; •; •; •; •; •; •; •; •]

This is a homogeneous module of constant Jordan type [3]^{2}[2] [1].

Be warned, however, that it is not easy to tell whether a module is homogeneous, or even whether it has constant Jordan type, just by looking at the diagram. See for example Proposition 5.1.6.

Example 5.2.9. Let p ≥ 3 and let M be the module with diagram

[figure: •; •; •; •; •; •; •]

Then M has constant Jordan type [3][2][1]^{2}. However, Rad(M) does not have constant Jordan type. It follows from Proposition 5.2.3 that M is not homogeneous. It also follows from this example and its dual that if M has constant Jordan type and j > 0 then Rad^{j}(M), M/Rad^{j}(M), Soc^{j}(M) and M/Soc^{j}(M) do not necessarily have constant Jordan type.

Remark 5.2.10. Another operation preserving homogeneity is as follows. Given a nonprojective indecomposable module M, there is an almost split sequence

0 → Ω^{2}M → M′ → M → 0,

unique up to isomorphism, and if M is homogeneous then so is M′. See for example Section 4.12 of [39]. Carlson, Friedlander and Pevtsova [81] observe furthermore that for r ≥ 2, the almost split sequence splits on every cyclic shifted subgroup. So if M has constant Jordan type then so does M′, and the Jordan type of M′ is just a direct sum of two copies of that of M.

Example 5.2.11. We end this section with an example of a module of constant Jordan type with a large group of automorphisms that is nonetheless not homogeneous. Let p = 2 and r = 4, so that

E = (ℤ/2)^{4} = ⟨g_{1}, …, g_{4}⟩.

Let M be the six dimensional module given by

g_{1} ↦ (\begin{matrix}1 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 & 1 \end{matrix}) g_{2} ↦ (\begin{matrix}1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 1 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 & 0 & 1 \end{matrix})

g_{3} ↦ (\begin{matrix}1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 1 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 1 & 0 & 0 & 1 \end{matrix}) g_{4} ↦ (\begin{matrix}1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 & 0 & 1 \end{matrix}) .

It is easy to check that M has constant Jordan type [2]^{2}[1]^{2}. The module is easier to visualise using the diagram

[figure: •; X_{1}; X_{2}; X_{3}; X_{4}; •; X_{4}; •; X_{3}; •; X_{2}; •; X_{1}; •]

This module is not homogeneous, because the elements ρ ∈ GL(4, k) satisfying ρ^{∗}(M) ≅ M have to fix a nondegenerate symplectic form in four variables corresponding to the matrix with ones down the trailing diagonal and zeros elsewhere. To see where this symplectic form comes from, note that M/Soc(M) ≅ kE/J^{2}(kE). Consider the diagram

[figure: 0; k; 0; Ω(M)/Soc^{2}Ω(M); kE/J^{3}(kE); M; 0; 0; J^{2}(kE)/J^{3}(kE); kE/J^{3}(kE); M/Soc(M); 0; k; 0]

in which the switchback map is an isomorphism. We have

J^{2}(kE)/J^{3}(kE) ≅ Λ^{2}(J(kE)/J^{2}(kE)) = Λ^{2}(𝔸^{r}(k)),

and so the map J^{2}(kE)/J^{3}(kE) → k gives us the symplectic form Λ^{2}(𝔸^{r}(k)) → k that has to be preserved by ρ.

Note that the module described in this way for p odd does not have constant Jordan type, because for example (X_{1} + X_{4})^{2} does not act as zero whereas X_{1}^{2} does.

This example will reappear in Section 9.6 in the context of the null correlation bundle on projective 3-space. and again in Section 10.5 concerning restrictions on possible Jordan types.

This example satisfies a weaker condition than homogeneity, which still has some of the consequences that are true for homogeneous modules.

Definition 5.2.12. A kE-module M is said to be n-homogeneous (1 ≤ n ≤ r) if given an elementary abelian group E′ of rank n and embeddings as shifted subgroups ϕ_{1}, ϕ_{2} : E′ → kE we have ϕ_{1}^{∗}(M) ≅ ϕ_{2}^{∗}(M).

For n = r, this definition is equivalent to the definition of homogeneity, while for n = 1 it is equivalent to contant Jordan type.

The proofs of the following properties are essentially the same as for homogeneous modules.

Proposition 5.2.13. The class of n-homogeneous kE-modules is closed under direct sums and summands. □

Proposition 5.2.14. If M is an n-homogeneous kE-module then Ω(M) and Ω^{−1}(M) are n-homogeneous. □

Proposition 5.2.15. If M is n-homogeneous with n ≥ 1 then M has constant Jordan type. □

Example 5.2.16. Example 5.2.11 is 1-homogeneous, but not 2-homogeneous.

Example 5.2.17. Let n ≤ r, and let M be the kE-module with generators m_{1}, …, m_{n} satisfying the following relations:

(i) For all i, j, k, X_{i}X_{j}m_{k} = 0; i.e., J^{2}(kE).M = 0.

(ii) X_{1}m_{1} + ⋯ + X_{n}m_{n} = 0. Then the restriction of M to a shifted subgroup E′ ≤ kE is isomorphic to the corresponding module for that subgroup plus some copies of kE′/J^{2}(kE′) and of the trivial module, if X_{1}, …, X_{n} ∈ E′, and to a direct sum of n copies of kE′/J^{2}(kE′) plus some copies of the trivial module otherwise.

It follows that M is (n − 1)-homogeneous but not n-homogeneous.

5.3. An Exact Category

Carlson and Friedlander [80] put a structure of exact category on the modules of constant Jordan type. In this section we describe this structure.

Lemma 5.3.1. Let A be a k-algebra. A short exact sequence of finite dimensional Amodules

0 → M_{1} → M_{2} → M_{3} → 0

splits if and only if M_{2} ≅ M_{1} ⊕ M_{3}.

Proof. One direction is obvious. For the other direction, suppose that M_{2} ≅ M_{1} ⊕ M_{3}. Consider the sequence

0 → Hom_{A}(M_{3}, M_{1}) → Hom_{A}(M_{3}, M_{2}) → Hom_{A}(M_{3}, M_{3}).

If M_{2} ≅ M_{1} ⊕ M_{3} then the dimension of the middle term is the sum of the dimensions of the outside terms, so the right-hand map is surjective. An inverse image in Hom_{A}(M_{3}, M_{2}) of the identity map in Hom_{A}(M_{3}, M_{3}) is a splitting for the sequence. □

Lemma 5.3.2. Let

0 → M_{1} → M_{2} → M_{3} → 0

be a short exact sequence of kE-modules of modules of constant Jordan type. Then the following are equivalent:

(1) The sequence splits on restriction to X_{α} for every closed point 0 ≠ α ∈ 𝔸^{r}.

(2) There exists a point 0 ≠ α of 𝔸^{r}(K) for some extension field K (i.e., a generic point for some irreducible subvariety other than {0}, cf. §3.2), such that the sequence splits on extending coefficients to K and restricting to X_{α}.

Proof. By Lemma 5.3.1, the sequence splits for 0 ≠ α ∈ 𝔸^{r} if and only if the Jordan type of X_{α} on M_{2} is the sum of the Jordan types on M_{1} and M_{3}. Since these are modules of constant Jordan type, this condition is independent of α. □

Definition 5.3.3. A short exact sequence

0 → M_{1} → M_{2} → M_{3} → 0

of kE-modules is locally split if for every 0 ≠ α ∈ 𝔸^{r} the restriction of the sequence to X_{α} splits.

Remark 5.3.4. Not every short exact sequence of modules of constant Jordan type is locally split. For example, if 0 ≤ i < j < ℓ ≤ p then the sequence

0 → J^{j}(kE)/J^{ℓ}(kE) → J^{i}(kE)/J^{ℓ}(kE) → J^{i}(kE)/J^{j}(kE)

is not locally split because the Jordan type of the middle is not the sum of the Jordan types of the outside terms.

On the other hand, there exist non-split short exact sequences of modules of constant Jordan type that are locally split. For example, the almost split sequences of Remark 5.2.10 have this property.

Example 5.3.5. Another class of examples comes from Tate cohomology. For any n, j ∈ ℤ we have

Ĥ^{n}(E, k) ≅ Ext_{kE}^{1}(Ω^{j+n−1}(k), Ω^{j}(k)).

If n < 0 then the corresponding sequence

0 → Ω^{j}(k) → M → Ω^{j+n−1}(k) → 0

is a locally split sequence of modules of constant Jordan type. This is because elements of negative degree restrict to zero on every proper shifted subgroup. In case n = −1 these are almost split sequences, but for n < −1 the module M is usually not homogeneous. We shall examine these examples in more detail in Section 8.12.

Definition 5.3.6 (Quillen [208]). An exact category is an additive category with a class of short exact sequences M_{1} → M_{2} → M_{3} satisfying the following properties:

(1) Any sequence isomorphic to a short exact sequence is a short exact sequence.

(2) If M_{1} → M_{2} → M_{3} is a short exact sequence then M_{1} → M_{2} is the kernel of M_{2} → M_{3} and M_{2} → M_{3} is the cokernel of M_{1} → M_{2}.

(3) For every pair of objects M_{1} and M_{3}, the split exact sequence M_{1} → M_{1}⊕M_{3} → M_{3} is a short exact sequence.

We say that a map M_{1} → M_{2} appearing as the first map in a short exact sequence is an admissible monomorphism, and that a map M_{2} → M_{3} appearing as the second map in a short exact sequence is an admissible epimorphism.

(4) The pushout of any map and an admissible monomorphism exists and is an admissible monomorphism.

(5) The pullback of any map and an admissible epimorphism exists and is an admissible epimorphism.

(6) The composite of two admissible monomorphisms is an admissible monomorphism,

(7) The composite of two admissible epimorphisms is an admissible epimorphism.

(8) If M → M_{1} is a morphism with a cokernel, and M_{1} → M_{2} and the composite M → M_{1} → M_{2} are both admissible monomorphisms, then so is M → M_{1}.

(9) If M_{3} → M is a morphism with a kernel, and M_{2} → M_{3} and the composite M_{2} → M_{3} → M are both admissible epimorphisms then so is M_{3} → M.

Remark 5.3.7. Keller [175, Appendix A] has shown that these axioms are somewhat redundant. It suffices to assume that (1), (2), (4), (5) and (7) hold, and that the identity map 0 → 0 is an admissible epimorphism; (3), (6), (8) and (9) follow from these statements. A survey of the theory of exact categories can be found in Bühler [68].

Proposition 5.3.8. The category of finite dimensional kE-modules and locally split exact sequences is an exact category. Similarly, the category of modules of constant Jordan type and locally split exact sequences is an exact category.

Proof. The requisite properties hold for split exact sequences, and therefore for locally split exact sequences. □

Definition 5.3.9. We write cJt(kE) for the exact category of modules of constant Jordan type and locally split exact sequences.

Definition 5.3.10. We write K_{0}(cJt(kE)) for the Grothendieck group whose generators are the symbols [M] corresponding to the isomorphism classes of modules M of constant Jordan type, and with relations [M_{1}] − [M_{2}] + [M_{3}] where 0 → M_{1} → M_{2} → M_{3} → 0 is a locally split exact sequence.

There is a homomorphism of abelian groups

JType: K_{0}(cJt(kE)) → ℤ^{p}

defined as follows. If M is a module of constant Jordan type [p]^{a_{p}} … [1]^{a_{1}} then

JType[M] = (a_{p}, …, a_{1}).

We can make K_{0}(cJt(kE)) and ℤ^{p} into commutative rings via tensor product; the tensor structure on ℤ^{p} being given by the tensor product of modules for a cyclic group of order p. Recall that this is described in Section 2.3 and does not depend on which Hopf algebra structure is used, by Theorem 2.3.2. With this definition, JType is a ring homomorphism.

The following is Proposition 4.1 of [80].

Proposition 5.3.11. The homomorphism JType is surjective.

Proof. This is easy to see using the modules kE/J^{j}(kE) with 1 ≤ j ≤ p and the description of their Jordan type given in Lemma 5.2.7. The only property we need of this is that the Jordan type of kE/J^{j}(kE) contains a single block of length j together with some smaller ones, and this property is easy to verify. □

5.4. Endotrivial Modules

In this section, we examine modules of stable constant Jordan type [1] and [p − 1], and find that these are exactly the endotrivial modules.

Definition 5.4.1. If G is a finite group, we say that a kG-module M is endotrivial if End_{k}(M) is stably isomorphic to the trivial module. In other words,

M ⊗_{k} M^{∗} ≅ End_{k}(M) ≅ k ⊕ (projective).

This definition comes from Dade [98], where he classified the endotrivial modules over an abelian p-group.

Theorem 5.4.2 (Dade, Theorem 10.1 of [98]). The endotrivial modules for an abelian p-group are precisely the modules which are stably isomorphic to Ω^{n}(k) for n ∈ ℤ.

Proposition 5.4.3 (Carlson, Friedlander and Pevtsova [81]). A kE-module M has stable constant Jordan type [1] if and only if M is stably isomorphic to Ω^{n}(k) for some integer n, and n is even in case p is odd.

Proof. One direction of this is given in Corollary 5.2.5. So assume that M has stable constant Jordan type [1]. By Theorem 5.1.9, M ⊗_{k} M^{∗} has constant Jordan type. By Lemma 5.1.13, End_{k}(M) ≅ M ⊗_{k} M^{∗} again has stable constant Jordan type [1]. There are maps

k → End_{k}(M) → k

given by the inclusion of the scalar matrices and the matrix trace. The dimension of M is congruent to 1 modulo p, so the composite of these two maps is the identity. Thus

M ⊗_{k} M^{∗} ≅ k ⊕ N

where N is a module of stable constant Jordan type ∅. By Proposition 5.1.14, N is a projective module, so M is endotrivial. By Theorem 5.4.2, it follows that M is stably isomorphic to Ω^{n}(k) for some n. If p is odd and n is odd, the stable constant Jordan type of Ω^{n}(k) is [p − 1] rather than [1]. So if p is odd then n is even. □

Proposition 5.4.4. If p is odd then a kE-module M has stable constant Jordan type [p − 1] if and only if M ≅ Ω^{n}(k) for some odd integer n.

Proof. The proof of this is essentially the same, after noting that for a cyclic group of order p we have

J_{p−1} ⊗_{k} J_{p−1} ≅ (p − 2)J_{p} ⊕ J_{1}.

This is because J_{p−1} ≅ Ω(k) and Ω(k) ⊗_{k} Ω(k) is stably isomorphic to Ω^{2}(k) ≅ k ≅ J_{1}. Alternatively, this formula is a special case of the formulas given at the end of Section 2.3. □

Putting these together, we obtain the following theorem.

Theorem 5.4.5. Let M be a kE-module. The following are equivalent:

(1) M has stable constant Jordan type [1] or [p − 1].

(2) M is endotrivial.

(3) M ≅ Ω^{n}(k) for some n ∈ ℤ. □

5.5. Wild Representation Type

In this section, we show that the modules of constant Jordan type usually have wild representation type. This has been independently proved by Bondarenko and Lytvynchuk [58]. See Section 1.2 for a discussion of representation type. We do this by taking a pair of endomorphisms ϕ and ψ of a finite dimensional vector space V and constructing a module of constant Jordan type M in such a way that V, ϕ and ψ can be recovered functorially from M. We begin with an observation about common eigenvectors.

Lemma 5.5.1. The problem of classifying configurations of a finite dimensional vector space V with two endomorphisms ϕ and ψ with no common eigenvector has wild representation type.

Proof. Given a finite dimensional vector space with endomorphisms ϕ and ψ, we construct another vector space W = V ⊕ V with endomorphisms Φ and Ψ with no common eigenvector as follows. In terms of matrices, we take

Φ = (\begin{matrix}1 & 0 \\ 0 & 0 \end{matrix}) Ψ = (\begin{matrix}ϕ & 1 \\ 1 & ψ \end{matrix}) .

The two copies of V can be recovered as W = Im Φ ⊕ Ker Φ, and so the off-diagonal components of Ψ allow us to identify these two copies, and the diagonal components allow us to recover the endomorphisms ϕ and ψ. □

Next we discuss the quiver

Q = \xymatrix{• \ar[r]^{β} \ar[r]_{δ} & • \ar[r]^{α} \ar[r]_{γ} & •}

with relation αβ = γδ. Representations of Q satisfying this relation are equivalent to representations of the quotient kQ/I of the quiver algebra kQ by the ideal I = (αβ − γδ).

If V is a finite dimensional vector space with endomorphisms ϕ and ψ, we have a representation of kQ/I given by

(5.5.1)  [figure: V; ^{( \begin{matrix}1 \\ 0 \\ 0 \end{matrix} )}; _{( \begin{matrix}0 \\ 1 \\ 0 \end{matrix} )}; V ⊕ V ⊕ V; ^{(\begin{matrix}1 & 0 & 0 \\ 0 & 1 & 0 \end{matrix})}; _{( \begin{matrix}ϕ & 1 & 0 \\ ψ & 0 & 1 \end{matrix} )}; V ⊕ V.]

The summands in the middle term can be identified as Im (β) ⊕ Im (δ) ⊕ ker(α), and the summands in the right-hand term can be identified as Im (αβ) ⊕ Im (αδ). The maps α, β, γ and δ induce isomorphisms between the five copies of V in these decompositions. It follows that the vector space V as well as the maps ϕ and ψ can be recovered from the representation of kQ/I.

Remark 5.5.2. It is easy to check that this construction has the following property. For all λ, μ ∈ k, not both zero, λβ + μδ is injective and λγ + μα is surjective. Furthermore, (λγ + μα)(λβ + μδ) acts as

(\begin{matrix}2λμ + λ^{2}ϕ \\ μ^{2} + λ^{2}ψ \end{matrix}) : V → V ⊕ V.

As long as ϕ and ψ have no common eigenvectors, this map is injective.

Next, let E = ⟨g_{1}, g_{2}⟩ ≅ ℤ/p × ℤ/p with p odd. Given a representation of kQ/I, we can make a kE-module with the same underlying vector space, and where X_{1} = g_{1} − 1 acts as β + γ and X_{2} = g_{2} − 1 acts as δ + α. The relation X_{1}X_{2} = X_{2}X_{1} follows from the relation αβ = γδ, and X_{1}^{3}, X_{2}^{3} act as zero.

More explicitly, the representations (5.5.1) have underlying vector space V^{⊕6} with the following matrices for X_{1} and X_{2}:

(5.5.2)  X_{1} ↦ ( \begin{matrix}0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & ϕ & 1 & 0 & 0 & 0 \\ 0 & ψ & 0 & 1 & 0 & 0 \end{matrix} )  X_{2} ↦ ( \begin{matrix}0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \end{matrix} ) .

Here is a diagram for such a module:

[figure: V; V; V; V; V; V; ϕ; ψ]

The representation of kQ/I can be recovered from the socle layers of the kE-module.

Theorem 5.5.3. Let p be an odd prime, k an algebraically closed field of characteristic p and E = ℤ/p × ℤ/p. As long as ϕ and ψ have no common eigenvectors, the matrices (5.5.2) define a kE-module of constant Jordan type [3]^{n}[2]^{n}[1]^{n} where n is the dimension of V.

Proof. This is easy to check using Remark 5.5.2. □

Corollary 5.5.4. Let p be an odd prime, k an algebraically closed field of characteristic p and E = ℤ/p × ℤ/p. Then kE-modules of constant Jordan type [3]^{n}[2]^{n}[1]^{n} are of wild representation type.

Proof. This follows from Theorem 5.5.3 and Lemma 5.5.1. □

Next we give a different construction which works in rank at least three for any prime. Again we begin with a vector space V with endomorphisms ϕ and ψ. Let

α_{1}, α_{2}, α_{3} : V^{⊕4} → V^{⊕2}

be given by the following matrices:

α_{1} = (\begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \end{matrix})  α_{2} = (\begin{matrix}0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{matrix})  α_{3} = (\begin{matrix}ϕ & 0 & 1 & ψ \\ 0 & 0 & 0 & 1 \end{matrix}) .

The summands of V^{⊕2} can be identified as

α_{1}Ker (α_{2}) ⊕ α_{2}Ker (α_{1}).

The summands of V^{⊕4} can be identified as

(α_{3}^{−1}α_{1}Ker α_{2} ∩ Ker α_{2}) ⊕ (α_{1}^{−1}α_{2}Ker α_{1} ∩ α_{2}^{−1}α_{1}Ker α_{2} ∩ α_{3}^{−1}α_{1}Ker α_{2})

⊕ (Ker α_{1} ∩ α_{3}^{−1}α_{1}Ker α_{2}) ⊕ (Ker α_{1} ∩ Ker α_{2}).

The maps α_{1}, α_{2}, α_{3} induce isomorphisms between the six copies of V in these decompositions. It follows that the vector space V and the maps ϕ and ψ can be recovered from the maps α_{1}, α_{2}, α_{3} : V^{⊕4} → V^{⊕2}.

If E = ⟨g_{1}, g_{2}, g_{3}⟩ ≅ (ℤ/p)^{3} then we form a kE-module M with underlying vector space V^{⊕6} and where the elements X_{i} = g_{i} − 1 acts via α_{i}. It is easy to check that every non-zero linear combination of α_{1}, α_{2}, α_{3} is surjective, so that M has constant Jordan type [2]^{2n}[1]^{2n}.

The matrices for this representation are

X_{1} ↦ ( \begin{matrix}0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \end{matrix} ) X_{2} ↦ ( \begin{matrix}0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \end{matrix} ) X_{1} ↦ ( \begin{matrix}0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ ϕ & 0 & 1 & ψ & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \end{matrix} ) .

This example generalises easily to groups of higher rank. If E has rank r then we use maps α_{1}, …, α_{r}: V^{⊕(r+1)} → V^{⊕2} given by

α_{1} = (\begin{matrix}1 & 0 & … & 0 \\ 0 & 1 & … & 0 \end{matrix}) , α_{2} = (\begin{matrix}0 & 1 & 0 & … & 0 \\ 0 & 0 & 1 & … & 0 \end{matrix}) , … α_{r} = (\begin{matrix}ϕ & 0 & … & 1 & ψ \\ 0 & 0 & … & 0 & 1 \end{matrix}) .

To conclude, we have proved the following theorem.

Theorem 5.5.5. If r ≥ 3 or if r ≥ 2 and p ≥ 3 then the kE-modules of constant Jordan type have wild representation type. □

Remark 5.5.6. This theorem is best possible, because modules for ℤ/2 × ℤ/2 in characteristic two have tame representation type and modules for ℤ/p in characteristic p have finite representation type.

5.6. The Constant Image Property

Modules with the constant image property were introduced by Carlson and Friedlander [80] and further investigated by Carlson, Friedlander and Suslin [82] and by Worch [250]. We continue with the convention that k is algebraically closed.

Definition 5.6.1. A finitely generated kE-module has the constant image property if for every non-zero α ∈ 𝔸^{r}(k) the image of X_{α} on M is equal to Rad(M).

Remark 5.6.2. The name comes from the fact that if the image of X_{α} on M is independent of α for 0 ≠ α ∈ 𝔸^{r}(k) then the image has to equal Rad(M).

Lemma 5.6.3. A kE-module M has the constant image property if and only if for all X ∈ J(kE) \ J^{2}(kE) we have X.M = Rad(M).

Proof. One direction is clear. For the other direction, suppose that M has the constant image property. Write X = X_{α} + Y with α ≠ 0 and Y ∈ J^{2}(kE). Then Y.M ⊆ Rad^{2}(M) and

Rad(M) = X_{α}.M ⊆ X.M + Y.M,

and so X.M = Rad(M). □

Lemma 5.6.4. A kE-module M has the constant image property if and only if the quotient module M/Rad^{2}(M) has the constant image property.

Proof. This follows immediately from Lemma 5.6.3, because X.M = Rad(M) if and only if

X.(M/Rad^{2}(M)) = Rad(M)/Rad^{2}(M) = Rad(M/Rad^{2}(M)).  □

Lemma 5.6.5. If M is a kE-module with the constant image property and K is an extension field of k then K ⊗_{k} M is a KE-module with the constant image property.

Proof. This follows immediately from Theorem 4.1.1. □

Example 5.6.6. The (2n − 1)-dimensional module for (ℤ/p)^{2} with diagram

[figure: •; •; …; •; •; •; •; …; •]

has the constant image property. A generalisation of this example is the W -modules of Section 5.11.

Proposition 5.6.7. If j ≥ (r−1)(p−1) then Rad^{j}(kE) has the constant image property.

Proof. Since the automorphism group of kE is transitive on the X_{α} for α ≠ 0, we may assume that X_{α} = X_{1}. So we must show that every element of Rad^{j+1}(kE) is in the image of X_{1}.

The submodule Rad^{j+1}(kE) has a basis consisting of the monomials X_{1}^{a_{1}} … X_{r}^{a_{r}} with 0 ≤ a_{i} < p and ∑_{i} a_{i} ≥ j + 1 > (r − 1)(p − 1). This forces all a_{1} to be greater than zero, and so this basis element is in the image of X_{1}. □

Remark 5.6.8. If r = 2, the modules of Proposition 5.6.7 are examples of the W -modules that we shall be discussing in Section 5.11.

Lemma 5.6.9. The class of modules with the constant image property is closed under finite sums, quotients (and, in particular, summands) and radicals.

Proof. This follows immediately from the definition. □

Lemma 5.6.10. If M has constant image property then for every non-zero α ∈ 𝔸^{r}(k) and for every 1 ≤ j ≤ p, the image of X_{α}^{j} is equal to Rad^{j}(M). In particular we have Rad^{p}(M) = 0.

Proof. Since kE is commutative we have

X_{α}.Rad^{j}(M) = X_{α}.J(kE)^{j}.M = J(kE)^{j}.X_{α}.M = J(kE)^{j}.Rad(M) = Rad^{j+1}(M).

The lemma now follows by induction. □

The following is Proposition 5.8 of [80].

Theorem 5.6.11. If M is a kE-module with the constant image property then M has constant Jordan type.

Proof. This follows from Lemma 5.6.10, since the Jordan canonical form of a nilpotent matrix is determined by the dimensions of the images of powers of the matrix. □

Theorem 5.6.12. If E has rank at least three then the collection of indecomposable modules with the constant image property has wild representation type.

Proof. The modules constructed in Section 5.5 for rank at least three have the constant image property. □

The following theorem shows that the radical layers of a module with the constant image property are decreasing in dimension by at least r − 1 at each stage.

Theorem 5.6.13. If M has the constant image property then

dim_{k} M/Rad(M) ≥ dim_{k} Rad(M)/Rad^{2}(M) + r − 1.

Proof. Let m = dim_{k} M/Rad(M) and n = dim_{k} Rad(M)/Rad^{2}(M). Consider the matrix A for the action of

X_{α} = ∑_{i=1}^{r} λ_{i}X_{i}: M/Rad(M) → Rad(M)/Rad^{2}(M).

This is an n × m matrix of homogeneous linear forms in the λ_{i}. By the definition of the constant image property, the matrix A has full row rank provided the λ_{i} are not all zero.

Let z_{1}, …, z_{n} be variables, and consider the row vector (z_{1}, …, z_{n}) times the matrix A. Then for all values of z_{1}, …, z_{n}, not all zero, and all values of λ_{1}, …, λ_{r}, not all zero, the product (z_{1}, …, z_{n})A is non-zero. So the m equations saying that the products of (z_{1}, …, z_{n}) with the columns of A are zero have no common solutions. These m equations define a subvariety of codimension at most m in the projective variety ℙ^{n−1} × ℙ^{r−1}. Since this subvariety is empty, it follows that

m ≥ (n − 1) + (r − 1) + 1 = n + r − 1.  □

Corollary 5.6.14. If M has the constant image property then for all j ≥ 1 we have

dim_{k} Rad^{j}(M)/Rad^{j+1}(M) ≤ dim_{k} M/Rad(M) − j(r − 1).

Proof. This follows from Lemma 5.6.9 and Theorem 5.6.13. □

Corollary 5.6.15. If M has the constant image property and radical length d then

dim_{k} M ≥ \binom{d}{2}(r − 1) + d.

Proof. If M has radical length d then dim_{k} Rad^{d−1}(M) ≥ 1 and so for 0 ≤ j ≤ d − 1 we have

dim_{k} Rad^{j}(M)/Rad^{j+1}(M) ≥ (d − 1 − j)(r − 1) + 1.

Now sum over j. □

Example 5.6.16. The following family of examples shows that Theorem 5.6.13 is sharp. Let n ≥ 1 and consider the n × (n + r − 1) matrix

U_{n,r} = (\begin{matrix}λ_{1} & λ_{2} & & ⋯ & λ_{r} & 0 & ⋯ & 0 \\ 0 & λ_{1} & λ_{2} & & ⋯ & λ_{r} & & 0 \\ ⋮ & & ... & ... & & & ... & ⋮ \\ 0 & 0 & ⋯ & λ_{1} & λ_{2} & & ⋯ & λ_{r} \end{matrix}) .

Let M be the representation of (ℤ/p)^{r} of dimension 2n + r − 1 for which λ_{1}X_{1} + ⋯ + λ_{r}X_{r} acts as

( \begin{matrix}0 & 0 \\ U_{n,r} & 0 \end{matrix}) .

Then M has the constant image property, with

dim M/Rad(M) = n + r − 1, dim Rad(M) = n.

In particular, M has constant Jordan type [2]^{n}[1]^{r−1}. Here is a diagram for the module M in the case r = 3, n = 4:

[figure: •; •; •; •; •; •; •; •; •; •]

Finally, we have the following property of modules with the constant image property.

Theorem 5.6.17. Suppose that E has rank at least two, and let M be a kE-module with the constant image property. Then

∑_{0≠α∈𝔸^{r}(k)} Ker (X_{α}, M) = M.

Proof. Consider the matrices A_{1}, A_{2} representing the maps induced by X_{1}, X_{2} from M/Rad(M) to Rad(M)/Rad^{2}(M). These matrices satisfy the conditions of Proposition 4.1.2 (1), and so M/Rad(M) is equal to the sum of the kernels of λA_{1} + μA_{2} for (λ, μ) ≠ (0, 0).

If m̄ ∈ M/Rad(M) is in the kernel of X_{α} then X_{α}m ∈ Rad^{2}(M), and so there exists m′ ∈ Rad(M) such that X_{α}m = X_{α}m′. Thus X_{α}(m − m′) = 0. It follows that every element of the kernel of X_{α}: M/Rad(M) → Rad(M)/Rad^{2}(M) lifts to an element of the kernel of X_{α} on M.

It follows that Rad(M) + ∑_{0≠α∈𝔸^{r}(k)} Ker (X_{α}) = M, and hence by Nakayama’s lemma we have ∑_{0≠α∈𝔸^{r}(k)} Ker (X_{α}) = M. □

5.7. The Generic Kernel

For rank two elementary abelian groups there is a notion closely related to the constant image property, namely the generic kernel of a module. Throughout this section, we set E = ⟨g_{1}, g_{2}⟩ ≅ ℤ/p × ℤ/p, X_{1} = g_{1} − 1 and X_{2} = g_{2} − 1 in kE. The results in this section are due to Carlson, Friedlander and Suslin [82]. We shall use the methods developed here in Section 5.13 to prove a special case of a conjecture of Rickard.

Definition 5.7.1. Let M be a kE-module. If

0 ≠ α = (λ_{1}, λ_{2}) ∈ 𝔸^{2}(k),

then the kernel of X_{α} = λ_{1}X_{1} + λ_{2}X_{2} on M only depends on the image ᾱ of α in ℙ^{1}(k). We define the generic kernel 𝔎(M) to be the intersection over all dense open subsets S of ℙ^{1}(k) (i.e., subsets whose complement is finite) of the sum of Ker (X_{α}, M) for ᾱ ∈ S.

Remark 5.7.2. Since M is finite dimensional, there exists a dense open subset S ⊆ ℙ^{1}(k) such that M = ∑_{ᾱ∈S} Ker (X_{α}, M). The subset S may be taken to exclude any given finite set of points.

Lemma 5.7.3. Let M be a kE-module. Then we have 𝔎(𝔎(M)) = 𝔎(M).

Proof. This is clear from the definition. □

Theorem 5.7.4. Let M be a kE-module. Then

(1) 𝔎(M) has the constant image property.

(2) If N is a submodule of M with the constant image property then N ⊆ 𝔎(M).

Proof. (1) By the lemma, without loss of generality we can assume that M = 𝔎(M). We need to show that for all 0 ≠ α, β ∈ 𝔸^{2}(k) we have X_{α}M = X_{β}M. Write

M = ∑_{γ∈S} Ker (X_{γ}, M)

where S is a dense open subset chosen to satisfy α, β ∉ S. For all γ ∈ S there exist λ_{1}, λ_{2} ∈ k, neither equal to zero, such that γ = λ_{1}α + λ_{2}β. So

X_{α}Ker (X_{γ}, M) = X_{β}Ker (X_{γ}, M).

It follows that X_{α}M = X_{β}M.

(2) This follows from Theorem 5.6.17. □

Corollary 5.7.5. Let M be a kE-module. Then 𝔎(M) has constant Jordan type.

Proof. This follows from Theorems 5.6.11 and 5.7.4 (1). □

Theorem 5.7.6. Let M be a kE-module. Then Ker (X_{α}, M) ⊆ 𝔎(M) if and only if X_{α} has maximal rank on M.

Proof. First note that the set of ᾱ ∈ ℙ^{1}(k) such that X_{α} has maximal rank on M is a dense open subset of ℙ^{1}(k). The subset S may therefore be chosen so that for all ᾱ ∈ S X_{α} has maximal rank on M. Now let α ∈ ℙ^{1}(k) and γ ∈ S. Then using the maximality of the rank of X_{γ} and Theorem 5.7.4, we have

dim Ker (X_{α}, M) ≥ dim Ker (X_{γ}, M) = dim Ker (X_{γ}, 𝔎(M)) = dim Ker (X_{α}, 𝔎(M)).

It follows that

Ker (X_{α}, M) ⊆ 𝔎(M) ⇔ dim Ker (X_{α}, M) = dim Ker (X_{α}, 𝔎(M))
⇔ dim Ker (X_{α}, M) = dim Ker (X_{γ}, M).  □

Corollary 5.7.7. If M has constant Jordan type then for all 0 ≠ α ∈ 𝔸^{2}(k) we have Ker (X_{α}, M) ⊆ 𝔎(M). □

Proposition 5.7.8. Let M be a kE-module of constant Jordan type. If 0 ≠ α ∈ 𝔸^{2}(k) and m ∈ M such that X_{α}m is in the generic kernel 𝔎(M) then for all 0 ≠ β ∈ 𝔸^{2}(k) we have X_{β}m ∈ 𝔎(M). Thus we have

J(kE).m ⊆ 𝔎(M).

Proof. If X_{α}m ∈ 𝔎(M) then X_{α}X_{β}m = X_{β}X_{α}m ∈ Rad 𝔎(M). So by Theorem 5.7.4 and Lemma 5.6.3 there exists m′ ∈ 𝔎(M) such that X_{α}m′ = X_{α}X_{β}m. Thus X_{α}(X_{β}m − m′) = 0. By Corollary 5.7.7, this implies that X_{β}m − m′ ∈ 𝔎(M). Since also m′ ∈ 𝔎(M), this implies that X_{β}m ∈ 𝔎(M). □

5.8. The Subquotient Rad^{−1}𝔎(M)/Rad^{2}𝔎(M)

In this section we show that for some purposes we can reduce to the case of a module of constant Jordan type of radical length three. We will use this construction in Section 5.13 to prove a special case of a conjecture of Rickard.

For the purposes of this section, we maintain the notation of the last section: E = ⟨g_{1}, g_{2}⟩ ≅ ℤ/p × ℤ/p, k is an algebraically closed field of characteristic p, and X_{1} = g_{1} − 1 and X_{2} = g_{2} − 1 in kE. If M is a kE-module, we write Rad^{−1}𝔎(M) for the submodule {m ∈ M | J(kE).m ∈ 𝔎(M)}. We are concerned with the subquotient Rad^{−1}𝔎(M)/Rad^{2}𝔎(M).

Example 5.8.1. If M = kE then Rad^{−1}𝔎(M)/Rad^{2}𝔎(M) is isomorphic to the module M_{p} discussed in Proposition 5.1.6.

Lemma 5.8.2. Let 0 ≠ α ∈ 𝔸^{2}(k) and let M be a kE-module of constant Jordan type. Then X_{α} has the same number of Jordan blocks of length one on Rad^{−1}𝔎(M)/Rad^{2}𝔎(M) as on M.

Proof. The number of Jordan blocks of X_{α} on M is the dimension of

\frac{Ker (X_{α}, M)}{Ker (X_{α}, M) ∩ X_{α}M} ≅ \frac{Ker (X_{α}, M) + X_{α}M}{X_{α}M} .

By Corollary 5.7.7 we have

Ker (X_{α}, M) ⊆ 𝔎(M) ⊆ Rad^{−1}𝔎(M)

and by Proposition 5.7.8 we have

Ker (X_{α}, M) ∩ X_{α}M = Ker (X_{α}, Rad^{−1}𝔎(M)) ∩ X_{α}Rad^{−1}𝔎(M).

So

\frac{Ker (X_{α}, M)}{Ker (X_{α}, M) ∩ X_{α}M} = \frac{Ker (X_{α}, Rad^{−1}𝔎(M))}{Ker (X_{α}, Rad^{−1}𝔎(M)) ∩ X_{α}Rad^{−1}𝔎(M)}

≅ \frac{Ker (X_{α}, Rad^{−1}𝔎(M)) + X_{α}Rad^{−1}𝔎(M)}{X_{α}Rad^{−1}𝔎(M)}.

Now if m ∈ Rad^{−1}𝔎(M) and X_{α}m ∈ Rad^{2}𝔎(M) then by Theorem 5.7.4 and Lemma 5.6.10 there exists m′ ∈ 𝔎(M) such that X_{α}m = X_{α}^{2}m′, and so X_{α}(m − X_{α}m′) = 0 and X_{α}m′ ∈ X_{α}𝔎(M) ⊆ X_{α}Rad^{−1}𝔎(M). It follows that the map

\frac{Ker (X_{α}, Rad^{−1}𝔎(M)) + X_{α}Rad^{−1}𝔎(M)}{X_{α}Rad^{−1}𝔎(M)} → \frac{Ker (X_{α}, \frac{Rad^{−1}𝔎(M)}{Rad^{2}𝔎(M)}) + X_{α} \frac{Rad^{−1}𝔎(M)}{Rad^{2}𝔎(M)}}{X_{α} \frac{Rad^{−1}𝔎(M)}{Rad^{2}𝔎(M)}}

is an isomorphism. The dimension of this is the number of Jordan blocks of length one of X_{α} on Rad^{−1}𝔎(M)/Rad^{2}𝔎(M). □

Theorem 5.8.3 (Baland [26]). Let M be a kE-module of constant Jordan type. Then the subquotient Rad^{−1}𝔎(M)/Rad^{2}𝔎(M) also has constant Jordan type, with the same number of Jordan blocks of length one, and the same total number of Jordan blocks.

Proof. Let 0 ≠ α ∈ 𝔸^{2}(k). Lemma 5.8.2 shows that the number of Jordan blocks of length one of X_{α} on M is the same as on Rad^{−1}𝔎(M)/Rad^{2}𝔎(M). Corollary 5.7.7, Theorem 5.7.4 and Lemma 5.6.10 show that the total number of Jordan blocks of X_{α} on M is the same as on Rad^{−1}𝔎(M)/Rad^{2}𝔎(M).

Since Rad^{−1}𝔎(M)/Rad^{2}𝔎(M) has radical length at most three, the Jordan canonical form of X_{α} on M has the form [3]^{a}[2]^{b}[1]^{c}. The quantity 3a + 2b + c is the dimension, and is therefore independent of α. The quantities a + b + c and c are also independent of α. Since the matrix

(\begin{matrix}3 & 2 & 1 \\ 1 & 1 & 1 \\ 0 & 0 & 1 \end{matrix}) ∈ GL(3, ℤ)

has determinant one, it follows that a, b and c are independent of α. □

Example 5.8.4. Carlson, Friedlander and Suslin [82, Question 8.11] ask the following question. If a kE-module (E is ℤ/p × ℤ/p as in the rest of this section) M has constant Jordan type, is it true that Rad^{−i}𝔎(M) has constant Jordan type for all i ≥ 0? Baland [26] found the following example to show that this is not the case. Let p ≥ 5 and let M be the 26-dimensional module given by the following diagram.

[figure: •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •]

Note that in this diagram, the top vertex is taken by X_{1} to the sum of the two vertices at the end of single lines from it, and by X_{2} to the sum of the two vertices at the end of double lines from it.

It is straightforward to check that M has constant Jordan type [4]^{2}[3] [2]^{5}[1]^{5}, by computing the ranks of powers of λ_{1}X_{1} + λ_{2}X_{2} for λ_{1}, λ_{2} ∈ k not both zero.

On the other hand, the codimension one submodule Rad^{−1}𝔎(M) decomposes as a direct sum of three modules, two of which do not have constant Jordan type, and therefore Rad^{−1}𝔎(M) does not have constant Jordan type.

5.9. The Constant Kernel Property

For the sake of completeness, in this and the next section we formulate the dual notions to the constant image property and the generic kernel. The proofs are mostly dual to those of Sections 5.6 and 5.7.

Definition 5.9.1. A finite dimensional kE-module M has the constant kernel property if for every non-zero α ∈ 𝔸^{r}(k) the kernel of X_{α} on M is equal to Soc(M).

For the purposes of the following lemma, we use the Lie theoretic dual rather than the group theoretic one, so that duality commutes with restriction to cyclic shifted subgroups.

Lemma 5.9.2. A kE-module M has the constant image property if and only if the dual module M ˜∗ has the constant kernel property.

Proof. The kernel of X_{α} on M ˜∗ is the annihilator in M ˜∗ of the image of X_{α} on M, and Soc(M ˜∗) is the annihilator of Rad(M). □

Lemma 5.9.3. A kE-module M has the constant kernel property if and only if for all X ∈ J(kE) \ J^{2}(kE) the kernel of X on M is equal to Soc(M).

Proof. This is dual to Lemma 5.6.3. □ Lemma 5.9.4. A kE-module M has the constant kernel property if and only if Soc^{2}(M)

has the constant kernel property.

Proof. This is dual to Lemma 5.6.4. □ Lemma 5.9.5. If M is a kE-module with the constant kernel property and K is an

extension field of k then K ⊗_{k} M is a KE-module with the constant kernel property. Proof. This is dual to Lemma 5.6.5. □ Example 5.9.6. The (2n − 1)-dimensional module for (ℤ/p)^{2} with diagram

[figure: •; •; …; •; •; •; …; •; •]

has the constant kernel property.

Proposition 5.9.7. If j ≤ p then kE/Rad^{j}(kE) has the constant kernel property. Proof. Noting that Rad^{j}(kE) = Soc^{r(p−1)+1−j}(kE), this is dual to Proposition 5.6.7. □ Lemma 5.9.8. The class of modules with the constant kernel property is closed under

finite sums, submodules and socle quotients. □

Remark 5.9.9. Every k(ℤ/p)^{2}-module M with the constant kernel property and with Rad^{d}(M) = 0 may be embedded in W_{n,d}^{∗} for some n ≥ d.

Lemma 5.9.10. If M has the constant kernel property then for every non-zero α ∈ 𝔸^{r}(k) and for every 1 ≤ j ≤ p, the kernel of X_{α}^{j} on M is equal to Soc^{j}(M).

Proof. This is dual to Lemma 5.6.10. □ Theorem 5.9.11. If M is a kE-module with the constant kernel property then M has

constant Jordan type.

Proof. This is dual to Theorem 5.6.11. □ Theorem 5.9.12. If E has rank at least three then the collection of indecomposable

modules with the constant kernel property has wild representation type.

Proof. This is dual to Theorem 5.6.12. □ Theorem 5.9.13. If M has the constant kernel property then

dim_{k} Soc(M) ≥ dim_{k} Soc^{2}(M)/Soc(M) + r − 1.

Proof. This is dual to Theorem 5.6.13. □

Corollary 5.9.14. If M has the constant kernel property then for all j ≥ 1 we have

dim_{k} Soc^{j+1}(M)/Soc^{j}(M) ≤ dim_{k} Soc(M) − j(r − 1).  □

Theorem 5.9.15. If M has constant Jordan type then M/𝔎(M) has the constant kernel property.

Proof. This follows from Proposition 5.7.8. □

Theorem 5.9.16. Suppose that E has rank at least two, and let M be a kE-module with the constant kernel property. Then

⋂_{0≠α∈𝔸^{r}(k)} Im (X_{α}, M) = 0.

Proof. This is dual to Theorem 5.6.17. □

5.10. The Generic Image

For this section we assume that E = ⟨g_{1}, g_{2}⟩, X_{1} = g_{1} − 1, X_{2} = g_{2} − 1 in kE, and we develop the concept of generic image of a module.

Definition 5.10.1. Let M be a kE-module. If

0 ≠ α = (λ_{1}, λ_{2}) ∈ 𝔸^{2}(k),

then the image of X_{α} = λ_{1}X_{1} + λ_{2}X_{2} on M only depends on the image ᾱ of α in ℙ^{1}(k). We define the generic image ℑ(M) to be the sum over all dense open subsets S of ℙ^{1}(k) of the intersection of Im (X_{α}, M) for ᾱ ∈ S.

Remark 5.10.2. Since M is finite dimensional, there exists a dense open subset S ⊆ ℙ^{1}(k) such that M = ⋂_{ᾱ∈S} Im (X_{α}, M). The subset S may be taken to exclude any given finite set of points.

Again, for the purposes of the following lemma, we use the Lie theoretic dual rather than the group theoretic one, so that duality commutes with restriction to cyclic shifted subgroups.

Lemma 5.10.3. Let M be a kE-module. Then ℑ(M ˜∗) is the annihilator in M ˜∗ of 𝔎(M) and 𝔎(M ˜∗) is the annihilator of ℑ(M). Thus we have

M ˜∗/ℑ(M ˜∗) ≅ 𝔎(M)˜∗, (M/ℑ(M))˜∗ ≅ 𝔎(M ˜∗),
ℑ(M ˜∗) ≅ (M/𝔎(M))˜∗,  ℑ(M)˜∗ ≅ M ˜∗/𝔎(M ˜∗).  □

Lemma 5.10.4. Let M be a kE-module. Then we have ℑ(M/ℑ(M)) = 0. □

Theorem 5.10.5. Let M be a kE-module. Then

(1) M/ℑ(M) has the constant kernel property.

(2) If N is a submodule of M such that M/N has the constant kernel property then N ⊇ ℑ(M).

Proof. This is dual to Theorem 5.7.4. □

Corollary 5.10.6. Let M be a kE-module. Then M/ℑ(M) has constant Jordan type. □

Theorem 5.10.7. Let M be a kE-module. Then ℑ(M) ⊆ Im (X_{α}, M) if and only if X_{α} has maximal rank on M.

Proof. This is dual to Theorem 5.7.6. □

Corollary 5.10.8. If M has constant Jordan type then for all 0 ≠ α ∈ 𝔸^{2}(k) we have ℑ(M) ⊆ Im (X_{α}). □

Proposition 5.10.9. Let M be a kE-module of constant Jordan type. If m ∈ ℑ(M) and 0 ≠ α, β ∈ 𝔸^{2}(k) then there exists m′ ∈ ℑ(M) such that X_{β}m′ = X_{α}m.

Proof. This is almost dual to Proposition 5.7.8, but requires separate proof. By Corollary 5.10.8 there exists m′′ ∈ M such that m = X_{β}m′′. We set m′ = X_{α}m′′. By Theorem 5.10.5 the image of m′′ in M/ℑ(M) lies in the socle, and so X_{α}m′′ ∈ ℑ(M) and X_{β}m′ = X_{β}X_{α}m′′ = X_{α}X_{β}m′′ = X_{α}m. □

Theorem 5.10.10. If M has constant Jordan type then ℑ(M) has the constant image property.

Proof. This is dual to Theorem 5.9.15. □

Corollary 5.10.11. If M has constant Jordan type then ℑ(M) ⊆ 𝔎(M).

Proof. This follows from Theorems 5.10.10 and 5.7.4 (2). □

Lemma 5.10.12. Suppose that M has constant Jordan type. Then the following are equal:

(1) The number of Jordan blocks of each X_{α} on M.

(2) The number of Jordan blocks of each X_{α} on 𝔎(M).

(3) The number of Jordan blocks of each X_{α} on 𝔎(M)/Rad^{2}𝔎(M).

(4) The dimension of 𝔎(M)/Rad 𝔎(M).

(5) The number of Jordan blocks of each X_{α} on M/ℑ(M).

(6) The number of Jordan blocks of each X_{α} on Soc^{2}(M/ℑ(M)).

(7) The dimension of Soc(M/ℑ(M)).

Proof. This follows from Corollaries 5.7.7 and 5.10.8 and Lemmas 5.6.3 and 5.9.3. □

5.11. W -Modules

If n ≥ d ≥ 1 and d ≤ p, we define a module W_{n,d} for (ℤ/p)^{2} with generators v_{1}, …, v_{n} and relations

X_{1}v_{1} = 0,  X_{2}v_{n} = 0,  X_{2}v_{i} = X_{1}v_{i+1} (1 ≤ i < n),
X_{1}^{d}v_{i} = 0, X_{2}^{d}v_{i} = 0 (1 ≤ i ≤ n).

Then W_{n,d} has the constant image property.

Example 5.11.1. Here is the diagram for W_{6,4} (p ≥ 5):

[figure: •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •]

The duals of the modules W_{n,d} have the constant kernel property. A presentation for W_{n,d}^{∗} is given by generators w_{1}, …, w_{n−d+1} and relations X_{2}w_{i} = X_{1}w_{i+1} (1 ≤ i ≤ n − d), X_{1}^{d}w_{i} = 0 and X_{2}^{d}w_{i} = 0 (1 ≤ i ≤ n − d + 1).

Lemma 5.11.2. The module W_{n,d} is of constant Jordan type

[d]^{n−d+1}, [d − 1], …, [2], [1].  □

Remark 5.11.3. Carlson, Friedlander and Suslin [82] have proved that every k(ℤ/p)^{2}- module M with the constant image property and with Rad^{d}(M) = 0 is a quotient of the module W_{n,d} of Example 5.6.6 for some n ≥ d.

It seems difficult to formulate an analogue of this statement for higher rank. The lefthand diagram below gives a one-parameter family of examples of modules for (ℤ/p)^{3} with the constant image property:

\xymatrix{• \ar[dr] &  & • \ar[dl] \ar[dr]^{λ} &  & • \ar[dl] & • \ar[dr] &  & • • \ar[dl] \ar[dr] &  & • \ar[dl] \\  & • &  & • &  &  & • &  & • \\  & • \ar[u] \ar[dr] &  & • \ar[u] \ar[dl] &  &  & • \ar[u] \ar[dr] &  & • \ar[u] \ar[dl] \\  &  & • &  &  &  &  & • \\  &  & • \ar[u] &  &  &  &  & • \ar[u]}

(0 ≠ λ ∈ k). Here, the actions of the elements X_{1}, X_{2}, X_{3} are indicated by up, down-right and down-left arrows respectively. Note that the modules in this family are not quotients of the module on the right. So it is unlikely that a discrete set of modules analogous to the W_{n,d} exists for higher rank, in such a way that they surject onto every module with the constant image property. This is probably related to the fact that for rank at least three the modules with the constant image property are of wild representation type, see Theorem 5.6.12.

5.12. Constant Jordan type with One Non-Projective Block

In Section 5.4, we proved that the only modules of stable constant Jordan type [1] and [p − 1] are the modules stably isomorphic to Ω^{n}(k).

Conjecture 9.5 of Carlson, Friedlander and Pevtsova [81] states that there cannot exist a module of stable constant Jordan type [2] for an elementary abelian p-group of rank at least two with p > 3. Such a module is characterised by the condition that its restriction to every cyclic shifted subgroup consists of a single Jordan block of length two together with some number of Jordan blocks of length p. In this section, we prove the following strengthened form of their conjecture, which is the main theorem of [42]. In Section 5.13 we prove a stronger statement (Corollary 5.14.2) by more complicated methods.

Theorem 5.12.1. Let E = (ℤ/p)^{r} with r ≥ 2, and let k be an algebraically closed field of characteristic p. If 2 ≤ a ≤ p − 2 then there does not exist a finite dimensional kE-module of stable constant Jordan type [a].

Proof. Let M be a kE-module of stable constant Jordan type [a] with 2 ≤ a ≤ p − 2, and let dim_{k} M = np + a. By Theorem 5.1.9, Λ^{a+1}(M) is of stable constant Jordan type ∅, so by Proposition 5.1.14 it is projective. It follows that dim_{k} Λ^{a+1}M is divisible by p^{2}. This dimension is

\binom{np + a}{a + 1} = \frac{(np + a)(np + a − 1) … (np)}{(a + 1)!}.

The only term in the numerator or the denominator of this fraction that is divisible by p is the last term on the top, namely np. So we deduce that n is a multiple of p.

On the other hand, by Theorem 2.8.1 and Proposition 5.1.14, S^{p−a+1}M is also projective, and its dimension is

\binom{np + p}{p − a + 1} = \frac{(np + p)(np + p − 1) … (np + a)}{(p − a + 1)!}.

The only term in the numerator or the denominator of this fraction that is divisible by p is the first term on the top, namely np+p. Since n is a multiple of p, this is not divisible by p^{2}. This contradicts the fact that this module is projective, and hence no such M can exist. □

Corollary 5.12.2. Let E = (ℤ/p)^{r} and let M be a finite dimensional kE-module with stable generic Jordan type [a], with 2 ≤ a ≤ p − 2. Then the generic Jordan set U_{max}(M) is the complement of a closed set of codimension one in affine space 𝔸^{r}(k).

Proof. Let V ⊆ 𝔸^{r}(k) be the complement of U_{max}(M). If V has codimension at least two then there is a linear subspace W of 𝔸^{r}(k) of dimension two such that V ∩ W = {0}. Choosing a basis for W gives a rank two shifted subgroup E′, and then kE′ is a subalgebra of kE. The restriction of M to kE′ has constant Jordan type [a], contradicting Theorem 5.12.1. □

5.13. Rickard’s Conjecture

The following conjecture was made by Rickard at MSRI in 2008, based on data from a large number of modules of constant Jordan type. The modules he was examining were all produced by the methods of Section 5.2, so they are all homogeneous, but the methods we know that produce inhomogeneous modules of constant Jordan type still seem to satisfy this conjecture. Throughout this section E is an elementary abelian p-group of rank at least two and k is an algebraically closed field of characteristic p.

Conjecture 5.13.1 (Rickard’s Conjecture). Suppose that M is a kE-module of constant Jordan type [a_{1}] … [a_{t}]. If there is a value of j with 1 ≤ j ≤ p − 1 such that none of the a_{i} is equal to j then the total number of a_{i} that are greater than j is a multiple of p.

In other words, if there is a missing size of Jordan block, then the total number of Jordan blocks of larger size is divisible by p.

This conjecture also rules out stable constant Jordan type [a] for 2 ≤ a ≤ p − 2 (see Theorem 5.12.1) as follows. If there were such a module, then the conjecture with j = a + 1 would imply that the number of Jordan blocks of length p is divisible by p. On the other hand, the conjecture with j = a−1 would imply that this number is congruent to −1 modulo p. These incompatible congruences would show that such a module cannot exist.

There is a strengthened version of Rickard’s conjecture which states that the number of a_{i} that are greater than j is a multiple not just of p but of p^{r−1}. Rickard’s conjecture reduces to the case r = 2, whereas the strengthened version does not. The following is easy to verify.

Theorem 5.13.2. If M is a module of constant Jordan type that satisfies Rickard’s conjecture then so is Ω(M). Similarly, if M satisfies the strengthened version then so does Ω(M). □

Following [44], we shall use the generic kernel to prove a special case of Rickard’s conjecture, namely the case j = 1. Motivation for the proof of this theorem may be found in the proof of Dade’s lemma in [98], and in the example of Proposition 5.1.6. Consequences of this theorem, and a related theorem in characteristic zero, are investigated in section 5.14.

Theorem 5.13.3. Let M be a kE-module of constant Jordan type [a_{1}] … [a_{t}], and suppose that none of the a_{i} is equal to 1. Then t is divisible by p.

Proof. Let 𝔎(M) be the generic kernel of M, as described in Section 5.7. By Lemma 5.10.12, it suffices to show that for each summand N of 𝔎(M)/Rad^{2}𝔎(M), N/RadN has dimension divisible by p.

Let 0 ≠ α ∈ 𝔸^{2}(k). Look at the maps of vector spaces induced by X_{α}

Rad^{−1}𝔎(M)/𝔎(M) ⟶^{X_{α}} 𝔎(M)/Rad𝔎(M) ⟶^{X_{α}} Rad𝔎(M)/Rad^{2}𝔎(M).

It follows from Lemma 5.6.3, Theorem 5.6.11 and Corollary 5.7.7 that the first of these maps is injective and the second is surjective.

It follows from Theorem 5.8.3 that Rad^{−1}𝔎(M)/Rad^{2}𝔎(M) has constant Jordan type, X_{α} has no Jordan blocks of length one on it, and the total number of Jordan blocks on it is equal to t.

In order to prove the theorem, we may therefore replace M by Rad^{−1}𝔎(M)/Rad^{2}𝔎(M). Thus we may assume that Rad^{2}𝔎(M) = 0 and RadM ⊆ 𝔎(M), which implies that Rad^{3}M = 0 so that every Jordan block of X_{α} on M has length two or three.

The indecomposable modules for kE/J^{2}(kE) are well understood in terms of the classification of modules for the Kronecker quiver. Since every X_{α} induces a surjective map from 𝔎(M)/Rad𝔎(M) to Rad𝔎(M), it follows that 𝔎(M) is a direct sum of W -modules for kE/J^{2} of the following form. Such a module has a basis v_{0}, …, v_{n}, w_{1}, …, w_{n} with

X_{1}v_{0} = 0, X_{1}v_{i} = w_{i}, X_{1}w_{i} = 0,

X_{2}v_{i−1} = w_{i}, X_{2}v_{n} = 0, X_{2}w_{i} = 0
(1 ≤ i ≤ n).

A diagram of such a module is as follows.

(5.13.1)  [figure: •^{v_{0}}; •^{v_{1}}; …; •^{v_{n}}; •_{w_{1}}; •_{w_{2}}; •_{w_{n}}]

Let N be an indecomposable summand of 𝔎(M). Thus N is a module of the form (5.13.1) with n ≥ 0. For notation, we write v̄_{0}, …, v̄_{n} for the images of v_{0}, …, v_{n} in 𝔎(M)/Rad𝔎(M). Let 0 ≠ α ∈ 𝔸^{2}(k). Since X_{α} has no Jordan blocks of length one on M, the kernel of X_{α} on N is in the image of X_{α} on M. Indeed, this remains true after field extension, see Theorem 5.1.8. Apply this to X_{α} = X_{1} + tX_{2} after extending the coefficients to the field of Laurent power series k[[t]][t^{−1}]. The kernel of X_{α} on N is spanned by the image of X_{α} on N together with the vector

v_{0} − tv_{1} + t^{2}v_{2} − ⋯ ± t^{n}v_{n}.

It follows that there is a vector ∑_{i}(−1)^{i}t^{i}ū_{i} ∈ M/𝔎(M) whose image under X_{1} + tX_{2} is

v̄_{0} − tv̄_{1} + t^{2}v̄_{2} − ⋯ ± t^{n}v̄_{n} ∈ 𝔎(M)/Rad𝔎(M).

Since X_{1} and X_{2} are injective as maps from M/𝔎(M) to 𝔎(M)/Rad𝔎(M), comparing coefficients of powers of t shows that ū_{i} is zero unless i ≥ 0. For i ≥ n we have X_{2}ū_{i} = X_{1}ū_{i+1}.

The argument at this stage follows the lines of the proof of Proposition 4.1.2. If ū_{n} ≠ 0 then let m > n be as small as possible so that there is a linear relation of the form ∑_{i=n}^{m} λ_{i}ū_{i} = 0. Such a linear relation exists because M is finite dimensional, and minimality of m implies that ū_{n}, …, ū_{m−1} are linearly independent and ū_{m} is a linear combination of them. So

X_{2}ū_{m} ∈ ⟨X_{2}ū_{n}, …, X_{2}ū_{m−1}⟩ = ⟨X_{1}ū_{n+1}, …, X_{1}ū_{m}⟩ ⊆ ⟨X_{1}ū_{n}, …, X_{1}ū_{m−1}⟩

as subspaces of 𝔎(M)/Rad𝔎(M). It follows that for all λ, μ not both zero,

λX_{1} + μX_{2} : ⟨ū_{n}, …, ū_{m−1}⟩ → ⟨X_{1}ū_{n}, …, X_{1}ū_{m−1}⟩

is injective, and hence an isomorphism. This is impossible because using X_{1} to identify these two spaces, X_{2} has to have a non-zero eigenvector for some eigenvalue. This contradiction shows that ū_{n} = 0.

Again using the fact that X_{1} is injective and X_{2}ū_{i} = X_{1}ū_{i+1} for i ≥ n, it follows that ū_{i} = 0 for i ≥ n. Thus we have

(X_{1} + tX_{2})(ū_{0} − tū_{1} + t^{2}ū_{2} − ⋯ ∓ t^{n−1}ū_{n−1}) = v̄_{0} − tv̄_{1} + t^{2}v̄_{2} − ⋯ ± t^{n}v̄_{n}.

In particular it follows that n > 0. Comparing coefficients we get

X_{1}ū_{0} = v̄_{0}, X_{1}ū_{1} − X_{2}ū_{0} = v̄_{1}, X_{1}ū_{2} − X_{2}ū_{1} = v̄_{2}, ⋯, −X_{2}ū_{n−1} = v̄_{n}.

Next, we lift ū_{0}, …, ū_{n−1} to elements u_{0}, …, u_{n−1} in M. Then the equation above holds without bars, modulo Rad𝔎(M). Since X_{1} is surjective as a map from 𝔎(M)/Rad𝔎(M) to Rad𝔎(M), the lifting can be done in such a way that

X_{1}u_{0} = v_{0}, X_{1}u_{1} − X_{2}u_{0} = v_{1}, X_{1}u_{2} − X_{2}u_{1} = v_{2}, ⋯, −X_{2}u_{n−1} = v_{n} + w

with w ∈ Rad𝔎(M). This shows that X_{1}u_{1} − 2v_{1} = X_{2}u_{0} − v_{1} is annihilated by X_{1}; for 2 ≤ i ≤ n − 1 we have

X_{2}(X_{1}u_{i−1} − iv_{i−1}) = X_{1}^{2}u_{i} − (i + 1)w_{i} = X_{1}(X_{1}u_{i} − (i + 1)v_{i}),

and

X_{2}(X_{1}u_{n−1} − nv_{n−1}) = −(n + 1)w_{n}.

We therefore have the following relationships between elements of 𝔎(M):

[figure: X_{1}u_{1} − 2v_{1}; X_{1}u_{2} − 3v_{2}; …; X_{1}u_{n−1} − nv_{n−1}; X_{1}^{2}u_{2} − 3w_{2}; X_{1}^{2}u_{3} − 4w_{3}; −(n + 1)w_{n}]

Here, we do not claim that the depicted elements are linearly independent, or even non-zero. However, it is easily checked that there cannot be such a chain of elements in 𝔎M unless the coefficient of w_{n} on the lower rightmost vector is zero. This is because

w_{n} ∉ (X_{2}X_{1}^{−1})^{n−1}{0}_{𝔎M},

as can be seen by projecting onto the summand N of 𝔎M (the notation here is that if U is a subspace of 𝔎M we denote by X_{1}^{−1}U its inverse image under X_{1} and by X_{2}U its image under X_{2}).

It follows that −(n + 1) is zero in k, which implies that n ≡ −1 (mod p). It follows that the dimension of N/RadN is divisible by p. Since this is true for every indecomposable summand N of 𝔎M, it follows that the dimension of 𝔎M/Rad𝔎M is divisible by p. □

5.14. Consequences and Variations

In this section we deduce some consequences of the special case of Rickard’s conjecture proved in Theorem 5.13.3, and then we investigate an analogue in characteristic zero. The first consequence is another special case of Rickard’s conjecture given in the following corollary.

Corollary 5.14.1. Let M be a kE-module of constant Jordan type [a_{1}] … [a_{t}], and suppose that none of the a_{i} is equal to p − 1. Then the number of a_{i} that are equal to p is divisible by p.

Proof. Let M be a module of constant Jordan type [a_{1}] … [a_{t}] with none of the a_{i} equal to p − 1. Then Ω(M) is a module of constant Jordan type where the number of blocks of length p−j (j ≥ 1) is equal to the number of blocks of M of length j. The number of Jordan blocks of length p in Ω(M) is congruent mod p to minus the total number of Jordan blocks of M. Therefore, Corollary 5.14.1 for M follows by applying Theorem 5.13.3 to Ω(M). □

Corollary 5.14.2. Let M be a kE-module of stable constant Jordan type [a_{1}] … [a_{t}], and suppose that none of the a_{i} is equal to 1 or p − 1. Then t is divisible by p.

Proof. If M has constant Jordan type with no blocks of length 1 or p − 1 then we may apply Theorem 5.13.3 to deduce that the total number of Jordan blocks is divisible by p, and we may also apply Corollary 5.14.1 to deduce that the number of Jordan blocks of length p is also divisible by p. Subtracting, we see that the number of Jordan blocks of length between 2 and p − 2 is divisible by p. □

Corollary 5.14.2 gives a new proof of Theorem 5.12.1. Moreover, combining our main theorem with the methods of Section 5.12 gives new information.

Theorem 5.14.3. Suppose that M has constant Jordan type with all stable Jordan blocks of distinct odd length or all of distinct even length. Then the stable Jordan type is either [1] or [p − 1] or has at least four Jordan blocks.

Proof. Suppose that all the stable Jordan blocks of M are distinct and odd. By Theorem 5.1.9, Λ^{2}(M) has constant Jordan type.

If M has one stable Jordan block then by Theorem 5.13.3 the length of the Jordan block is equal to one.

If M has two stable Jordan blocks then again applying Theorem 5.13.3, the restriction to a cyclic group is J_{a} ⊕ J_{1} plus projectives, with a > 1. So Λ^{2}(M) has stable constant Jordan type Λ^{2}(J_{a} ⊕ J_{1}) = Λ^{2}(J_{a}) ⊕ J_{a}. By Proposition 2.8.6, this has at most (p − 1)/4 non-projective summands, none of which is isomorphic to J_{1}, contradicting Theorem 5.13.3.

If M has three stable Jordan blocks then the restriction to a cyclic group is J_{a} ⊕ J_{b} ⊕ J_{1} plus projectives, with a < b. We have

Λ^{2}(J_{a} ⊕ J_{b} ⊕ J_{1}) ≅ Λ^{2}(J_{a}) ⊕ Λ^{2}(J_{b}) ⊕ J_{a} ⊗ J_{b} ⊕ J_{a} ⊕ J_{b}.

It follows from Proposition 2.8.6 that Λ^{2}(J_{a}) and Λ^{2}(J_{b}) each has at most (p − 3)/4 nonprojective indecomposable summands, and J_{a} ⊗ J_{b} has at most (p − 3)/2. So Λ^{2}(M) has at most p − 1 non-projective summands, and that none of them is isomorphic to J_{1}. Again this contradicts Theorem 5.13.3.

It follows that if M has more than one stable Jordan block then it has at least four. Finally, if all the stable Jordan blocks are distinct and even, then Ω(M) has constant

Jordan type with all stable Jordan blocks distinct and odd. So we apply the previous arguments to Ω(M). □

Theorem 5.14.4. If p/2 < n < p − 1 there is no module of stable constant Jordan type [n] [1].

Proof. If M has stable constant Jordan type [n] [1] then because of the restriction on the value of n, there are no Jordan blocks of length p−1. So by Corollary 5.14.1, the number of Jordan blocks of length p is divisible by p. By Proposition 2.8.6, S^{2}(J_{n}) has n − (p − 1)/2 copies of J_{p}. So S^{2}(M) has no Jordan blocks of length p − 1 and the number of Jordan blocks of length p is congruent to n − (p − 1)/2 modulo p. This contradicts Corollary 5.14.1 for S^{2}(M). □

The same technique does not seem to dispose of stable constant Jordan type [n] [1] for 4 ≤ n < p/2 even, so for example [4] [1] for p = 11 is the smallest case that remains undecided.

Some further cases can be ruled out by the use of other Schur functors. For example, by applying the Schur functor S^{[2][1]}, we can rule out the case of stable constant Jordan type [6] [1] for p = 13 using the method of proof of Theorem 5.14.4. More generally the Schur functor S^{[2][1]} rules out modules of stable constant Jordan type [2n] [1] when p = 6n − 5 or p = 6n − 7. This is because using the technique explained in Section 2.9 we see that for p = 6n − 5 we have

S^{[2][1]}(J_{2n}) = 2J_{p} ⊕ J_{p−3} ⊕ ⋯

while for p = 6n − 7 we have

S^{[2][1]}(J_{2n}) = 4J_{p} ⊕ J_{p−3} ⊕ ⋯

where the dots indicate smaller Jordan blocks.

Theorem 5.14.5. Let M be a kE-module of constant Jordan type [a_{1}] … [a_{t}] with none of the a_{i} equal to one. Then at most 2t/p of the a_{i} are equal to 2.

Proof. Looking back at the proof of Theorem 5.13.3, we see that 𝔎(M)/J^{2}𝔎(M) is a direct sum of W -modules of the form (5.13.1) with n + 1 divisible by p. There is at most one a_{i} equal to 2 and n of the a_{i} greater than 2 for each such summand. □

As an application of Theorem 5.14.5, we completely classify the Jordan types that can occur for (ℤ/3)^{2}. The corresponding problem for elementary abelian 2-groups is studied in Chapter 10 using the theory of vector bundles.

Theorem 5.14.6. Let E = (ℤ/3)^{2} and k be an algebraically closed field of characteristic 3. Then there is a kE-module of constant Jordan type [3]^{a}[2]^{b}[1]^{c} if and only if the following conditions hold:

(1) If a = 0 then c ≥ 1.

(2) If b = 0 then a is divisible by 3.

(3) If c = 0 then a + b is divisible by 3 and b ≤ 2a.

Proof. Condition (1) holds by Theorem 5.1.18, condition (2) holds by Corollary 5.14.1, and condition (3) holds by Theorems 5.13.3 and 5.14.5.

To prove that these conditions are sufficient, we display the following modules of constant Jordan type:

(a) kE of type [3]^{3},

(b) k of type [1],
(c) Rad^{2}(kE) of type [3] [2] [1],
(d) Rad(kE) of type [3]^{2}[2],
(e) Rad(kE)/Soc(kE) of type [3] [2]^{2},
(f) W -modules W_{n,2} (see Section 5.11) of type [2]^{n−1}[1] (n ≥ 2).

If a = 0 then we use direct sums of modules of type (f). If b = 0 then we use direct sums of modules of type (a) and (b). If c = 0 then we use direct sums of modules of type (a), (d) and (e). Finally, if a, b, c are all non-zero then we use direct sums of modules of all these types; first we use modules of type (a) to reduce to a ≤ 2, then we use modules of type (b) and (f) to reduce to the cases (c), (d) and (e). □

Now we turn to a characteristic zero analogue of Rickard’s conjecture, and again we prove a special case of it.

Conjecture 5.14.7. Let k be an algebraically closed field of characteristic zero. Let A and B be commuting nilpotent d × d matrices over k, and assume that all non-zero linear combinations of A and B have the same Jordan canonical form. If there is a Jordan block of length j then there are Jordan blocks of all lengths up to and including j.

Theorem 5.14.8. Let k be an algebraically closed field of characteristic zero. Let A and B be commuting nilpotent d × d matrices over k with d ≥ 1, and assume that all non-zero linear combinations of A and B have the same Jordan canonical form. Then there are Jordan blocks of length one.

Proof. The concept of generic kernel works in this situation just as well as for representations of ℤ/p × ℤ/p in characteristic p. For the proof of the theorem, assume that there are no Jordan blocks of length one, and argue as in the proof of Theorem 5.13.3 up to the the last paragraph, where −(n+1)w_{n} is shown to be equal to zero in k. Since w_{n} is non-zero, and k has characteristic zero, it follows that n = −1, which contradicts the fact that n ≥ 0. □

5.15. Further Conjectures

Let E be an elementary abelian p-group of rank r ≥ 2 and let k be an algebraically closed field of characteristic p. In Section 5.13 we discussed Rickard’s conjecture on the possible constant Jordan types of kE-modules. In this section we discuss some further conjectures.

If p = 3 then k and Ω(k) have stable constant Jordan type [1] and [2] respectively, and therefore sums of these realise all possible stable constant Jordan types. So we shall assume for the sake of this section that p ≥ 5.

For (ℤ/p)^{2}, the only known restrictions on stable constant Jordan type are given in Section 5.13, although restrictions for larger rank will be given in the next chapter. It is not known, for example, whether there can exist a module of stable constant Jordan type [4] [1] if p ≥ 11. This is the simplest type where the answer is not known.

The following conjecture is recorded in [81].

Conjecture 5.15.1 (Suslin’s Conjecture). Suppose that M is a kE-module of constant Jordan type. If there is a Jordan block of length j with 2 ≤ j ≤ p − 1 then there is either a Jordan block of length j + 1 or one of length j − 1.

This conjecture does rule out a module of stable constant Jordan type [a] for 2 ≤ a ≤ p−2, as in Theorem 5.12.1, as there is no Jordan block of length a − 1 or a + 1 in this case. It also rules out [a]^{n} for all n > 0, for example.

Rickard’s conjecture does not rule out [a]^{p} for 2 ≤ a ≤ p − 2, whereas Suslin’s conjecture does. On the other hand, Rickard’s conjecture rules out stable constant Jordan type [3] [2] for p ≥ 5 whereas Suslin’s conjecture does not. So neither of these conjectures implies the other.

At the moment, we have no plausible conjecture which precisely characterises which set of stable constant Jordan types we expect to occur, even in the case r = 2.

We mention one more conjecture which is stated in [81]. This should be compared with Example 5.2.6.

Conjecture 5.15.2. If a module has stable constant Jordan type [2] [1]^{j} then j ≥ r − 1.

In Section 10.4 we prove a weak version of this conjecture, namely that j ≥ r − 2 as long as p is large enough, see Corollary 10.4.2. The proof involves vector bundles on projective space and their Chern numbers, which is our next topic.

6 Vector Bundles on Projective Space

This chapter is by way of introduction to algebraic vector bundles on projective space for modular representation theorists. Background material can be found in the books of Hartshorne [147], Okonek, Schneider and Spindler [202] and Fulton [129]. The literature concerning vector bundles on projective space is extensive; see the references for a selection of relevant papers.

6.1. Definitions and First Properties

Definition 6.1.1. Let 𝒪 be a sheaf of rings on a space X. An 𝒪-module is said to be locally free if X has an open cover {U_{α}} such that ℱ|_{U_{α}} is a free 𝒪|_{U_{α}}-module for all α. If X is connected then the rank of ℱ|_{U_{α}} as a free 𝒪|_{U_{α}}-module does not depend on α, and is called the rank of ℱ.

Definition 6.1.2. If V is a quasi-projective variety then a vector bundle is defined to be a locally free 𝒪_{V} -module which has finite rank on each connected component. A homomorphism of vector bundles ℱ_{1} → ℱ_{2} is just a homomorphism of sheaves from ℱ_{1} to ℱ_{2}.

Definition 6.1.3. The dual of a vector bundle ℱ is the vector bundle ℱ^{∨} = ℋom_{𝒪_{V}} (ℱ, 𝒪_{V}). There are natural isomorphisms

ℱ → ℱ^{∨∨}
ℱ_{1}^{∨} ⊗_{𝒪_{V}} ℱ_{2} → ℋom_{𝒪_{V}} (ℱ_{1}, ℱ_{2}).

In particular we have

Hom_{𝒪_{V}} (ℱ_{1}, ℱ_{2}) ≅ Γ(V, ℱ_{1}^{∨} ⊗_{𝒪_{V}} ℱ_{2}).

Definition 6.1.4. A line bundle on V is a vector bundle of rank one. For example, the twists 𝒪_{v}(i) of the structure sheaf are line bundles on V. Under tensor product − ⊗_{𝒪_{V}} −, isomorphism classes of vector bundles on V form an abelian group called the Picard group Pic(V).

The following definitions are analogous to the corresponding definition for modules.

Definition 6.1.5. A non-zero vector bundle ℱ on X is said to be decomposable if it can be written as ℱ_{1} ⊕ ℱ_{2} with ℱ_{1} and ℱ_{2} non-zero vector bundles. Otherwise it is said to be indecomposable.

Theorem 6.1.6 (Atiyah [14]). The Krull–Schmidt theorem holds for vector bundles over a connected complete algebraic variety (for example projective space). □

Definition 6.1.7. A non-zero vector bundle ℱ on X is said to be irreducible if given a short exact sequence of vector bundles

0 → ℱ′ → ℱ → ℱ′′ → 0,

either ℱ′ = 0 or ℱ′′ = 0.

However, the following definition is not the analogue of the corresponding module theoretic notion of simplicity; rather, it is the analogue of the conclusion of Schur’s lemma.

Definition 6.1.8. A non-zero vector bundle ℱ on X is said to be simple if every endomorphism of ℱ is a scalar multiple of the identity. In other words, the natural map

k → End_{X}(ℱ) ≅ H^{0}(X, ℱ^{∨} ⊗ ℱ)

is an isomorphism. This is equivalent to the statement that

dim_{k} H^{0}(X, ℱ^{∨} ⊗ ℱ) = 1.

Lemma 6.1.9. For j ∈ ℤ, a vector bundle ℱ is simple if and only if ℱ(j) is simple.

Proof. We have

ℱ(j)^{∨} ⊗ ℱ(j) ≅ (ℱ^{∨} ⊗ 𝒪(−j)) ⊗ (ℱ ⊗ 𝒪(j)) ≅ ℱ^{∨} ⊗ ℱ.  □

Remark 6.1.10. The notion of algebraic vector bundle behaves quite differently from the notion of topological vector bundle. For example, every short exact sequence of topological vector bundles on a paracompact space splits, whereas the same is not at all true of algebraic vector bundles on a variety. Every complex algebraic vector bundle induces a topological one, but there are topological vector bundles which do not come from algebraic ones, and homomorphisms of topological ones not coming from algebraic homomorphisms.

A theorem of Serre [225] states that if k = ℂ, the field of complex numbers, then algebraic vector bundles over a complex projective variety are essentially the same as holomorphic vector bundles, regarding the variety as a complex manifold. This theorem usually goes under the name GAGA, after the title of the paper. An extended discussion is contained in the book of Neeman [198].

6.2. Tests for Vector Bundles

The following test for when a coherent 𝒪_{V} -module is a vector bundle comes from Exercise 5.8 in Section II.5 of Hartshorne [147] and Proposition 4.11 of Friedlander–Pevtsova [127]. Since we shall use it extensively, we provide a proof.

Theorem 6.2.1. Let X be a reduced connected Noetherian scheme.

(a) If ℱ is a coherent sheaf of 𝒪_{X}-modules, then the following are equivalent.

(1) ℱ is a vector bundle.

(2) The dimension of the fibre dim_{k(x)} ℱ_{x} ⊗_{𝒪_{x}} k(x) is constant for x ∈ X.

(b) If f: ℱ → ℱ′ is a map of vector bundles on X then the following are equivalent.

(1) The cokernel of f is a vector bundle of rank equal to rank ℱ′ − n.

(2) The induced map of fibres

f̄: ℱ_{x} ⊗_{𝒪_{x}} k(x) → ℱ′_{x} ⊗_{𝒪_{x}} k(x)

has constant rank n for x ∈ X.

If these hold then the image of f is a vector bundle of rank n.

(c) If f: ℱ → ℱ′ is a map of vector bundles on X then the following are equivalent.

(1) f is a locally split injection.

(2) The induced map of fibres

f̄: ℱ_{x} ⊗_{𝒪_{x}} k(x) → ℱ′_{x} ⊗_{𝒪_{x}} k(x)

is injective for all x ∈ X.

Proof. Consider the function ϕ: X → ℤ given by ϕ(x) = dim_{k(x)} ℱ_{x} ⊗_{𝒪_{x}} k(x). We shall show that for any coherent sheaf ℱ, the function ϕ is upper semicontinuous, i.e., that for any n ∈ ℤ the set X_{≥n} = {x ∈ X | ϕ(x) ≥ n} is closed, or equivalently, the complementary set X_{<n} = {x ∈ X | ϕ(x) < n} is open.

Since a set is open if and only if its intersection with each set from an open cover is open, we may assume that X is an affine variety, X = Spec R with R Noetherian and ℱ = M^{∼} for some finitely generated R-module M, see Theorem 3.8.5. If x ∈ X_{<n} corresponds to a prime ideal 𝔭 of R, then k(x) is the field of fractions k(𝔭) of R/𝔭 and ℱ_{x} ⊗_{𝒪_{x}} k(x) = M_{𝔭}/𝔭M_{𝔭}. Let m = dim_{k(𝔭)}(M_{𝔭}/𝔭M_{𝔭}), so m < n and let ã_{1}, …, ã_{m} ∈ M_{𝔭} be elements such that ā_{1}, …, ā_{m} form a k(𝔭)-basis for M_{𝔭}/𝔭M_{𝔭}. By Nakayama’s lemma, ã_{1}, …, ã_{m} generate M_{𝔭}. Clearing denominators, we may assume that ã_{1}, …, ã_{m} are the images of a_{1}, …, a_{m} ∈ M under the map M → M_{𝔭}.

Let b_{1}, …, b_{t} be a set of generators for M. The image of each b_{i} in M_{𝔭} is a linear combination of the ã_{j} with coefficients in R_{𝔭}. There are only a finite number of denominators in these equations, and so there exists a single element r ∈ R with r ∉ 𝔭 such that each rb_{i} is a linear combination of the a_{j} with coefficients in R. Consider the open set U of X consisting of those primes 𝔮 not containing r. Then for 𝔮 ∈ U the images of b_{i} are linear combinations of the a_{j} with coefficients in R_{𝔮}. So M_{𝔮} is generated by at most m elements. It follows that U ⊆ X_{<n}. So we have shown that each point 𝔭 ∈ X_{<n} has an open neighbourhood U ⊆ X_{<n}, and it follows that X_{<n} is open.

(a) (1) ⇒ (2): If ℱ is a vector bundle then for each x ∈ X there is an open neighbourhood of x on which ℱ is free, and therefore on which ϕ is constant. It follows that for each n ∈ ℤ the set X_{≥n} is open. We already showed that it is closed, so since X is connected it follows that ϕ is constant on X.

(2) ⇒ (1): Assume that ϕ is constant on X, say with value n. We must prove that each point x ∈ X has an open neighbourhood on which ℱ is free of finite rank. So again we may assume that X = Spec R is affine, ℱ = M^{∼} for some finitely generated R-module M, and x corresponds to some prime ideal 𝔭 of R. As before, choose a_{1}, …, a_{n} ∈ M such that ā_{1}, …, ā_{n} generate M_{𝔭}/𝔭M_{𝔭}. Let b_{1}, …, b_{t} be a set of generators for M. Then there exists r ∈ R with r ∉ 𝔭 such that the image of each rb_{i} in M_{𝔭} is a linear combination of the images of the a_{j}. Writing R_{r} for R with r inverted, the images of b_{1}, …, b_{t} in M_{r} are linear combinations of the images of a_{1}, …, a_{n}. It follows that M_{r} is generated by the images of a_{1}, …, a_{n}. This gives us a surjective module homomorphism from a free module of rank n over R_{r} to M_{r}. Write K for the kernel, so that we have a short exact sequence

0 → K → R_{r}^{n} → M_{r} → 0.

Localisation is exact, so for each prime ideal 𝔮 not containing r we have an exact sequence

0 → K_{𝔮} → R_{𝔮}^{n} → M_{𝔮} → 0.

Since ϕ is constant, M_{𝔮}/𝔮M_{𝔮} has dimension n over R_{𝔮}/𝔮R_{𝔮} and so K_{𝔮} ⊆ 𝔮R_{𝔮}^{n}. This implies that the coordinates of an element of K are in 𝔮. Since X is reduced, R_{r} has no nilpotent elements, so the intersection of the prime ideals of R_{r} is zero. Thus K = 0 and ℱ is free on the open subset of X defined by r.

(b) Consider the exact sequence

(6.2.1)  ℱ → ℱ′ → Coker f → 0.

Exactness of stalks and right exactness of tensor product shows that

ℱ_{x} ⊗_{𝒪_{x}} k(x) → ℱ′_{x} ⊗_{𝒪_{x}} k(x) → (Coker f)_{x} ⊗_{𝒪_{x}} k(x) → 0

is also exact. By part (a), Coker f is a vector bundle of rank equal to rank ℱ′ − n if and only if (Coker f)_{x} ⊗_{𝒪_{x}} k(x) has constant dimension equal to rank ℱ′ − n for x ∈ X, namely if and only if ℱ_{x} ⊗_{𝒪_{x}} k(x) → ℱ′ ⊗_{𝒪_{x}} k(x) has constant rank n.

If Coker f is a vector bundle then the sequence (6.2.1) is locally split and so the image of f is also a vector bundle.

(c) It is clear that (1) implies (2). To see that (2) implies (1), let f: ℱ → ℱ′ be such that the induced map of fibres is injective for all x ∈ X. Then by part (b), the cokernel is a vector bundle, the sequence ℱ ⟶^{f} ℱ′ → Coker f → 0 is locally split, and so the image of f is a subbundle of ℱ′. Thus

0 → Ker f → ℱ → Im f → 0

is locally split, and (Ker f)_{x} ⊗_{𝒪_{x}} k(x) = 0 for all x ∈ X. So (Ker f)_{x} = 0 for all x ∈ X and hence Ker f = 0. □

Lemma 6.2.2. Let

0 → ℱ ⟶^{α} ℱ′ ⟶^{β} ℱ′′ → 0

be a short exact sequence of sheaves. If ℱ′′ is a vector bundle then it follows that ℱ is a vector bundle if and only if ℱ′ is a vector bundle.

Proof. Since ℱ′′ is locally free, the sequence is locally split. So ℱ′ is locally a direct sum of ℱ and ℱ′′. Since locally projective is the same as locally free, ℱ is a vector bundle if and only if ℱ′ is a vector bundle. □

Remark 6.2.3. In the lemma, it is not true that if ℱ and ℱ′ are vector bundles then so is ℱ′′. For example if x is a non-zero function of degree one on projective space then we have a short exact sequence

0 → 𝒪(−1) ⟶^{x} 𝒪 → 𝒪/x → 0

and 𝒪/x is not a vector bundle.

Definition 6.2.4. We shall say that the image of α is a subbundle of ℱ′ if ℱ and ℱ′ are bundles and the sequence is locally split. Under this condition, ℱ′′ is also a vector bundle, called the quotient bundle.

Under the assumptions of Theorem 6.2.1, this condition may be checked using part (b) of that theorem.

We shall also need the following extended version of Lemma 6.2.2 in the proof of Theorem 8.4.11.

Proposition 6.2.5. Let ℱ be a coherent sheaf of 𝒪_{X}-modules with a filtration

0 = ℱ_{0} ⊆ ℱ_{1} ⊆ ⋯ ⊆ ℱ_{n} = ℱ.

If the quotients ℱ_{i}/ℱ_{i−1} are vector bundles for 1 ≤ i ≤ n then ℱ_{i}/ℱ_{j} is a vector bundle for 0 ≤ j < i ≤ n. In particular, ℱ is a vector bundle.

Proof. The proof is by induction on n. If n < 2 there is nothing to prove. If n = 2 it says that for the short exact sequence

0 → ℱ_{1} → ℱ → ℱ/ℱ_{1} → 0,

if both end terms are locally free then so is the middle. This follows from Lemma 6.2.2.

For n > 2 the inductive hypothesis shows that ℱ_{i}/ℱ_{j} is locally free for 0 ≤ j < i ≤ n − 1. Now examine the sequence

0 → ℱ_{n−1}/ℱ_{j} → ℱ/ℱ_{j} → ℱ/ℱ_{n−1} → 0

and apply the case n = 2, to deduce the remaining cases. □

6.3. Vector Bundles on Projective Space

Let k be an algebraically closed field, and let ℙ^{r−1} = Proj k[Y_{1}, …, Y_{r}], projective space of dimension r − 1 over k. We use r − 1 to denote the dimension because in later chapters r will be the rank of a finite elementary abelian p-group E, and k will be an algebraically closed field of characteristic p.

Theorem 6.3.1. The only line bundles on ℙ^{r−1} are the twists of the structure sheaf 𝒪(i). Thus Pic(ℙ^{r−1}) ≅ ℤ.

Vector bundles on ℙ^{1} were classified by Grothendieck [140].

Theorem 6.3.2. Every vector bundle on ℙ^{1} is a direct sum of line bundles

𝒪(i_{1}) ⊕ ⋯ ⊕ 𝒪(i_{d}).

The integers i_{1}, …, i_{d} are uniquely determined up to reordering.

If r ≥ 3, it is not difficult to produce indecomposable vector bundles on ℙ^{r−1} of every rank at least r − 2; see, for example, the examples of Tango described in Section 6.8. But it turns out to be much harder to produce any of smaller dimension. So there is a great deal of interest in vector bundles of rank d satisfying 2 ≤ d ≤ r − 3. Here is a complete list of the known vector bundles with such ranks.

(1) The Horrocks–Mumford bundle of rank 2 on ℙ^{4} with 15, 000 symmetries [161].

(2) Horrocks’ Parent bundle of rank 3 on ℙ^{5} [160].

(3) The Tango bundle of rank 2 on ℙ^{5} in characteristic two (Tango [235]; see also [116, 170]).

(4) Some further rank 2 vector bundles on ℙ^{4} and rank 3 bundles on ℙ^{5} constructed by Kumar [182], Kumar, Peterson and Rao [183], in positive characteristic only.

(5) Bundles obtained from the above by twisting and by pulling back through self-maps of projective space.

[figure: •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; → 𝒪(n); •; •; ↑ Lots; ℙ^{1}; ℙ^{2}; ℙ^{3}; ℙ^{4}; ℙ^{5}; ℙ^{6}; 1; 2; 3; 4; 5; 6; ?; rank]

Remark 6.3.3. There are no known restrictions on the possible ranks of indecomposable bundles on ℙ^{r−1} for r ≥ 3. For example, it is not known whether there exists an indecomposable vector bundle of rank 2 on ℙ^{5} in characteristic not equal to 2, or on ℙ^{r−1} (r ≥ 7) in any characteristic.

Remark 6.3.4. Sato [222] and Tyurin [238] (reprinted in [239]) proved that a vector bundle on ℙ^{r−1} which extends to a vector bundle on ℙ^{r+n−1} for all n ≥ 0 has to be a direct sum of line bundles. A short proof of this “Babylonian tower theorem” using ideas from the proof of Kempf’s splitting theorem [177] may be found in Coandă and Trautmann [93]. A closely related theorem on stably extendable bundles may be found in Coandă [92].

We return to the subject of Babylonian towers in the context of modules of constant Jordan type in Section 10.11.

In the next few sections, we discuss some of the vector bundles appearing in this section. The proofs of simplicity (and hence indecomposability) depend on cohomological computations from Sections 6.9 and 6.10, and so this will not happen until Section 6.11. The Horrocks–Mumford, Horrocks Parent, and Tango bundles will appear in Chapter 9.

6.4. The Tangent Bundle and the Euler Sequence

The first example of a vector bundle on ℙ^{r−1} which is not a sum of line bundles is the tangent bundle or sheaf of derivations 𝒯.

Definition 6.4.1. The tangent bundle on ℙ^{r−1} is defined as the cokernel of the map 𝒪 → 𝒪(1)^{⊕r} whose ith coordinate is given by multiplication by Y_{i}.

Thus we have a short exact sequence

(6.4.1)  0 → 𝒪 → 𝒪(1)^{⊕r} → 𝒯 → 0.

On the affine patch given by Y_{i} ≠ 0, the r − 1 local coordinate functions are Y_{j}/Y_{i} with j ≠ i and the map is split by (f_{1}, …, f_{r}) ↦ f_{i}/Y_{i}. Thus the sequence is locally split, and so 𝒯 is a vector bundle (cf. Definition 6.2.4).

We identify the fibre of 𝒯 at a point a ∈ ℙ^{r−1} with the tangent space T_{ℙ^{r−1},a} as follows. If a is in the affine patch Y_{i} ≠ 0 then an element g ∈ 𝔪_{ℙ^{r−1},a} is a polynomial in the Y_{j}/Y_{i} with j ≠ i, and so g is a rational function of Y_{1}, …, Y_{r} of degree zero. Given f_{1}, …, f_{r} of degree one, we evaluate ∑_{i} \frac{∂g}{∂Y_{i}} at a. This expression has degree zero, so its value at a is well defined. It vanishes on products, so it only depends on the image of g in 𝔪_{ℙ^{r−1},a}/𝔪_{ℙ^{r−1},a}^{2}. It also only depends on the value of the f_{i} at a. We have ∑_{i} Y_{i} \frac{∂g}{∂Y_{i}} = 0 so it is well defined on the fibre at a of the cokernel of 𝒪 → 𝒪(1)^{⊕r}. It is easy to check that this is then a perfect pairing, so it identifies the fibre of 𝒯 at a with the vector space dual of 𝔪_{ℙ^{r−1},a}/𝔪_{ℙ^{r−1},a}^{2}, which is the tangent space T_{ℙ^{r−1},a}.

Definition 6.4.2. The Euler sequence is the short exact sequence obtained by tensoring the sequence (6.4.1) with 𝒪(−1):

(6.4.2)  0 → 𝒪(−1) → 𝒪^{⊕r} → 𝒯(−1) → 0.

Definition 6.4.3. The cotangent bundle or sheaf of differentials is the dual of the tangent bundle

Ω = Ω_{ℙ^{r−1}} = 𝒯^{∨}.

It is the kernel of the map 𝒪(−1)^{⊕r} → 𝒪 whose ith coordinate is given by multiplication by Y_{i}. The dual of the sequence (6.4.1) is the short exact sequence

(6.4.3)  0 → Ω → 𝒪(−1)^{⊕r} → 𝒪 → 0.

6.5. Homogeneity and Uniformity

The tangent and cotangent bundles are examples of homogeneous bundles on ℙ^{r−1}. These are the bundles that are invariant under the action of the general linear group on projective space, as we now explain.

The general linear group GL(r, k) acts on the vector space spanned by Y_{1}, …, Y_{r} and hence on the graded polynomial ring k[Y_{1}, …, Y_{r}] and the projective space ℙ^{r−1}. The scalar matrices k^{×} act trivially on ℙ^{r−1}, so we get an action of P GL(r, k) = GL(r, k)/k^{×}. Since k is algebraically closed, every element has an rth root. So GL(r, k) is generated by the special linear group SL(r, k) and k^{×}, and we have

P GL(r, k) ≅ P SL(r, k) = SL(r, k)/(k^{×} ∩ SL(r, k)).

Definition 6.5.1. A vector bundle ℱ on ℙ^{r−1} is said to be homogeneous if for every ρ ∈ GL(r, k) (or equivalently for every ρ ∈ SL(r, k)) we have ρ^{∗}(ℱ) ≅ ℱ.

Lemma 6.5.2. We have H^{2}(SL(r, k), k^{×}) = 0.

Proof. See Steinberg [229]. □

Proposition 6.5.3. If ℱ is a simple homogeneous vector bundle then the isomorphisms i_{ρ}: ρ^{∗}(ℱ) ≅ ℱ can be chosen so that if ρ, ρ′ ∈ SL(r, k) then the following diagram commutes:

[figure: ℱ; i_{ρ}; i_{ρρ′}; ρ^{∗}(ℱ); i_{ρ′}; (ρρ′)^{∗}(ℱ); (ρ′)^{∗}ρ^{∗}(ℱ)]

Proof. Since ℱ is simple, if we’re given a system of isomorphisms i_{ρ} then there are scalars λ_{ρ,ρ′} such that i_{ρρ′} = λ_{ρ,ρ′}i_{ρ′}i_{ρ}. These form a 2-cocycle on SL(r, k) for the bar resolution with coefficients in k^{×}. By Lemma 6.5.2, this is a coboundary, so there are scalars μ_{ρ} such that λ_{ρ,ρ′} = μ_{ρ}μ_{ρ′}μ_{ρρ′}^{−1}. Then replacing i_{ρ} by μ_{ρ}i_{ρ} makes the diagram commute. □

Given a vector bundle ℱ on ℙ^{r−1}, for each line L ⊆ ℙ^{r−1} (i.e., linearly embedded copy of ℙ^{1}) we may restrict ℱ to L and decompose as a sum of line bundles

ℱ|L ≅ ⨁_{i=1}^{s} 𝒪_{L}(a_{i}).

The unordered set of a_{i} with multiplicities is called the splitting type of ℱ on L.

Definition 6.5.4. A vector bundle ℱ on ℙ^{r−1} is said to be uniform if the splitting type is the same on all lines.

Proposition 6.5.5. If ℱ is a homogeneous bundle on ℙ^{r−1} then ℱ is uniform.

Proof. The group GL(r, k) is transitive on lines in ℙ^{r−1}. □

Remark 6.5.6. Homogeneous vector bundles on ℙ^{2} are studied in the paper of Gorodentsev and Rudakov [137], in terms of solutions of the Markov equation

x^{2} + y^{2} + z^{2} = 3xyz,

see also Rudakov [221].

More generally, homogeneous vector bundles on ℙ^{r−1} are discussed in detail by Bondal and Kapranov [56], Hille [154] and Ottaviani and Rubei [205], where they are related to Higgs bundles and to representations of certain quivers with relations.

6.6. Monads and Subquotients

In the literature, vector bundles are often described using monads, which are nothing more than a convenient method of describing subquotients of a bundle.

Definition 6.6.1. A monad over ℙ^{r−1} (or over any other variety) is a complex

0 → ℱ_{1} ⟶^{α} ℱ_{2} ⟶^{β} ℱ_{3} → 0

of vector bundles such that α is injective, β is surjective, βα = 0, and the image of α is a subbundle of ℱ_{2}.

A monad gives rise to a commutative diagram with exact rows and columns as follows:

[figure: 0; 0; 0; ℱ_{1}; Ker β; ℱ; 0; 0; ℱ_{1}; α; ℱ_{2}; Coker α; 0; ℱ_{3}; ℱ_{3}; 0; 0]

Lemma 6.6.2. The subquotient ℱ = Ker β/Im α is a vector bundle.

Proof. Since by assumption Im α is a subbundle of ℱ_{2}, it follows that Coker α is a vector bundle. Applying Lemma 6.2.2 to the right-hand column of the display, we see that ℱ is a vector bundle. □

Definition 6.6.3. The vector bundle ℱ = Ker β/Im α is called the cohomology of the monad.

We shall see examples of monads in the next section, as well as in Sections 9.8 and 9.9.

6.7. The Null Correlation Bundle

Our next example is the null correlation bundle ℱ_{N}. This is an example of an indecomposable bundle of rank r − 2 on ℙ^{r−1} for r even.

We form a monad

(6.7.1)  0 → 𝒪(−1) → 𝒪^{⊕r} → 𝒪(1) → 0

where the first map is the same as in the Euler sequence, while the second map is given by multiplication by (Y_{r}, −Y_{r−1}, …, Y_{2}, −Y_{1}). It is easy to see by direct computation that the composite is zero:

(Y_{r}, −Y_{r−1}, …, Y_{2}, −Y_{1})(Y_{1}, Y_{2}, …, Y_{r−1}, Y_{r})^{t} = 0.

So we do indeed have a monad.

The second map above may be interpreted as being obtained from the first by means of the symplectic form on 𝔸^{r} defined by

⟨X_{i}, X_{j}⟩ = \begin{cases}(−1)^{i} & if i + j = r + 1 \\ 0 & otherwise. \end{cases}

Any choice of non-degenerate symplectic form gives a monad.

The null correlation bundle ℱ_{N} is defined to be the cohomology of this monad:

[figure: 0; 0; 0; 𝒪(−1); Ω(1); ℱ_{N}; 0; 0; 𝒪(−1); 𝒪^{⊕r}; 𝒯(−1); 0.; 𝒪(1); 𝒪(1); 0; 0]

It is clear from the construction that ℱ_{N} is a self-dual bundle:

ℱ_{N}^{∨} ≅ ℱ_{N}.

6.8. The Examples of Tango

In the last section we saw examples of indecomposable bundles of rank r − 2 on ℙ^{r−1} when r is even. Tango [234] constructed indecomposable bundles of rank r−2 on ℙ^{r−1} for all values of r. We present here a simplified version of his construction. Let V be a vector space of dimension r, and consider the map V × V → Λ^{2}V given by (x, y) ↦ x ∧ y. The image is the closed homogeneous subvariety corresponding to the cone on the Grassmannian of 2dimensional subspaces of V, and hence has dimension 2r − 3 (one more than the dimension of the Grassmannian). So there exists a linear subspace W of Λ^{2}V of codimension 2r − 3 whose intersection with the image is equal to the origin. In other words, W contains no non-zero elements of the form x ∧ y.

Now look at the beginning of the Koszul complex on ℙ^{r−1}, suitably twisted, and let ℰ be the cokernel:

0 → 𝒪(−2) → 𝒪(−1)^{⊕r} → 𝒪^{⊕\binom{r}{2}} → ℰ → 0.

Thus

ℰ ≅ (Λ^{2}𝒯)(−2) ≅ Ω^{r−2}(r − 2)

of rank

\binom{r}{2} − (r − 1) = \binom{r − 1}{2}.

Then thinking of 𝒪^{⊕\binom{r}{2}} as the bundle Λ^{2}(V) ⊗_{k} 𝒪, the subspace W of Λ^{2}(V) defines a trivial subbundle W ⊗_{k} 𝒪 which injects into ℰ. Letting ℱ_{W} be the quotient of ℰ by the image of this subbundle, we have a diagram with exact rows and columns

(6.8.1)  [figure: 0; 0; 𝒪^{⊕(\binom{r}{2}−(2r−3))}; 𝒪^{⊕(\binom{r}{2}−(2r−3))}; 0; 𝒪(−2); 𝒪(−1)^{⊕r}; 𝒪^{⊕\binom{r}{2}}; ℰ; 0; 0; 𝒪(−2); 𝒪(−1)^{⊕r}; 𝒪^{⊕(2r−3)}; ℱ_{W}; 0.; 𝒯(−2); 0; 0; 0; 0]

This bundle ℱ_{W} is the example of Tango.

Remark 6.8.1. The construction depends on the choice of a subspace W of Λ^{2}V not containing any non-zero wedges, and so we obtain a large family of examples this way. The symmetry of the bundle is the same as the symmetry of the subspace W in Λ^{2}V.

Cascini [84] has shown that if we take V to be the (r − 1)st symmetric power of a vector space U of dimension two then for p large, decomposing Λ^{2}V as

Λ^{2}V = S^{2(r−2)}U ⊕ S^{2(r−4)}U ⊕ S^{2(r−6)}U ⊕ ⋯ ⊕ S^{2}U

we may take

W = S^{2(r−4)}U ⊕ S^{2(r−6)}U ⊕ ⋯ ⊕ S^{2}U

of codimension 2r −3. Thus there are examples of Tango’s construction with SL(2, k) acting as symmetries if p is large compared with r.

6.9. Cohomology of Projective Space

We shall need to make use of the cohomology of ℙ^{r−1} with coefficients in twists of the structure sheaf in order to understand vector bundles. The following is Theorem III.5.1 in Hartshorne [147].

Theorem 6.9.1. Let R = k[Y_{1}, …, Y_{r}] and ℙ^{r−1} = Proj R.

(i) The natural map R → ⨁_{j∈ℤ} H^{0}(ℙ^{r−1}, 𝒪(j)) is an isomorphism of graded R-modules. (ii) H^{i}(ℙ^{r−1}, 𝒪(j)) = 0 for 0 < i < r − 1 and all j ∈ ℤ. (iii) H^{r−1}(ℙ^{r−1}, 𝒪(−r − j)) is isomorphic to the k-vector space dual of H^{0}(ℙ^{r−1}, 𝒪(j)).

Corollary 6.9.2. The dimension of H^{i}(ℙ^{r−1}, 𝒪(j)) is equal to \binom{r+j−1}{j} if i = 0 and j ≥ 0, \binom{−j−1}{−r−j} if i = r − 1 and j ≤ −r, and zero otherwise.

We illustrate this with a diagram for r = 4:

j↑ 14 10 14 10 . .. .. . i →

6.10. Differential Forms and Bott’s Theorem

The exterior mth power of the cotangent bundle Ω is the bundle of differential m-forms

Ω^{m} = Λ^{m}Ω.

The exterior powers of the sequence (6.4.3) give sequences

(6.10.1)  0 → Ω^{m} → 𝒪(−m)^{⊕\binom{r}{m}} → Ω^{m−1} → 0.

In particular for m = r − 1, the top exterior power is the canonical bundle and we obtain

(6.10.2) ω = Ω^{r−1} ≅ 𝒪(−r), Ω^{r−2} ≅ 𝒯(−r).

The differential forms are the kernels in the (exact) Koszul complex:

[figure: 0; 𝒪(−r); 𝒪(−r + 1)^{⊕r}; 𝒪(−r + 2)^{⊕\binom{r}{r−2}}; 𝒪(−3)^{⊕\binom{r}{3}}; 𝒪(−2)^{⊕\binom{r}{2}}; 𝒪(−1)^{⊕r}; 𝒪; 0; ω; _{Ω_{r−2}}; _{Ω_{2}}; Ω]

The r-fold shift of this complex is isomorphic to its dual:

[figure: 0; 𝒪; 𝒪(1)^{⊕r}; 𝒪(2)^{⊕\binom{r}{2}}; 𝒪(3)^{⊕\binom{r}{3}}; 𝒪(r − 2)^{⊕\binom{r}{r−2}}; 𝒪(r − 1)^{⊕r}; 𝒪(r); 0; 𝒯; Λ^{2}𝒯; Λ^{r−2}𝒯]

and we have

(6.10.3)  (Ω^{m})^{∨} ≅ Ω^{r−1−m}(r) ≅ Λ^{m}𝒯.

Theorem 6.10.1 (Bott’s formula).

dim_{k} H^{i}(ℙ^{r−1}, Ω^{m}(j)) = \begin{cases}\binom{j + r − m − 1}{j}\binom{j − 1}{m} & for i = 0, 0 ≤ m < r, j > m \\ 1 & for j = 0, 0 ≤ m = i < r \\ \binom{m − j}{−j}\binom{−j − 1}{r − m − 1} & for i = r − 1, 0 ≤ m < r, j ≤ m − r \\ 0 & otherwise. \end{cases}

Proof. Bott [64] proved a more general statement in characteristic 0 (see also [202]), but this theorem also holds in any characteristic. We prove this by induction using the long exact cohomology sequences of the short exact sequences of sheaves (6.4.3) and (6.10.1). The cases m = 0 and m = r − 1 come from Corollary 6.9.2 and the isomorphism Ω^{r−1} ≅ 𝒪(−r). By downwards induction on m we have H^{i}(ℙ^{r−1}, Ω^{m}(j)) = 0 for i > 0 and j > 0. It follows that we have short exact sequences

0 → H^{0}(ℙ^{r−1}, Ω^{m}(j)) → H^{0}(ℙ^{r−1}, 𝒪(j − m)^{⊕\binom{r}{m}}) → H^{0}(ℙ^{r−1}, Ω^{m−1}(j)) → 0

for j > 0, which inductively gives the dimension of H^{0}(ℙ^{r−1}, Ω^{m}(j)) for j > 0.

Dually, by (upwards) induction on m we have H^{i}(ℙ^{r−1}, Ω^{m}(j)) = 0 for i < r − 1 and j < 0. So we have short exact sequences

0 → H^{r−1}(ℙ^{r−1}, Ω^{m}(j)) → H^{r−1}(ℙ^{r−1}, 𝒪(j − m)^{⊕\binom{r}{m}}) → H^{r−1}(ℙ^{r−1}, Ω^{m−1}(j)) → 0

for j < 0, which inductively gives the dimension of H^{r−1}(ℙ^{r−1}, Ω^{m}(j)) for j < 0.

Finally, in the remaining case j = 0, it again follows by induction on m that the connecting homomorphisms give isomorphisms

H^{i}(ℙ^{r−1}, Ω^{m}) → H^{i+1}(ℙ^{r−1}, Ω^{m+1})

so that these groups are isomorphic to k when i = m and zero otherwise. □

We illustrate Bott’s formula with a diagram for r = 4:

i → j↑ −5 −4 −3 −2 −1 0 1 2 3 4 5 1 4 10 20 35 56  1 4 0 1 2 3 m = 0  0 1 2 3 6 20 45 84 1 4 15 36 m = 1  0 1 2 3 4 15 36 1 6 20 45 84 m = 2  0 1 2 3 1 4  1 4 10 20 35 56 m = 3

6.11. Simplicity

Next, we use Bott’s formula to prove the simplicity of the bundles Ω^{m}(j) (these include the tangent and cotangent bundles), the null correlation bundle and the examples of Tango.

Theorem 6.11.1. H^{0}(ℙ^{r−1}, Ω ⊗ 𝒯) ≅ k.

Proof. Tensor Ω with the sequence (6.4.1) to get a sequence

0 → Ω → Ω(1)^{⊕r} → Ω ⊗ 𝒯 → 0.

Taking cohomology, we obtain an exact sequence

⋯ → H^{0}(ℙ^{r−1}, Ω(1)^{⊕r}) → H^{0}(ℙ^{r−1}, Ω ⊗ 𝒯) → H^{1}(ℙ^{r−1}, Ω) → H^{1}(ℙ^{r−1}, Ω(1)^{⊕r}) → ⋯ By Theorem 6.10.1, we have

H^{0}(ℙ^{r−1}, Ω(1)) = H^{1}(ℙ^{r−1}, Ω(1)) = 0,  H^{1}(ℙ^{r−1}, Ω) ≅ k.  □

Corollary 6.11.2. The tangent bundle 𝒯 and the cotangent bundle Ω of ℙ^{r−1} are simple.

Proof. Using the theorem, we have

End_{ℙ^{r−1}}(𝒯) ≅ H^{0}(ℙ^{r−1}, 𝒯^{∨} ⊗ 𝒯) = H^{0}(ℙ^{r−1}, Ω ⊗ 𝒯) ≅ k.

Similarly End_{ℙ^{r−1}}(Ω) ≅ H^{0}(ℙ^{r−1}, Ω ⊗ 𝒯) ≅ k. □

More generally, we have the following.

Proposition 6.11.3. The bundles Ω^{m}(j) are simple for m ≥ 0 and j ∈ ℤ.

Proof. By Lemma 6.1.9, we may assume that j = 0. The case m = 0 is clear, and the case m = 1 is dealt with in Corollary 6.11.2. The general case is proved by induction on m using the exact sequence (6.10.1). Namely, we tensor (Ω^{m−1})^{∨} with the sequence (6.10.1), take the long exact sequence in cohomology and use Bott’s formula (Theorem 6.10.1) to deduce that

H^{0}(ℙ^{r−1}, (Ω^{m−1})^{∨} ⊗ Ω^{m−1}) ≅ H^{1}(ℙ^{r−1}, (Ω^{m−1})^{∨} ⊗ Ω^{m}).

Then tensor the dual of the sequence (6.10.1), take the long exact sequence in cohomology and use Bott’s formula to deduce that

H^{0}(ℙ^{r−1}, (Ω^{m})^{∨} ⊗ Ω^{m}) ≅ H^{1}(ℙ^{r−1}, (Ω^{m−1})^{∨} ⊗ Ω^{m}).

Thus by the inductive hypothesis

H^{0}(ℙ^{r−1}, (Ω^{m})^{∨} ⊗ Ω^{m}) ≅ H^{0}(ℙ^{r−1}, (Ω^{m−1})^{∨} ⊗ Ω^{m−1}) ≅ k.  □

Theorem 6.11.4. The null correlation bundle is simple.

Proof. Tensor ℱ_{N} with the right hand column of the above diagram to get an exact sequence

0 → ℱ_{N} ⊗ ℱ_{N} → ℱ_{N} ⊗ 𝒯(−1) → ℱ_{N}(1) → 0

which shows that

(6.11.1)  dim_{k} H^{0}(ℙ^{r−1}, ℱ_{N} ⊗ ℱ_{N}) ≤ dim_{k} H^{0}(ℙ^{r−1}, ℱ_{N} ⊗ 𝒯(−1)).

On the other hand, tensoring the top row with 𝒯(−1) gives an exact sequence

0 → 𝒯(−2) → Ω ⊗ 𝒯 → ℱ_{N} ⊗ 𝒯(−1) → 0.

The long exact sequence in cohomology for this gives

⋯ → H^{0}(ℙ^{r−1}, Ω ⊗ 𝒯) → H^{0}(ℙ^{r−1}, ℱ_{N} ⊗ 𝒯(−1)) → H^{1}(ℙ^{r−1}, 𝒯(−2)) → ⋯

Now by (6.10.2) we have 𝒯(−2) ≅ Ω^{r−2}(r − 2), so that by Bott’s formula (Theorem 6.10.1) we have

H^{1}(ℙ^{r−1}, 𝒯(−2)) = 0.

Thus

(6.11.2)  dim_{k} H^{0}(ℙ^{r−1}, ℱ_{N} ⊗ 𝒯(−1)) ≤ dim_{k} H^{0}(ℙ^{r−1}, Ω ⊗ 𝒯).

Combining Theorem 6.11.1 with (6.11.1) and (6.11.2) and using the self-duality of ℱ_{N}, we get

End_{ℙ^{r−1}}(ℱ_{N}) = H^{0}(ℙ^{r−1}, ℱ_{N}^{∨} ⊗ ℱ_{N}) ≅ H^{0}(ℙ^{r−1}, ℱ_{N} ⊗ ℱ_{N}) ≅ k.  □

Next, we examine the examples of Tango.

Lemma 6.11.5. Suppose that there is a short exact sequence of vector bundles

0 → 𝒪^{⊕m} → 𝒜 → ℬ → 0

with m ≥ 0. If 𝒜 is simple and H^{1}(ℙ^{r−1}, 𝒜^{∨}) = 0 then ℬ is simple.

Proof. Tensor the dual of the given short exact sequence with ℬ to get

0 → ℬ^{∨} ⊗ ℬ → 𝒜^{∨} ⊗ B → ℬ^{⊕m} → 0.

Thus

dim_{k} H^{0}(ℙ^{r−1}, ℬ^{∨} ⊗ ℬ) ≤ dim_{k} H^{0}(ℙ^{r−1}, 𝒜^{∨} ⊗ ℬ).

Similarly, we tensor 𝒜^{∨} with the given short exact sequence to get

0 → (𝒜^{∨})^{⊕m} → 𝒜^{∨} ⊗ 𝒜 → 𝒜^{∨} ⊗ ℬ → 0.

Since H^{1}(ℙ^{r−1}, 𝒜^{∨}) = 0, we obtain

dim_{k} H^{0}(ℙ^{r−1}, 𝒜^{∨} ⊗ ℬ) ≤ dim_{k} H^{0}(ℙ^{r−1}, 𝒜^{∨} ⊗ 𝒜) = 1.

Thus

dim_{k} H^{0}(ℙ^{r−1}, ℬ^{∨} ⊗ ℬ) = 1.  □

Proposition 6.11.6. Tango’s vector bundle ℱ_{W} is simple.

Proof. Apply Lemma 6.11.5 to the last column of diagram (6.8.1), and use the fact that

H^{1}(ℙ^{r−1}, ℰ^{∨}) = H^{1}(ℙ^{r−1}, Ω^{2}(2)) = 0

by Bott’s formula (Theorem 6.10.1). □

Remark 6.11.7. The simplicity of the bundle (Proposition 6.11.6) does not depend on the dimension of W; using a smaller dimensional subspaces produces simple bundles of larger rank.

6.12. Hilbert’s Syzygy Theorem

We shall make repeated use of Hilbert’s Syzygy Theorem, so it seems wise to give a brief exposition here for the sake of the uninitiated. We begin with the Koszul complex.

Let R be a commutative k-algebra and x_{1}, …, x_{n} be a sequence of elements generating an ideal I of R. We say that x_{1}, …, x_{n} is a regular sequence if each x_{i} is not a zero divisor in R/(x_{1}, …, x_{i−1}). The Koszul complex over R for the sequence x_{1}, …, x_{n} is the tensor product over R of the complexes

0 → R ⟶^{x_{i}} R → 0.

For example, if n = 2, this is the complex

0 → R ⟶^{(x_{2},−x_{1})} R^{⊕2} \underrightarrow{(\begin{matrix}x_{1} \\ x_{2} \end{matrix} )} R → 0.

It follows from the fact that x_{1}, …, x_{n} is a regular sequence that the homology of the Koszul complex is concentrated in degree zero, and equal to R/I. Thus the following is an exact sequence:

0 → R → R^{⊕n} → R^{⊕\binom{n}{n−2}} → ⋯ → R^{⊕\binom{n}{2}} → R^{⊕n} → R → R/I → 0.

Now let R = k[Y_{1}, …, Y_{r}] and let

S = R ⊗_{k} R = k[Y_{1}, …, Y_{r}, Y′_{1}, …, Y′_{r}].

Then S-modules are the same as (R, R)-bimodules. Let I be the ideal in S generated by the regular sequence Y_{1} − Y′_{1}, …, Y_{r} − Y′_{r}. Then S/I ≅ R as an (R, R)-bimodule. So the Koszul complex for this regular sequence is a resolution of R as an (R, R)-bimodule, and we have an exact sequence

(6.12.1)  0 → S → S^{⊕r} → ⋯ → S^{⊕\binom{r}{2}} → S^{⊕r} → S → R → 0.

Such a complex is called a Hochschild resolution of R.

Now if M is a left R-module, we may tensor M over R with this Hochschild resolution to obtain a complex whose terms are

S^{⊕\binom{r}{j}} ⊗_{R} M ≅ R^{⊕\binom{r}{j}} ⊗_{k} M.

So we obtain a free resolution of M as a left R-module. This proves the following theorem.

Theorem 6.12.1 (Hilbert’s Syzygy Theorem). Every left module over R = k[Y_{1}, …, Y_{r}] has a free resolution of length at most r over R. □

Corollary 6.12.2. Every finitely generated R-module has a resolution of length at most r over R by finitely generated free modules.

Proof. By Hilbert’s basis theorem, R is a Noetherian ring. So if M is a finitely generated R-module then M has a free resolution by finitely generated R-modules. By the extended Schanuel lemma it follows from the Hilbert Syzygy Theorem that the rth kernel in such a resolution is a finitely generated projective module.

By a theorem of Quillen and Suslin [209, 230] projective R-modules are free. An easier theorem of Serre [226] states that finitely generated projective modules are stably finitely generated free, which is enough here. In all our applications, the modules are graded with respect to the standard grading on the polynomial ring, in which case we can use the much more obvious statement that a direct summand of a free graded module is again free. □

Example 6.12.3. The Koszul complex for the regular sequence Y_{1}, …, Y_{r} in R gives such a resolution for k, considered as an R-module with all Y_{i} acting as zero. So we have an exact sequence:

0 → R → ⋯ → R^{⊕\binom{r}{2}} → R → k → 0.

The analogue for graded modules is as follows. We say that a graded R-module is free if it is isomorphic to a direct sum of shifts of R.

Theorem 6.12.4. Every left graded module over R = k[Y_{1}, …, Y_{r}] has a resolution of length at most r by free graded modules.

Every finitely generated graded R-module has a resolution of length at most r over R by finitely generated free graded modules.

Proof. The Hochschild resolution (6.12.1) for R = k[Y_{1}, …, Y_{r}] is a resolution by graded free bimodules, provided we shift the ith term S^{⊕\binom{r}{i}} so that the generators are in degree i. If M is a graded module then tensoring with the Hochschild resolution gives a resolution by free graded R-modules.

Now when we apply Schanuel’s lemma in the proof of Corollary 6.12.2, the rth kernel is a finitely generated projective graded module. For graded R-modules, it is very easy to show that projective graded modules are free, without recourse to the theorem of Quillen and Suslin, so this part of the proof is easier. □

Corollary 6.12.5. Let 𝒪 be the structure sheaf on ℙ^{r−1}. Every coherent sheaf of 𝒪modules has a resolution of length at most r in which each term is a finite direct sum of line bundles 𝒪(j).

Given a morphism ℱ → ℱ′ of coherent sheaves, the resolutions may be chosen so that it lifts to a morphism of resolutions, unique up to chain homotopy.

Proof. Let ℱ be a coherent sheaf of 𝒪-modules. Then by Theorem 3.9.12, there is a finitely generated graded R-module M such that ℱ ≅ M^{∼}. By Theorem 6.12.4, M has a resolution of the form

0 → F_{r} → ⋯ → F_{1} → F_{0} → M → 0

where the F_{i} are free graded modules. This gives an exact sequence of sheaves

0 → F_{r}^{∼} → ⋯ → F_{1}^{∼} → F_{0}^{∼} → ℱ → 0

where each F_{i}^{∼} is a direct sum of line bundles 𝒪(j).

By the definition of morphisms in the quotient by a Serre subcategory, given a morphism ℱ → ℱ′, we may choose R-modules M and M′ with M^{∼} ≅ ℱ and M^{′∼} ≅ ℱ′, and a morphism of R-modules M → M′ inducing the given morphism of sheaves. The last statement of the corollary now follows from the usual comparison theorem for free resolutions. □

7 Chern Classes

In this chapter we give a purely algebraic development of the theory of Chern classes for coherent sheaves on projective space. Our treatment is inspired by the book of Hirzebruch [156].

7.1. Chern Classes of Graded Modules

Recall from Theorem 3.9.12 that if 𝒪 is the structure sheaf on ℙ^{r−1} then there is an equivalence of categories between coherent sheaves of 𝒪-modules and finitely generated graded modules over R = k[Y_{1}, …, Y_{r}] modulo finite length modules. So finding invariants of coherent sheaves of 𝒪-modules is equivalent to finding invariants of finitely generated graded R-modules that respect the equivalence relation.

We define the Poincaré series (also known as the Hilbert series) of a finitely generated graded R-module M = ⨁_{j∈ℤ} M_{j} to be

p_{M}(t) = ∑_{j∈ℤ} t^{j} dim_{k} M_{j}.

Note that since M is finitely generated, M_{j} = 0 for j sufficiently large negative, and so p_{M}(t) is a Laurent power series in t.

Lemma 7.1.1 (Hilbert, Serre). The Poincaré series of a finitely generated graded Rmodule M takes the form

p_{M}(t) = \frac{f(t)}{(1 − t)^{r}}

where f(t) is a Laurent polynomial.

If M is a finite length module then p_{M}(t) is itself a Laurent polynomial, so that the Laurent polynomial f(t) = (1 − t)^{r}p_{M}(t) is divisible by (1 − t)^{r}.

Proof. See for example Theorem 11.1 of Atiyah and Macdonald [16]. □

Definition 7.1.2. The number f(1), namely the sum of the coefficients of the numerator of the Poincaré series, is called the rank of M. It is not hard to see that this is equal to the dimension of the (ungraded) vector space

k(Y_{1}, …, Y_{r}) ⊗_{k[Y_{1},...,Y_{r}]} M

over the field k(Y_{1}, …, Y_{r}).

Lemma 7.1.3. If 0 → M_{1} → M_{2} → M_{3} → 0 is a short exact sequence of finitely generated graded R-modules then

p_{M_{2}}(t) = p_{M_{1}}(t) + p_{M_{3}}(t).

Definition 7.1.4. The Chow ring of R = k[Y_{1}, …, Y_{r}] is defined to be the truncated polynomial ring A^{∗}(R) = ℤ[h]/(h^{r}).

Remark 7.1.5. The Chow ring [90] A^{∗}(X) of an algebraic variety X is the analogue of the cohomology ring of a topological space. The formal definition involves algebraic cycles and rational equivalence. The multiplication is given by intersection of algebraic cycles in general position. More precise details may be found, for example, in Appendix A of Hartshorne [147], or in Fulton [129]. We do not need this general definition; our purpose is to give a formal algebraic definition of the Chern polynomial of a vector bundle on projective space, as an element of the truncated polynomial ring A^{∗}(ℙ^{r−1}) = A^{∗}(R) = ℤ[h]/(h^{r}).

Definition 7.1.6. If M is a finitely generated graded R-module with Poincaré series

p_{M}(t) = \frac{∑_{j} a_{j}t^{j}}{(1 − t)^{r}}

where the numerator is a Laurent polynomial, then the Chern polynomial of M is defined to be

c(M) = ∏_{j}(1 + jh)^{a_{j}} ∈ A^{∗}(R) = ℤ[h]/(h^{r}).

If we want to emphasise the role of the variable h, we write c(M, h).

The Chern numbers are the coefficients c_{i}(M) ∈ ℤ of powers of h in this polynomial:

c(M) = 1 + c_{1}(M)h + ⋯ + c_{r−1}(M)h^{r−1};

we also use the convention c_{0}(M) = 1.

The Chern character of M is defined to be

Ch(M) = ∑_{j} a_{j}e^{jh} ∈ A_{ℚ}^{∗}(R) = ℚ ⊗_{ℤ} A^{∗}(R) = ℚ[h]/(h^{r}).

From the definition, it follows that the Chern character is determined by the rank and the Chern numbers:

Ch(M) = rank (M) + c_{1}h + \frac{1}{2}(c_{1}^{2} − 2c_{2})h^{2} + \frac{1}{6}(c_{1}^{3} − 3c_{1}c_{2} + 3c_{3})h^{3}
+ \frac{1}{24}(c_{1}^{4} − 4c_{1}^{2}c_{2} + 4c_{1}c_{3} + 2c_{2}^{2} − 4c_{4}) + ⋯

Proposition 7.1.7. If 0 → M_{1} → M_{2} → M_{3} → 0 is a short exact sequence of finitely generated graded R-modules then

(i) c(M_{2}) = c(M_{1})c(M_{3}) as elements of ℤ[h]/(h^{r}); in other words,

c_{j}(M_{2}) = ∑_{i=0}^{j} c_{i}(M_{1})c_{j−i}(M_{3}).

(ii) Ch(M_{2}) = Ch(M_{1}) + Ch(M_{3}) as elements of ℚ[h]/(h^{r}).

Proof. Both statements follow from Lemma 7.1.3. □

7.2. Chern Classes of Coherent Sheaves on ℙ^{r−1}

Recall from Section 3.9 that there is an equivalence of categories between coherent sheaves on ℙ^{r−1} and the quotient category of finitely generated graded modules over k[Y_{1}, …, Y_{r}] modulo finite length modules. So let us regard finitely generated k[Y_{1}, …, Y_{r}]-modules M and M′ as equivalent if for d large enough M_{≥d} and M′_{≥d} are isomorphic. Our goal is to show that equivalent modules have the same Chern polynomial and Chern character. This will enable us to define Chern polynomial and Chern character for coherent sheaves on ℙ^{r−1}. In the case of the Chern character this is easy, so we treat this first.

Lemma 7.2.1. If M and M′ are equivalent finitely generated graded R-modules for the equivalence relation of Theorem 3.9.12 then there are finite length modules M_{1} and M_{2}, a graded R-module M′′ and short exact sequences

0 → M′′ → M → M_{1} → 0
0 → M′′ → M′ → M_{2} → 0.

Proof. Recall that M and M′ are equivalent if and only if there exists d ∈ ℤ such that M_{≥d} ≅ M′_{≥d}. So we set M′′ = M_{≥d}. □

Theorem 7.2.2. If M and M′ are equivalent finitely generated graded R-modules then

Ch(M) = Ch(M′) ∈ ℚ[h]/(h^{r}).

Proof. By Proposition 7.1.7 and Lemma 7.2.1, it suffices to show that if M is a finite length module then Ch(M) = 0. Again applying Proposition 7.1.7, it suffices to show this for M equal to a copy of k in degree n. In this case p_{M}(t) = t^{n}, f(t) = t^{n}(1 − t)^{r} and

Ch(M) = ∑_{j}(−1)^{j}\binom{r}{j}e^{(j+n)h} = e^{nh}(1 − e^{h})^{r}.

Since 1 − e^{h} is divisible by h, Ch(M) is divisible by h^{r} and is hence zero in ℚ[h]/(h^{r}). □

To prove that equivalent modules have the same Chern polynomial, we use an interesting congruence for polynomials, namely

∏_{j=0}^{r}(1 + jh)^{(−1)^{j} \binom{r}{j}} ≡ 1 (mod h^{r}).

We prove a slight generalisation of this.

Proposition 7.2.3. For r ≥ 1 we have

∏_{j=0}^{r}(1 + jh)^{(−1)^{j} \binom{r}{j}} ≡ 1 − (r − 1)! h^{r} (mod h^{r+1}).

Proof. Let f_{r}(h) = ∏_{j=0}^{r}(1+jh)^{(−1)^{j} \binom{r}{j}}. Then using the identities \binom{r}{j} = \binom{r−1}{j}+\binom{r−1}{j−1} and

∑_{j=0}^{r−1}(−1)^{j}\binom{r−1}{j} = 0, it is easy to check that f_{r}(h) satisfies the following recurrence relation:

f_{r}(h) = \frac{f_{r−1}(h)}{f_{r−1}(\frac{h}{1 + h})}.

We work by induction on r. If r = 1 the formula states that 1/(1 + h) ≡ 1 − h (mod h^{2}), which is clearly true. So assume that r ≥ 2 and the statement is true for r − 1. Then for some value of a we have

f_{r−1}(h) = 1 − (r − 2)! h^{r−1} + ah^{r} + …

and so

f_{r−1}(\frac{h}{1 + h}) = f_{r−1}(h − h^{2} + h^{3} − ⋯)
= 1 − (r − 2)! (h − h^{2} + h^{3} − …)^{r−1} + a(h − h^{2} + h^{3} − ⋯)^{r} + ⋯
= 1 − (r − 2)! h^{r−1} + (a + (r − 1)!)h^{r} + ⋯

Dividing, we get

f_{r}(h) = 1 − (r − 1)! h^{r} + ⋯

as required. □

Corollary 7.2.4. For r ≥ 1, we have

(i) for any value of a,

∏_{j=0}^{r}(a + jh)^{(−1)^{j} \binom{r}{j}} ≡ 1 − (r − 1)!a^{−r}h^{r} (mod h^{r+1}).

(ii) for any value of n,

∏_{j=0}^{r}(1 + (j + n)h)^{(−1)^{j} \binom{r}{j}} ≡ 1 − (r − 1)! h^{r} (mod h^{r+1}).

Proof. (i) Replace h by a^{−1}h in Proposition 7.2.3 and notice that

∏_{j=0}^{r} a^{(−1)^{j} \binom{r}{j}} = 1

so that we can multiply every term in the product by a without changing the answer.

(ii) This follows by putting a = 1+nh in part (i) and noticing that (1+nh)^{−1} is congruent to 1 modulo h. □

Theorem 7.2.5. If M and M′ are equivalent finitely generated graded R-modules then

c(M) = c(M′).

Proof. As in the proof of Theorem 7.2.2, it suffices to prove that if M is a copy of k in degree n. In this case, we have

c(M) = ∏_{j=0}^{r}(1 + (j + n)h)^{(−1)^{j} \binom{r}{j}} ≡ 1 (mod h^{r})

by Corollary 7.2.4 (ii). □

Definition 7.2.6. Let 𝒪 be the structure sheaf of ℙ^{r−1}. If ℱ is a coherent sheaf of 𝒪-modules, we define

c(ℱ) = c(M), Ch(ℱ) = Ch(M),

where M is any finitely generated graded R-module satisfying M^{∼} ≅ ℱ, see Theorem 3.9.12. By Theorems 7.2.2 and 7.2.5, this is well defined. We thus talk about the Chern polynomial, Chern numbers and Chern character of a coherent sheaf.

7.3. Some Computations

In this section, we compute Chern polynomial and Chern character of some vector bundles on ℙ^{r−1}. Our starting point is the following example.

Example 7.3.1. It follows from the definitions that we have c(𝒪(a)) = 1+ah ∈ ℤ[h]/(h^{r}) and Ch(𝒪(a)) = e^{ah} ∈ ℚ[h]/(h^{r}).

The following is a direct consequence of Proposition 7.1.7.

Proposition 7.3.2. If 0 → ℱ_{1} → ℱ_{2} → ℱ_{3} → 0 is a short exact sequence of coherent sheaves on ℙ^{r−1} then

(i) c(ℱ_{2}) = c(ℱ_{1})c(ℱ_{3}) as elements of ℤ[h]/(h^{r}). (ii) Ch(ℱ_{2}) = Ch(ℱ_{1}) + Ch(ℱ_{3}) as elements of ℚ[h]/(h^{r}). □

Example 7.3.3. Let 𝒯 and Ω be the tangent and cotangent bundes. Applying Proposition 7.3.2 to the short exact sequences (6.4.1) and (6.4.3), we obtain c(𝒯) = (1 + h)^{r}, c(Ω) = (1 − h)^{r} in ℤ[h]/(h^{r}), and Ch(𝒯) = re^{h} − 1, Ch(Ω) = re^{−h} − 1 in ℚ[h]/(h^{r}).

Example 7.3.4. Tango’s vector bundle ℱ_{W} has rank r − 2 and Chern polynomial

(1 − 2h)/(1 − h)^{r}.

Lemma 7.3.5. Given a monad on ℙ^{r−1} as in Definition 6.6.1, we have

(1) rank ℱ = rank ℱ_{2} − rank ℱ_{1} − rank ℱ_{3}.

(2) c(ℱ) = c(ℱ_{2})c(ℱ_{1})^{−1}c(ℱ_{3})^{−1} as elements of ℤ[h]/(h^{r}).

(3) Ch(ℱ) = Ch(ℱ_{2}) − Ch(ℱ_{1}) − Ch(ℱ_{3}) as elements of ℚ[h]/(h^{r}).

Proof. This follows immediately from Proposition 7.3.2. □

For example, the null correlation bundle of Section 6.7 is defined in terms of a monad, so we may now compute its Chern polynomial.

Proposition 7.3.6. The Chern polynomial of the null correlation bundle is equal to

(1 − h^{2})^{−1} = 1 + h^{2} + ⋯ + h^{r−2}.

Proof. It follows from Lemma 7.3.5 that

c(ℱ_{N}) = c(𝒪)c(𝒪(−1))^{−1}c(𝒪(1))^{−1} = (1 − h)^{−1}(1 + h)^{−1}.  □

Theorem 7.3.7. Let 𝒪 be the structure sheaf of ℙ^{r−1}. There exists a unique function assigning to each coherent sheaf ℱ of 𝒪-modules a polynomial c(ℱ) ∈ ℤ[h]/(h^{r}) satisfying the following properties.

(1) If 0 → ℱ_{1} → ℱ_{2} → ℱ_{3} → 0 is a short exact sequence of coherent sheaves then

c(ℱ_{2}) = c(ℱ_{1})c(ℱ_{3}).

(2) c(𝒪(a)) = 1 + ah.

Proof. The existence of such a function is proved in Definition 7.2.6 via Theorem 7.2.5. For the uniqueness, let c be such a function. By (1), c is multiplicative over direct sums. Let ℱ = M^{∼}. By Corollary 6.12.5 to Hilbert’s Syzygy Theorem 6.12.1 there is an exact sequence of sheaves

(7.3.1)  0 → F_{r}^{∼} → ⋯ → F_{0}^{∼} → ℱ → 0

where each F_{i}^{∼} is a direct sum of line bundles. We then have

c(ℱ) = ∏_{i=0}^{r} c(F_{i}^{∼})^{(−1)^{i}}.

Since each F_{i}^{∼} is a direct sum of twists of 𝒪, this determines c(ℱ) using (2) and multiplicativity over direct sums. □

Remark 7.3.8. If ℱ is a vector bundle then c_{i}(ℱ) = 0 for i larger than the rank of ℱ. It does not seem easy to see this from the algebraic point of view we have developed here.

In applying Proposition 7.3.2 (i) and Remark 7.3.8, it is useful to have the following elementary lemma.

Lemma 7.3.9. A polynomial in ℤ[h]/(h^{r}) factorises as a product of polynomials such that the sum of the degrees is less than r if and only if its unique lift to a polynomial of degree less than r factorises non-trivially in ℤ[h].

Proof. Such a factorisation f(h) = f_{1}(h)f_{2}(h) in ℤ[h]/(h^{r}) lifts to a congruence modulo h^{r} between polynomials of degree less than r in ℤ[h]. This congruence is therefore an equality. □

7.4. Restriction of Vector Bundles

Definition 7.4.1. Let f: X → Y be a morphism of schemes. If 𝒢 is a sheaf on X then f_{∗}𝒢 is the sheaf on Y given by

f_{∗}𝒢(U) = 𝒢(f^{−1}(U))

for U ⊆ Y open. If 𝒢 is an 𝒪_{X}-module then f_{∗}𝒢 is an 𝒪_{Y} -module via the natural map 𝒪_{Y} → f_{∗}𝒪_{X} of sheaves of rings on Y. We call f_{∗}𝒢 the direct image of 𝒢 under the morphism f.

The direct image functor is left exact, but usually not right exact. So it has right derived functors R^{i}f_{∗}𝒢.

Definition 7.4.2. If ℱ is a sheaf on Y then f^{−1}ℱ is the sheaf on X given by

f^{−1}ℱ(U) = lim_{V ⊇f(U)^{⟶}} ℱ(V).

Here, the colimit is taken over the directed system of open sets V containing f(U).

If ℱ is a sheaf of 𝒪_{Y} -modules then f^{∗}ℱ is the sheaf of 𝒪_{X}-modules given by

f^{∗}ℱ = f^{−1}ℱ ⊗_{f^{−1}𝒪_{Y}} 𝒪_{X}.

We call f^{∗}ℱ the inverse image of ℱ along the morphism f. If f: X → Y is the inclusion of a subscheme then f^{∗}ℱ is the restriction of ℱ to X.

The following lemma is straightforward, and may be found in any of the standard textbooks.

Lemma 7.4.3. The functors f_{∗} and f^{∗} are adjoint:

Hom_{𝒪_{X}}(f^{∗}ℱ, 𝒢) ≅ Hom_{𝒪_{Y}} (ℱ, f_{∗}𝒢).

While f^{−1} is an exact functor from sheaves on Y to sheaves on X, tensor product is only right exact. So the functor f^{∗} is right exact and has left derived functors

L_{i}f^{∗}ℱ = 𝒯or_{i}^{f^{−1}𝒪_{Y}} (f^{−1}ℱ, 𝒪_{X}).

On the other hand, since f^{∗}(𝒪_{Y}^{⊕n}) ≅ 𝒪_{X}^{⊕n}, f^{∗} takes vector bundles of rank n to vector bundles of rank n. Furthermore, since exactness is a local property, f^{∗} is exact on vector bundles.

Let r′ < r, and let i: ℙ^{r′−1} ↪ ℙ^{r−1} be the inclusion induced by the map of graded rings

π: k[Y_{1}, …, Y_{r}] → k[Y_{1}, …, Y_{r′}]

with π(Y_{i}) = Y_{i} (1 ≤ i ≤ r′), π(Y_{i}) = 0 (r′ < i ≤ r). Then

i^{∗}(𝒪_{ℙ^{r−1}}(a)) ≅ 𝒪_{ℙ^{r′−1}}(a).

If ℱ is a vector bundle over ℙ^{r−1} then by Corollary 6.12.5 it has a finite resolution by direct sums of twists of 𝒪_{ℙ^{r−1}}. By the exactness of restriction on vector bundles, i^{∗}(ℱ) has a resolution by direct sums of the same twists of 𝒪_{ℙ^{r′−1}}. It follows using the proof of Theorem 7.3.7 that c(i^{∗}(ℱ)) ∈ ℤ[h]/(h^{r′}) is the image of c(ℱ) ∈ ℤ[h]/(h^{r}) under the quotient map taking h to h. Similarly, Ch(i^{∗}(ℱ)) is the image of Ch(ℱ) under the corresponding map ℚ[h]/(h^{r}) → ℚ[h]/(h^{r′}).

The way to express this is to write

i^{∗}: A^{∗}(ℙ^{r−1}) → A^{∗}(ℙ^{r′−1})

for the map of Chow rings ℤ[h]/(h^{r}) → ℤ[h]/(h^{r′}) taking h to h. In terms of this we have

c(i^{∗}(ℱ)) = i^{∗}(c(ℱ)).

We also write

i^{∗}: A_{ℚ}^{∗}(ℙ^{r−1}) → A_{ℚ}^{∗}(ℙ^{r′−1})

for the map of rational Chow rings ℚ[h]/(h^{r}) → ℚ[h]/(h^{r′}) taking h to h. In terms of this we have

Ch(i^{∗}(ℱ)) = i^{∗}(Ch(ℱ)).

For more general coherent sheaves on ℙ^{r−1} this formula does not hold, because restriction is not exact. However, what this means is that we should instead use the total left derived functor

Li^{∗}: D^{b}(Coh(ℙ^{r−1})) → D^{b}(Coh(ℙ^{r′−1})).

In this case the total left derived functor may be applied by replacing a sheaf (or a bounded complex) with a finite resolution by direct sums of line bundles before restricting.

We summarise in the following theorem.

Theorem 7.4.4. Suppose that r′ < r and let i: ℙ^{r′−1} → ℙ^{r−1} be the inclusion. Let

i^{∗}: ℤ[h]/(h^{r}) → ℤ[h]/(h^{r′})

be the ring homomorphism taking h to h.

(i) If ℱ is a vector bundle on ℙ^{r−1} then c(i^{∗}(ℱ)) = i^{∗}(c(ℱ)) and Ch(i^{∗}(ℱ)) = i^{∗}(Ch(ℱ)). (ii) If ℱ is a coherent sheaf on ℙ^{r−1} then c(Li^{∗}(ℱ)) = i^{∗}(c(ℱ)) and Ch(Li^{∗}(ℱ)) = i^{∗}(Ch(ℱ)). □

7.5. Chern Numbers of Twists and Duals

Lemma 7.5.1. The Chern polynomial of a twisted vector bundle ℱ(i) = ℱ⊗_{𝒪} 𝒪(i) is given by

(7.5.1)  c(ℱ(i), h) = (1 + ih)^{rank (ℱ)}c (ℱ, \frac{h}{1 + ih}) = ∑_{n=0}^{rank (ℱ)} c_{n}(ℱ)h^{n}(1 + ih)^{rank (ℱ)−n}.

Thus the Chern numbers of ℱ(i) are given by

c_{m}(ℱ(i)) = ∑_{j=0}^{m} i^{j}\binom{rank (ℱ) − m + j}{j}c_{m−j}(ℱ).

The Chern character of ℱ(i) is given by

Ch(ℱ(i)) = e^{ih}Ch(ℱ).

Proof. Let ℱ = M^{∼}, where M is a finitely generated graded R-module with Poincaré series ∑_{j} a_{j}t^{j}/(1 − t)^{r}. Then using the definition of the Chern polynomial we have

c(ℱ(i), h) = ∏_{j}(1 + (j + i)h)^{a_{j}}

= (1 + ih)^{∑_{j} a_{j}} ∏_{j} (\frac{1 + (j + i)h}{1 + ih})^{a_{j}}

= (1 + ih)^{∑_{j} a_{j}} ∏_{j} (1 + \frac{jh}{1 + ih})^{a_{j}}

= (1 + ih)^{rank (ℱ)}c (ℱ, \frac{h}{1 + ih}) .

This gives the formula for the Chern numbers.

For the Chern character we have

Ch(ℱ(i)) = ∑_{j} a_{j}e^{(j+i)h} = e^{ih} ∑_{j} a_{j}e^{jh} = e^{ih}Ch(ℱ).  □

More explicitly, the lemma says that if ℱ has rank s then

c_{1}(ℱ(i)) = c_{1}(ℱ) + is
c_{2}(ℱ(i)) = c_{2}(ℱ) + i(s − 1)c_{1}(ℱ) + i^{2}\binom{s}{2}
c_{3}(ℱ(i)) = c_{3}(ℱ) + i(s − 2)c_{2}(ℱ) + i^{2}\binom{s − 1}{2}c_{1}(ℱ) + i^{3}\binom{s}{3}

and so on.

Lemma 7.5.2. The Chern polynomial and Chern character of the dual bundle ℱ^{∨} are given by

c(ℱ^{∨}, h) = c(ℱ, −h), Ch(ℱ^{∨}, h) = Ch(ℱ, −h).

Proof. By Corollary 6.12.5 there is a resolution of ℱ by sums of line bundles

0 → ⨁_{j=1}^{m_{r}} 𝒪(a_{r,j}) → ⋯ → ⨁_{j=1}^{m_{1}} 𝒪(a_{1,j}) → ⨁_{j=1}^{m_{0}} 𝒪(a_{0,j}) → ℱ → 0.

Dualising, and using local freeness, we get an exact sequence

0 → ℱ^{∨} → ⨁_{j=1}^{m_{0}} 𝒪(−a_{0,j}) → ⨁_{j=1}^{m_{1}} 𝒪(−a_{1,j}) → ⋯ → ⨁_{j=1}^{m_{r}} 𝒪(−a_{r,j}) → 0.

Thus

c(ℱ, h) = ∏_{i=0}^{r} ∏_{j=1}^{m_{i}}(1 + a_{i,j}h)^{(−1)^{i}} while c(ℱ^{∨}, h) = ∏_{i=0}^{r} ∏_{j=1}^{m_{i}}(1 − a_{i,j}h)^{(−1)^{i}}.

Similarly,

Ch(ℱ, h) = ∑_{i=0}^{r}(−1)^{i} ∑_{j=1}^{m_{i}} e^{a_{i,j}h} while Ch(ℱ^{∨}, h) = ∑_{i=0}^{r}(−1)^{i} ∑_{j=1}^{m_{i}} e^{−a_{i,j}h}.  □

7.6. Chern Roots

Throughout this section, let 𝒪 be the structure sheaf of ℙ^{r−1}, and let ℱ be a coherent sheaf of 𝒪-modules with Chern polynomial c(ℱ) = c(ℱ, h) ∈ ℤ[h]/(h^{r}). Then c(ℱ) lifts uniquely to a polynomial in ℤ[h] of degree at most r − 1. By abuse of notation we denote this element of ℤ[h] and its image in ℂ ⊗_{ℤ} ℤ[h] = ℂ[h] also by c(ℱ) or c(ℱ, h). In ℂ[h] it factors as a product of linear factors

c(ℱ) = ∏_{j}(1 + α_{j}h).

The complex numbers α_{j} obtained this way are called the Chern roots of ℱ. It is clear that the α_{j} are algebraic integers.

Example 7.6.1. If ℱ is a direct sum of line bundles,

ℱ = ⨁_{j} 𝒪(a_{j})

then

c(ℱ) = ∏_{j}(1 + a_{j}h)

and so the Chern roots are the a_{j}.

Remarks 7.6.2. (i) We do not regard the number of Chern roots as well defined, since we can always pad them out with a list of zeros. If ℱ is a vector bundle then it is sometimes convenient to regard the number of Chern roots as being equal to rank (ℱ), as this makes many formulas easier. But this is not possible for coherent sheaves, as the degree of the polynomial c(ℱ) may be larger than the rank of ℱ.

(ii) In the case where ℱ is a vector bundle, there is a way of regarding ℱ as formally having a filtration by line bundles with first Chern classes α_{j}. The precise way to formulate this is called the splitting principle, see for example Remark 3.2.3 of Fulton [129]. We do not need to use this point of view in our development of the subject.

The following way to rewrite Lemma 7.5.1 is the content of Exercise 3.2.2 of [129], where it is regarded as a consequence of the splitting principle.

Lemma 7.6.3. The Chern roots of a twist of a coherent sheaf are given by

(7.6.1)  c(ℱ(i)) = (1 + ih)^{m} ∏_{j}(1 + (α_{j} + i)h),

where m is equal to rank (ℱ) minus the number of Chern roots used in the product.

Proof. By Lemma 7.5.1 we have

c(ℱ(i), h) = (1 + ih)^{rank (ℱ)}c (ℱ, \frac{h}{1 + ih})

= (1 + ih)^{rank (ℱ)} ∏_{j} (1 + α_{j} \frac{h}{1 + ih})

= (1 + ih)^{m} ∏_{j}(1 + ih) (1 + α_{j} \frac{h}{1 + ih})

= (1 + ih)^{m} ∏_{j}(1 + (α_{j} + i)h).  □

7.7. Power Sums

We continue with the notation of the last section. In this section we define invariants s_{i}(ℱ) (1 ≤ i ≤ r − 1) which will turn out to be the sum of the i th powers of the Chern roots.

Definition 7.7.1. Let ℱ be a coherent sheaf on ℙ^{r−1}. We define s(ℱ, h) ∈ ℤ[h]/(h^{r}) by the formula

−s(ℱ, −h) = \frac{hc′(ℱ, h)}{c(ℱ, h)},

where c′(ℱ, h) denotes the (formal) derivative of c(ℱ, h) with respect to h. The numbers s_{i}(ℱ) (1 ≤ i ≤ r − 1) are defined by

s(ℱ, h) = ∑_{i≥i} s_{i}(ℱ)h^{i} = s_{1}(ℱ)h + s_{2}(ℱ)h^{2} + · · · .

Remark 7.7.2. The definition (7.7.1) for s(ℱ, h) can be rewritten as

jc_{j} = ∑_{i=1}^{j−1}(−1)^{i−1}c_{j−i}s_{i}.

This equation is called Newton’s identity.

Thus each s_{i} = s_{i}(ℱ) is some polynomial in the Chern numbers with integer coefficients. For example,

s_{1} = c_{1}
s_{2} = c_{1}^{2} − 2c_{2}
s_{3} = c_{1}^{3} − 3c_{1}c_{2} + 3c_{3}
s_{4} = c_{1}^{4} − 4c_{1}^{2}c_{2} + 4c_{1}c_{3} + 2c_{2}^{2} − 4c_{4}.

Inverting this process allows us to express the Chern numbers as polynomials in the s_{i} with rational coefficients:

c_{1} = s_{1}
c_{2} = \frac{1}{2}(s_{1}^{2} − s_{2})
c_{3} = \frac{1}{6}(s_{1}^{3} − 3s_{1}s_{2} + 2s_{3})
c_{4} = \frac{1}{24}(s_{1}^{4} − 6s_{1}^{2}s_{2} + 3s_{2}^{2} + 8s_{1}s_{3} − 6s_{4}).

The coefficients here may be read off from the conjugacy class sizes and signs of permutations in the corresponding symmetric group. The same expressions without the signs give the complete symmetric functions of the Chern roots instead of the elementary symmetric functions.

Lemma 7.7.3. If 0 → ℱ_{1} → ℱ_{2} → ℱ_{3} → 0 is a short exact sequence of vector bundles then for 1 ≤ i ≤ r − 1 we have

s_{i}(ℱ_{2}) = s_{i}(ℱ_{1}) + s_{i}(ℱ_{3}).

Proof. We have

c(ℱ_{2}, h) = c(ℱ_{1}, h)c(ℱ_{3}, h).

Differentiating with respect to h, we get

c′(ℱ_{2}, h) = c′(ℱ_{1}, h)c(ℱ_{3}, h) + c(ℱ_{1}, h)c′(ℱ_{3}, h).

Multiply by h and divide by c(ℱ_{2}, h) to get

\frac{hc′(ℱ_{2}, h)}{c(ℱ_{2}, h)} = \frac{hc′(ℱ_{1}, h)}{c(ℱ_{1}, h)} + \frac{hc′(ℱ_{3}, h)}{c(ℱ_{3}, h)}.

Thus by Definition 7.7.1 we have

−s(ℱ_{2}, −h) = −s(ℱ_{1}, −h) − s(ℱ_{3}, −h)

and so

s(ℱ_{2}, h) = s(ℱ_{1}, h) + s(ℱ_{3}, h).

Now compare coefficients of h^{j}. □

Proposition 7.7.4. Let ℱ = M^{∼} with M a finitely generated graded R-module with Poincaré series

\frac{∑_{j} a_{j}t^{j}}{(1 − t)^{r}}

where the numerator is a Laurent polynomial. Then for 1 ≤ i ≤ r − 1 we have

s_{i}(ℱ) = ∑_{j} a_{j}j^{i}.

Proof. By definition we have

c(ℱ, h) = ∏_{j}(1 + jh)^{a_{j}}
log c(ℱ, h) = ∑_{j} a_{j} log(1 + jh).

Differentiating both sides with respect to h, we obtain

\frac{c′(ℱ, h)}{c(ℱ, h)} = ∑_{j} a_{j} \frac{j}{1 + jh}

−s(ℱ, −h) = \frac{hc′(ℱ, h)}{c(ℱ, h)} = ∑_{j} a_{j} \frac{jh}{1 + jh}

s(ℱ, h) = ∑_{j} a_{j} \frac{jh}{1 − jh}

= ∑_{j} a_{j}(jh + j^{2}h^{2} + …)

s_{i}(ℱ) = ∑_{j} a_{j}j^{i}.  □

Proposition 7.7.5. For 1 ≤ i ≤ r − 1 the s_{i}(ℱ) are given in terms of the Chern roots as the power sums

s_{i}(ℱ) = ∑_{j} α_{j}^{i}.

Proof. Working in ℂ[h] we have

c(ℱ, h) = ∏_{j}(1 + α_{j}h)
log c(ℱ, h) = ∑_{j} log(1 + α_{j}h).

Differentiating both sides with respect to h, we obtain

\frac{c′(ℱ, h)}{c(ℱ, h)} = ∑_{j} \frac{α_{j}}{1 + α_{j}h}

−s(ℱ, −h) = \frac{hc′(ℱ, h)}{c(ℱ, h)} = ∑_{j} \frac{α_{j}h}{1 + α_{j}h}

s(ℱ, h) = ∑_{j} \frac{α_{j}h}{1 − α_{j}h}

= ∑_{j}(α_{j}h + α_{j}^{2}h^{2} + …)

s_{i}(ℱ) = ∑_{j} α_{j}^{i}.  □

Theorem 7.7.6. Let p(x) ∈ ℂ[x] be a polynomial of degree < r with p(0) = 0. Then (i) ∑_{j} p(α_{j}) = ∑_{j} a_{j}p(j). (ii) If p(x) satisfies n ∈ ℤ ⇒ p(n) ∈ ℤ, then we have ∑_{j} p(α_{j}) ∈ ℤ.

Proof. (i) Both sides are linear in p(x), so it suffices to prove the equation for p(x) = x^{i} (i > 0). In this case it follows from Propositions 7.7.4 and 7.7.5.

(ii) This follows from (i), since the coefficients a_{j} are integers. □

Corollary 7.7.7. Let ℱ be a coherent sheaf on ℙ^{r−1} with Chern roots α_{j}. Then the Chern character is given by

Ch(ℱ) = rank (ℱ) + ∑_{j}(e^{α_{j}h} − 1).

Proof. By definition

Ch(ℱ) = ∑_{j} a_{j}e^{jh}

= rank (ℱ) + ∑_{j} a_{j}(e^{jh} − 1)

= rank (ℱ) + ∑_{j} ∑_{n=1}^{r−1} \frac{a_{j}j^{n}h^{n}}{n!} .

Applying Theorem 7.7.6 (i) to the polynomials x^{n}/n! (1 ≤ n ≤ r − 1), we deduce that

Ch(ℱ) = rank (ℱ) + ∑_{j} ∑_{n=1}^{r−1} \frac{α_{j}^{n}h^{n}}{n!}
= rank (ℱ) + ∑_{j}(e^{α_{j}h} − 1).  □

Remark 7.7.8. We remarked before (Remark 7.6.2 (i)) that the number of Chern roots is not regarded as well defined. If ℱ is a vector bundle and we insist that there are exactly rank (ℱ) Chern roots then the formula in the corollary reads

Ch(ℱ) = ∑_{j} e^{α_{j}h}.

Example 7.7.9. Let m be a positive integer. Then the polynomial

p(x) = \binom{x}{m} = \frac{x(x − 1) … (x − m + 1)}{m!}

satisfies n ∈ ℤ ⇒ p(n) ∈ ℤ and p(0) = 0. More generally, so does the polynomial \binom{x+s}{m} for s ∈ ℤ. So these are candidates for use in Theorem 7.7.6 (ii).

Corollary 7.7.10. Let ℱ be a coherent sheaf on ℙ^{r−1} with Chern roots α_{j}. If s and m are integers with 1 ≤ m ≤ r − 1 then

∑_{j} \binom{α_{j} + s}{m} ∈ ℤ.

Example 7.7.11. For coherent sheaves on ℙ^{1} and ℙ^{2}, this gives no condition at all on the Chern numbers. For any coherent sheaf on ℙ^{3} the conditions are equivalent to

c_{1}c_{2} + c_{3} ≡ 0 (mod 2).

Horrocks [159] has shown that given any c_{1} and c_{2} with c_{1}c_{2} even, there exists a vector bundle of rank two on ℙ^{3} with these Chern numbers.

For a bundle of rank two on ℙ^{4} the conditions are equivalent to

c_{2}(c_{2} + 1 − 3c_{1} − 2c_{1}^{2}) ≡ 0 (mod 12).

These congruences are usually regarded as consequences of the Hirzebruch–Riemann– Roch theorem, which we discuss in the next section. They will be used in Sections 10.5–10.9 to obtain restrictions on modules of constant Jordan type.

The following proposition tells us that Corollary 7.7.10 contains all the information to be obtained from the theorem.

Proposition 7.7.12. Let p(x) ∈ ℂ[x]. Then the following are equivalent:

(i) n ∈ ℤ ⇒ p(n) ∈ ℤ.

(ii) p(x) is in the ℤ-span of the polynomials

\binom{x}{m} = \frac{x(x − 1) … (x − m + 1)}{m!}

(0 ≤ m ≤ deg(p)).

Remark 7.7.13. Here, \binom{x}{0} is taken to be 1, and if we exclude this from (ii) and take 1 ≤ m ≤ deg(p) instead, then we obtain the polynomials satisfying condition (i) with p(0) = 0.

Proof. Clearly (ii) ⇒ (i). To prove that (i) ⇒ (ii), we work by induction on deg(p) using the calculus of finite differences as follows. If p(x) ∈ ℂ[x], we define

Δp(x) = p(x + 1) − p(x).

Then deg(Δp) = deg(p) − 1. We also have

Δ\binom{x}{m + 1} = \binom{x + 1}{m + 1} − \binom{x}{m + 1} = \binom{x}{m}.

By the inductive hypothesis we have

Δp(x) = ∑_{i=0}^{deg(p)−1} b_{m}\binom{x}{m}

with b_{m} ∈ ℤ. We also have

Δ ∑_{m=0}^{deg(p)−1} b_{m}\binom{x}{m + 1} = ∑_{m=0}^{deg(p)−1} b_{m}\binom{x}{m}

and so

Δ (p(x) − ∑_{m=0}^{deg(p)−1} b_{m}\binom{x}{m + 1}) = 0.

It follows that

p(x) − ∑_{m=0}^{deg(p)−1} b_{m}\binom{x}{m + 1}

is a constant, and this constant must be an integer. So p(x) is of the required form. □

7.8. The Hirzebruch–Riemann–Roch Theorem

Hirzebruch [156] developed a generalised version of the Riemann–Roch theorem that applies to vector bundles on a non-singular algebraic variety. We shall provide a quick proof of the Hirzebruch–Riemann–Roch theorem for vector bundles on ℙ^{r−1}. As remarked in the previous section, the congruences on Chern numbers coming from this theorem are already contained in Corollary 7.7.10.

Definition 7.8.1. The Euler characteristic of a vector bundle ℱ on ℙ^{r−1} is defined to be

χ(ℱ) = ∑_{i≥0}(−1)^{i} dim_{k} H^{i}(ℙ^{r−1}, ℱ).

Since H^{i}(ℙ^{r−1}, ℱ) = 0 for i ≥ r, this is a finite sum.

Lemma 7.8.2. If 0 → ℱ_{1} → ℱ_{2} → ℱ_{3} → 0 is a short exact sequence of vector bundles on ℙ^{r−1} then

χ(ℱ_{2}) = χ(ℱ_{1}) + χ(ℱ_{3}).

Proof. This follows from the long exact sequence in cohomology of this short exact sequence of vector bundles. □

Theorem 7.8.3 (Schwarzenberger, Appendix I, §22.4 of Hirzebruch [156]; see also §I.6.1 of Okonek, Schneider and Spindler [202]). Let ℱ be a coherent sheaf on ℙ^{r−1}. The Euler characteristic of ℱ is given in terms of the Chern roots by the formula

χ(ℱ) = rank (ℱ) + ∑_{j} (\binom{α_{j} + r − 1}{r − 1} − 1) .

Remark 7.8.4. Provided the number of Chern roots is taken to be equal to the rank of ℱ (see Remark 7.6.2 (i)), this simplifies to

χ(ℱ) = ∑_{j} \binom{α_{j} + r − 1}{r − 1}.

Proof. First we treat the case of a line bundle 𝒪(a). In this case the statement is that

χ(𝒪(a)) = \binom{a + r − 1}{r − 1}.

This is true by Corollary 6.9.2.

By Lemmas 7.7.3 and 7.8.2 both sides are additive over short exact sequences. So it remains true for direct sums of line bundles.

By Corollary 6.12.5 to Hilbert’s Syzygy Theorem 6.12.1, ℱ has a resolution

0 → F_{n}^{∼} → ⋯ → F_{0}^{∼} → ℱ → 0

where each F_{i}^{∼} is a direct sum of line bundles and n ≤ r. We proceed by induction on n. If n = 0 then ℱ is a direct sum of line bundles, and this case is already treated. If n > 0, let ℱ′ be the kernel of F_{0}^{∼} → ℱ, so that we have a short exact sequence

0 → ℱ′ → F_{0}^{∼} → ℱ → 0.

Then ℱ′ has such a resolution of length n − 1, so by the inductive hypothesis the theorem is true for ℱ′. It is also true for F_{0}^{∼}, since this is a direct sum of line bundles. So by additivity of both sides over short exact sequences, it is true for ℱ. □

Remark 7.8.5. Schwartzenberger’s theorem can be seen as telling us how to recover the Poincaré series of a graded k[Y_{1}, …, Y_{r}]-module M from the Chern roots of the coherent sheaf M^{∼}. Of course, this can only be achieved up to the Poincaré series of a finite length module, namely up to the addition of a polynomial.

The way the theorem tells us this information is as follows. As long as n is large enough, we have H^{i}(ℙ^{r−1}, M^{∼}(n)) = 0 for i > 0 and so χ(M^{∼}(n)) = dim H^{0}(ℙ^{r−1}, M^{∼}(n)). Furthermore, for n large enough we have H^{0}(ℙ^{r−1}, M^{∼}(n)) ≅ M_{n}. It follows that provided we can take the number of Chern roots to be the rank, then the Poincaré series p_{M}(t) differs from the power series

∑_{n≥0} ∑_{j} \binom{α_{j} + n + r − 1}{r − 1}t^{n}

by a polynomial.

Example 7.8.6. An instanton bundle (see Section 9.8) ℱ of quantum number k on ℙ^{r−1} (r even) has rank r − 2 and Chern polynomial 1/(1 − t^{2})^{k}. We can behave as though the Chern roots are +1 and −1, each with multiplicity −k, and 0 with multiplicity r − 2 + 2k, even though this doesn’t make literal sense. We then have

χ(ℱ(n)) = −k\binom{n + r}{r − 1} + (r − 2 + 2k)\binom{n + r − 1}{r − 1} − k\binom{n + r − 2}{r − 1}.

The Poincaré series of a corresponding graded module therefore differs by a polynomial from

\frac{−kt^{−1} + (r − 2 + 2k) − kt}{(1 − t)^{r}}.

Theorem 7.8.7 (Hirzebruch–Riemann–Roch for ℙ^{r−1}). Let ℱ be a coherent sheaf on ℙ^{r−1}. Then χ(ℱ) is equal to the coefficient of h^{r−1} in the power series

(\frac{h}{1 − e^{−h}} )^{r} Ch(ℱ).

Proof. By Cauchy’s integral formula, the coefficient of h^{r−1} in this power series is

\frac{1}{2πi} ∮ (\frac{h}{1 − e^{−h}} )^{r} Ch(ℱ) \frac{dh}{h^{r}}

where the integral is over a small anticlockwise circle around the origin in the complex plane. Using Corollary 7.7.7, we get

\frac{1}{2πi} ∮ \frac{rank (ℱ) + ∑_{j}(−1 + e^{α_{j}h})}{(1 − e^{−h})^{r}} dh.

Now substitute z = 1 − e^{−h}. We have

dz = e^{−h} dh,  dh = dz/(1 − z),  e^{α_{j}h} = 1/(1 − z)^{α_{j}}.

First we compute

\frac{1}{2πi} ∮ \frac{dh}{(1 − e^{−h})^{r}} = \frac{1}{2πi} ∮ \frac{dz}{z^{r}(1 − z)} = \frac{1}{2πi} ∮ (z^{−r} + z^{−r+1} + …) dz = 1.

Therefore the constant term of the Chern polynomial comes out of the integral, and we get

rank (ℱ)+ ∑_{j} (−1 + \frac{1}{2πi} ∮ \frac{dz}{z^{r}(1 − z)^{α_{j}+1}} )

= rank (ℱ) + ∑_{j} (−1 + \frac{1}{2πi} ∮ z^{−r}(1 + (α_{j} + 1)z + \binom{α_{j} + 2}{2}z^{2} + ⋯) dz)

= rank (ℱ) + ∑_{j} (−1 + \binom{α_{j} + r − 1}{r − 1}) .

By Theorem 7.8.3 this is equal to χ(ℱ). □

Remark 7.8.8. It is worth making some remarks about this substitution z = 1 − e^{−h} in this proof. Since z = h + O(h^{2}), if h traverses a small anticlockwise circle around the origin, z traverses a deformation of this circle that is small compared with the radius. So for small enough radius, the winding number of this contour around the origin is still equal to one. Since the integrand has an isolated pole at z = 0, by Cauchy’s theorem this contour gives the same integral as the anticlockwise circle in the z plane.

Remark 7.8.9. The power series

(\frac{h}{1 − e^{−h}} )^{r}

in the Hirzebruch–Riemann–Roch theorem for ℙ^{r−1} is playing the role of the Todd class of the tangent bundle. We discuss the tangent bundle in Section 6.4. Note that whatever the coherent sheaf ℱ is, the Todd class of the tangent bundle is what appears in the formula. But the Todd class is indeed defined for any vector bundle ℱ, and is given in terms of the Chern roots or Chern numbers by the formula

Td(ℱ) = ∏_{j} \frac{α_{j}h}{1 − e^{−α_{j}h}}
= 1 + \frac{1}{2}c_{1}h + \frac{1}{12}(c_{2} + c_{1}^{2})h^{2} + \frac{1}{24}c_{1}c_{2}h^{3}

+ \frac{1}{720}(−c_{4} + c_{1}c_{3} + 3c_{2}^{2} + 4c_{1}^{2}c_{2} − c_{1}^{4})h^{4}
+ \frac{1}{1440}c_{1}(−c_{4} + c_{1}c_{3} + 3c_{2}^{2} − c_{1}^{2}c_{2})h^{5} + · · · .

Note that we have

\frac{x}{1 − e^{−x}} = ∑_{j=0}^{∞} \frac{B_{j}x^{j}}{j!} .

where the B_{j} are the Bernoulli numbers:

B_{0} = 1, B_{1} = \frac{1}{2}, B_{2} = \frac{1}{6}, B_{4} = −\frac{1}{30}, B_{6} = \frac{1}{42}, etc.,

and B_{j} is zero for j odd, j > 1.^{1}

The general statement of the Hirzebruch–Riemann–Roch theorem is that if X is a nonsingular projective variety (or a compact complex manifold) and ℱ is a holomorphic vector bundle over X then χ(ℱ) is the evaluation of Td(𝒯)Ch(ℱ) at the fundamental class of X.

Grothendieck produced a relative version of this theorem, known as the Grothendieck– Riemann–Roch theorem. This was written up and published by Borel and Serre [62]. The proof of Grothendieck–Riemann–Roch in its full generality starts by proving Hirzebruch– Riemann–Roch for projective space, see Proposition 10 of [62].

7.9. Chern Numbers and the Frobenius Map

Let k be an algebraically closed field of characteristic p > 0 and let R = k[Y_{1}, …, Y_{r}]. The Frobenius map F: ℙ^{r−1} → ℙ^{r−1} is the map of projective varieties induced by the map of k-algebras R → R sending Y_{i} to Y_{i}^{p}. In this section we examine the inverse image of a coherent sheaf along F and its effect on the Chern polynomial.

Definition 7.9.1. We write ΦR for the subring k[Y_{1}^{p}, …, Y_{r}^{p}] of R. If M be a finitely generated graded R-module, we write ΦM for the graded vector space given by

(ΦM)_{j} = \begin{cases}M_{j/p} & p|j \\ 0 & otherwise. \end{cases}

If m ∈ M_{i}, we write Φ(m) for the corresponding element of (ΦM)_{ip}. We make ΦM into a ΦR-module via

Y_{i}^{p}Φ(m) = Φ(Y_{i}m).

Finally, we write F M for the graded R-module

F M = R ⊗_{ΦR} ΦM.

If ℱ is a coherent sheaf of 𝒪-modules, where 𝒪 is the structure sheaf of ℙ^{r−1}, then according to Theorem 3.9.12 there is a finitely generated graded R-module M with ℱ = M^{∼}. The inverse image F^{∗}(ℱ) of ℱ along the Frobenius map F: ℙ^{r−1} → ℙ^{r−1} is then defined by

F^{∗}(ℱ) = (F M)^{∼}.

Example 7.9.2. If ℱ = 𝒪(a) then F^{∗}ℱ = 𝒪(pa).

Lemma 7.9.3. R is a free ΦR-module on the basis consisting of monomials Y_{1}^{i_{1}} … Y_{r}^{i_{r}} with 0 ≤ i_{j} ≤ p − 1 for 1 ≤ j ≤ r.

1Beware that Hirzebruch [156] uses a somewhat unconventional indexing for the Bernoulli numbers.

Proof. This is clear by examining the basis of monomials in R. □

Proposition 7.9.4. If

0 → M_{1} → M_{2} → M_{3} → 0

is a short exact sequence of R-modules then so is

0 → F M_{1} → F M_{2} → F M_{3} → 0.

If

0 → ℱ_{1} → ℱ_{2} → ℱ_{3} → 0

is a short exact sequence of sheaves of 𝒪-modules, then so is

0 → F^{∗}(ℱ_{1}) → F^{∗}(ℱ_{2}) → F^{∗}(ℱ_{3}) → 0.

Proof. This follows from Lemma 7.9.3, since tensoring with a free module is exact. □

Remark 7.9.5. Recall from Corollary 6.12.5 that every coherent sheaf ℱ of 𝒪-modules has a resolution of length at most r by sums of line bundles. It follows from the Proposition that replacing each line bundle 𝒪(a) by 𝒪(pa) and replacing each Y_{i} by Y_{i}^{p} in the matrices defining the maps, we obtain a resolution of F^{∗}(ℱ).

Theorem 7.9.6. Let ℱ be a coherent sheaf of 𝒪-modules, and let F be the Frobenius map.

(i) The Chern polynomial of F^{∗}(ℱ) is given by

c(F^{∗}(ℱ), h) = c(ℱ, ph).

(ii) The Chern numbers of F^{∗}(ℱ) are given by

c_{i}(F^{∗}(ℱ)) = p^{i}c_{i}(ℱ).

(iii) If the Chern roots of ℱ are α_{j} then the Chern roots of F^{∗}(ℱ) are pα_{j}.

(iv) The Chern character of F^{∗}(ℱ) is given by

Ch(F^{∗}(ℱ), h) = Ch(ℱ, ph).

Proof. (i) Let ℱ = M^{∼}, where M is a finitely generated graded R-module. Write the Poincaré series for M in the form

p_{M}(t) = \frac{∑_{j} a_{j}t^{j}}{(1 − t)^{r}}

where ∑_{j} a_{j}t^{j} is a Laurent polynomial in t. Then

p_{ΦM}(t) = \frac{∑_{j} a_{j}t^{pj}}{(1 − t^{p})^{r}} .

By Lemma 7.9.3, we have

p_{F M}(t) = (1 + t + ⋯ + t^{p−1})^{r}p_{ΦM}(t) = \frac{∑_{j} a_{j}t^{pj}}{(1 − t)^{r}} .

Thus

c(F^{∗}(ℱ), h) = ∏_{j}(1 + pjh)^{a_{j}} = c(ℱ, ph).

(ii) and (iii) follow immediately from (i), and then (iv) follows using Corollary 7.7.7. □

8 Modules of Constant Jordan Type and Vector Bundles

8.1. Introduction

The construction of algebraic vector bundles (i.e., locally free sheaves) on projective space from modules of constant Jordan type over an elementary abelian p-group was introduced by Friedlander and Pevtsova [127]. In this chapter we give a refined version of their construction and investigate its properties.

Let E be an elementary abelian p-group of rank r and let k be an algebraically closed field of characteristic p. In this chapter we introduce functors ℱ_{i} from finitely generated kE-modules of constant Jordan type to vector bundles over projective space ℙ^{r−1}. We prove that given any vector bundle ℱ of rank s on ℙ^{r−1}, there is a kE-module M of stable constant Jordan type [1]^{s} such that ℱ_{1}(M) ≅ ℱ if p = 2, and such that ℱ_{1}(M) ≅ F^{∗}(ℱ) if p is odd. Here, F: ℙ^{r−1} → ℙ^{r−1} is the Frobenius map. We prove that the theorem cannot be improved if p is odd, because if M is any module of stable constant Jordan type [1]^{s} then the Chern numbers c_{1}, …, c_{p−2} of ℱ_{1}(M) are divisible by p. We also prove various formulas such as ℱ_{p−i}(ΩM) ≅ ℱ_{i}(M)(−p + i), and in Chapter 9 we shall give a number of explicit examples of vector bundles obtained from modules of constant Jordan type. In Chapter 11, we shall return to this topic, and give another interpretation of the main construction of this chapter in terms of the Orlov correspondence relating the stable module category stmod(kE) to a certain category of maximal Cohen–Macaulay modules over a certain hypersurface singularity.

In more detail, we shall define functors ℱ_{i} (1 ≤ i ≤ p) from finitely generated kE-modules of constant Jordan type to vector bundles on projective space ℙ^{r−1}, capturing the sum of the socles of the length i Jordan blocks. We prove the following realisation theorem in Section 8.9.

Theorem 8.1.1 (Benson and Pevtsova [53]). Given any vector bundle ℱ of rank s on ℙ^{r−1}, there exists a finitely generated kE-module M of stable constant Jordan type [1]^{s} such that

(i) if p = 2, then ℱ_{1}(M) ≅ ℱ.

(ii) if p is odd, then ℱ_{1}(M) ≅ F^{∗}(ℱ), the inverse image of ℱ along the Frobenius morphism F: ℙ^{r−1} → ℙ^{r−1}.

For p = 2 the theorem may be thought of as a version of the Bernstein–Gelfand–Gelfand correspondence [55], since the group algebra of an elementary abelian 2-group in characteristic two is isomorphic to an exterior algebra. See Section 8.13 for a clarification of this. But for p odd it says something new and interesting. In particular, it is striking that part (ii) of Theorem 8.1.1 cannot be strengthened to say that ℱ_{1}(M) ≅ ℱ. We shall prove in Section 10.2 that for a module M of stable constant Jordan type [1]^{s}, the Chern numbers c_{m}(ℱ_{1}(M)) are divisible by p for 1 ≤ m ≤ p − 2.

8.2. The Operator θ

Recall that k is an algebraically closed field of characteristic p, E = ⟨g_{1}, …, g_{r}⟩ is an elementary abelian p-group of rank r and X_{i} = g_{i} − 1 ∈ J(kE) for 1 ≤ i ≤ r. If

0 ≠ α = (λ_{1}, …, λ_{r}) ∈ 𝔸^{r},

we define

X_{α} = λ_{1}X_{1} + ⋯ + λ_{r}X_{r} ∈ kE,

an element of J(kE) satisfying X_{α}^{p} = 0.

We write k[Y_{1}, …, Y_{r}] for the coordinate ring k[𝔸^{r}], where the Y_{i} are the linear functions defined by Y_{i}(X_{j}) = δ_{ij} (Kronecker delta). We write ℙ^{r−1} for the corresponding projective space. Let 𝒪 be the structure sheaf on ℙ^{r−1}. If ℱ is a sheaf of 𝒪-modules and j ∈ ℤ, we write ℱ(j) for the jth twist ℱ ⊗_{𝒪} 𝒪(j). If M is a finitely generated kE-module, we write M̃ for the trivial vector bundle M ⊗_{k} 𝒪, so that M̃(j) = M ⊗_{k} 𝒪(j).^{1} Friedlander and Pevtsova [127] define maps of vector bundles θ_{M}: M̃(j) → M̃(j + 1), for all j ∈ ℤ, by the formula

θ_{M}(m ⊗ f) = ∑_{i=1}^{r} X_{i}(m) ⊗ Y_{i}f.

If the module M is clear from the context, we write θ for θ_{M}.

Remark 8.2.1. If we think of M as given by a degree n matrix representation with coefficients in k, then the space of homomorphisms from M̃ to M̃(1) is equal to Mat_{n}(Hom(𝒪, 𝒪(1))). Since Hom(𝒪, 𝒪(1)) is the vector space spanned by Y_{1}, …, Y_{r}, it follows that θ_{M} can be thought of as a matrix whose entries are linear forms in Y_{1}, …, Y_{r}. Namely, if ϕ_{M}: kE → Mat_{n}(k) gives the matrix representation for M then the matrix representing θ_{M} is

∑_{i} Y_{i}ϕ_{M}(X_{i}) ∈ Mat_{n}(k[Y_{1}, …, Y_{r}]).

Example 8.2.2. Let E = ⟨g_{1}, g_{2}⟩ ≅ (ℤ/p)^{2}, kE = k[X_{1}, X_{2}]/(X_{1}^{p}, X_{2}^{p}) and let M be the three dimensional kE-module given by

g_{1} ↦ (\begin{matrix}1 & 0 & 0 \\ 1 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}) g_{2} ↦ (\begin{matrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 1 & 0 & 1 \end{matrix}) .

Then

X_{1} ↦ (\begin{matrix}0 & 0 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \end{matrix}) X_{2} ↦ (\begin{matrix}0 & 0 & 0 \\ 0 & 0 & 0 \\ 1 & 0 & 0 \end{matrix}) ,

and the map θ_{M} is represented by the matrix

(\begin{matrix}0 & 0 & 0 \\ Y_{1} & 0 & 0 \\ Y_{2} & 0 & 0 \end{matrix}) ∈ Mat_{3}(k[Y_{1}, Y_{2}]).

1The use of a wide tilde here is to distinguish it from the tilde construction of a sheaf from a graded module as given in Section 3.8.

It can be seen that in some sense θ_{M} represents the action of the generic cyclic shifted subgroup of E.

Remark 8.2.3. More generally, a linear variety of matrices of constant rank, such as for example a nilvariety of constant Jordan type (see Definition 5.1.3) gives rise to an operator θ via exactly the same formula. So Example 4.1.6 gives rise to the following matrix:

θ = (\begin{matrix}0 & Y_{1} & 0 \\ Y_{2} & 0 & Y_{1} \\ 0 & −Y_{2} & 0 \end{matrix}) .

The reason this matrix does not represent the operator θ for a representation of an elementary abelian p-group is that its specialisations, for different values of Y_{1} and Y_{2}, do not commute.

8.3. The Action of θ on Fibres

Let M be a finitely generated kE-module and let

0 ≠ α = (λ_{1}, …, λ_{r}) ∈ 𝔸^{r}

with image

ᾱ = (λ_{1} : ⋯: λ_{r}) ∈ ℙ^{r−1}.

To find the fibre ℱ ⊗_{𝒪} k(ᾱ) of a sheaf ℱ at ᾱ, we must choose ℓ with λ_{ℓ} ≠ 0 and look at the affine patch U_{ℓ} ⊆ ℙ^{r−1} defined by the non-vanishing of the ℓth coordinate. Then the fibre of ℱ at ᾱ is

ℱ(U_{ℓ}) ⊗_{𝒪(U_{ℓ})} k

where

𝒪(U_{ℓ}) = k[U_{ℓ}] = k[Y_{1}Y_{ℓ}^{−1}, … ↑^{ℓ} …, Y_{r}Y_{ℓ}^{−1}]

and k is regarded as a k[U_{ℓ}]-module via evaluation at ᾱ.

Applying this to 𝒪, the fibre of 𝒪 at ᾱ is

k[U_{ℓ}] ⊗_{k[U_{ℓ}]} k = k.

On the other hand, for 𝒪(j) the sections are the degree j functions

𝒪(j)(U_{ℓ}) = k[Y_{1}Y_{ℓ}^{−1}, … ↑^{ℓ} …, Y_{r}Y_{ℓ}^{−1}]Y_{ℓ}^{j}

and the fibre of 𝒪(j) at ᾱ is

k[Y_{1}Y_{ℓ}^{−1}, … ↑^{ℓ} …, Y_{r}Y_{ℓ}^{−1}]Y_{ℓ}^{j} ⊗_{k[U_{ℓ}]} k.

If we normalise by choosing a particular value for λ_{ℓ}, for example λ_{ℓ} = 1, we obtain an identification of the left side of this tensor product with k[U_{ℓ}] and hence an identification of the fibre with k. Changing the normalisation to some other value of λ_{ℓ} changes this identification by the jth power of this factor.

To summarise, identifying the fibre of 𝒪(j) at ᾱ with k, we need to choose a point α ∈ 𝔸^{r} lying above ᾱ.

Since M̃(j) = M ⊗_{k} 𝒪(j), a choice of α ∈ 𝔸^{r} lying above ᾱ ∈ ℙ^{r−1} lets us identify the fibre of M̃(j) at ᾱ with M. In terms of this identification the action of θ on the fibre at ᾱ sends

m ⊗ 1 ⊗ 1 ∈ M ⊗_{k} k[U_{ℓ}] ⊗_{k[U_{ℓ}]} k ≅ M

to

∑_{i} X_{i}m ⊗ Y_{i} ⊗ 1 = ∑_{i} X_{i}m ⊗ 1 ⊗ λ_{i}

= ∑_{i} λ_{i}X_{i}m ⊗ 1 ⊗ 1

= X_{α}m ⊗ 1 ⊗ 1.

We summarise in a theorem.

Theorem 8.3.1. Let M be a kE-module and let θ: M̃(j) → M̃(j + 1) be the operator defined in Section 8.2. If ᾱ = (λ_{1} : ⋯: λ_{r}) ∈ ℙ^{r−1} then a choice of α = (λ_{1}, …, λ_{r}) ∈ 𝔸^{r} lying over ᾱ gives an identification of the fibre of M̃(j) at ᾱ with M. After making this identification, the action of θ on the fibre at ᾱ is given by the action of X_{α} on M. □

Remark 8.3.2. The same technique can be used to find the action on the fibre at the generic point for an irreducible subvariety of ℙ^{r−1}. Let ᾱ be the generic point for a homogeneous prime ideal 𝔭 ⊆ k[Y_{1}, …, Y_{r}]. As in Section 3.2, we get a closed point β̄ ∈ ℙ_{k(ᾱ)}^{r−1} whose

image is ᾱ. The fibre of M̃(j) at ᾱ is the same as the fibre of the bundle { \atop M ⊗_{k} k(ᾱ)}(j) at β̄, namely M ⊗_{k} k(ᾱ). Lift β̄ to an element β = (λ_{1}, …, λ_{r}) ∈ 𝔸_{k(ᾱ)}^{r}. Then the action of θ on the fibre of M̃(j) at ᾱ is given by the action of X_{β} = λ_{1}X_{1} + ⋯ + λ_{r}X_{r} on M ⊗_{k} k(ᾱ). By abuse of notation we shall write X_{α} for this X_{β}.

8.4. The Functors ℱ_{i} and ℱ_{i,j}

We define functors ℱ_{i,j} for 0 ≤ j < i ≤ p, from finitely generated kE-modules (or nilvarieties) M to coherent sheaves on ℙ^{r−1}, by taking the following subquotients of M̃:

ℱ_{i,j}(M) = \frac{Ker θ_{M}^{j+1} ∩ Im θ_{M}^{i−j−1}}{(Ker θ_{M}^{j+1} ∩ Im θ_{M}^{i−j}) + (Ker θ_{M}^{j} ∩ Im θ_{M}^{i−j−1})}.

We then define

ℱ_{i}(M) = ℱ_{i,0}(M) = \frac{Ker θ_{M} ∩ Im θ_{M}^{i−1}}{Ker θ_{M} ∩ Im θ_{M}^{i}} .

For a point 0 ≠ α ∈ 𝔸^{r} with residue field k(ᾱ) and the corresponding operator

X_{α}: M ⊗_{k} k(ᾱ) → M ⊗_{k} k(ᾱ),

we also define

ℱ_{i,α}(M) = \frac{Ker X_{α} ∩ Im X_{α}^{i−1}}{Ker X_{α} ∩ Im X_{α}^{i}} .

Note that ℱ_{i,α}(M) only depends on the image ᾱ ∈ ℙ^{r−1} of α ∈ 𝔸^{r}.

Example 8.4.1. Suppose that M has the constant image property. Then Im θ^{i} = M̃_{i} where M_{i} = Rad^{i}M. Thus

ℱ_{i}(M) ≅ \frac{(Ker θ ∩ Im θ^{i−1}) + Im θ^{i}}{Im θ^{i}} = \frac{(Ker θ ∩ { \atop M_{i−1}}) + M̃_{i}}{M̃_{i}}

can be thought of as the kernel of the map induced by θ from { \atop M_{i−1}/M_{i}} to { \atop M_{i}/M_{i+1}}. So we have a short exact sequence

0 → ℱ_{i}(M) → 𝒪^{⊕a_{i−1}} ⟶^{θ} 𝒪(1)^{⊕a_{i}} → 0

where a_{i} = dim(Rad^{i}M/Rad^{i+1}M). Thus the Chern polynomials are given by

c(ℱ_{i}(M)) = \frac{1}{(1 + h)^{a_{i}}} .

Example 8.4.2. Suppose that M has the constant kernel property. Then Ker θ = { \atop Soc M} is a direct sum of copies of 𝒪, and Ker θ ∩ Im θ^{i} = θ^{i}({ \atop M_{i+1}/M_{i}}) is a direct sum of copies of 𝒪(−i), where M_{i} = Soc^{i}M. So there is a short exact sequence

0 → 𝒪(−i)^{⊕b_{i+1}} ⟶^{θ} 𝒪(−i + 1)^{⊕b_{i}} → ℱ_{i}(M) → 0

where b_{i} = dim(Soc^{i}M/Soc^{i−1}M). Thus the Chern polynomials are given by

c(ℱ_{i}(M)) = \frac{(1 − (i − 1)h)^{b_{i}}}{(1 − ih)^{b_{i+1}}} .

The relationship between these two examples will be explained by Theorem 8.5.5.

In the next Proposition we show that functors ℱ_{i} take modules of constant Jordan type to algebraic vector bundles and that they commute with specialisation.

Proposition 8.4.3.

(1) Let M be a kE-module of constant Jordan type [p]^{a_{p}} … [1]^{a_{1}}. Then the sheaf ℱ_{i}(M) is a vector bundle of rank a_{i}.

(2) Let f: M → N be a map of modules of constant Jordan type. For any point ᾱ = (λ_{1} : ⋯: λ_{r}) ∈ ℙ^{r−1} with residue field k(ᾱ) we have a commutative diagram

\xymatrix{ℱ_{i}(M) ⊗_{𝒪} k(ᾱ) \ar[d]^{≃} \ar[r]^{ℱ_{i}(f)} & ℱ_{i}(N) ⊗_{𝒪} k(ᾱ) \ar[d]^{≃} \\ ℱ_{i,α}(M) \ar[r] & ℱ_{i,α}(N).}

Proof. (1) Since the module M is fixed throughout the proof, we shall use θ to denote θ_{M}.

Note that Ker θ∩Im θ^{i} = Ker {θ: Im θ^{i} → Im θ^{i+1}}. Hence, we have a short exact sequence

(8.4.1)  \xymatrix{0 \ar[r] & Ker θ ∩ Im θ^{i} \ar[r] & Im θ^{i} \ar[r]^{θ} & Im θ^{i+1} \ar[r] & 0}

Since M has constant Jordan type, Im θ^{i} is a vector bundle by Theorem 6.2.1. Therefore, specialisation of the sequence (8.4.1) at any point ᾱ = (λ_{1} : ⋯: λ_{r}) of ℙ^{r−1} yields a short exact sequence of vector spaces

(8.4.2)  0 → (Ker θ ∩ Im θ^{i}) ⊗_{𝒪} k(ᾱ) → Im θ^{i} ⊗_{𝒪} k(ᾱ) → Im θ^{i+1} ⊗_{𝒪} k(ᾱ) → 0.

By Theorem 6.2.1, Im θ^{i} ⊗_{𝒪} k(ᾱ) ≅ Im {X_{α}^{i}: M → M}. In particular, the dimension of fibres

of Im θ^{i} is constant and equals ∑_{j=i+1}^{p} a_{j}(j − i). We can rewrite the sequence (8.4.2) as

\xymatrix{0 \ar[r] & (Ker θ ∩ Im θ^{i}) ⊗_{𝒪} k(ᾱ) \ar[r] & Im X_{α}^{i} \ar[r]^{X_{α}} & Im X_{α}^{i+1} \ar[r] & 0}.

Hence the fibre of Ker θ ∩Im θ^{i} at a point ᾱ equals Ker X_{α} ∩Im X_{α}^{i}. In particular, Ker θ ∩Im θ^{i} has fibres of constant dimension, equal to

∑_{j=i+1}^{p} a_{j}(j − i) − ∑_{j=i+2}^{p} a_{j}(j − i − 1) = ∑_{j=i+1}^{p} a_{j}.

Applying Theorem 6.2.1 again, we conclude that Ker θ ∩ Im θ^{i} is a vector bundle of rank

∑_{j=i+1}^{p} a_{j}.

Consider the short exact sequence that defines ℱ_{i}(M):

\xymatrix{0 \ar[r] & Ker θ ∩ Im θ^{i} \ar[r] & Ker θ ∩ Im θ^{i−1} \ar[r] & ℱ_{i}(M) \ar[r] & 0.}

Specialising at ᾱ, we get

[figure: (Ker θ ∩ Im θ^{i}) ⊗_{𝒪} k(ᾱ); (Ker θ ∩ Im θ^{i−1}) ⊗_{𝒪} k(ᾱ); ℱ_{i}(M) ⊗_{𝒪} k(ᾱ); 0; Ker X_{α} ∩ Im X_{α}^{i}; Ker X_{α} ∩ Im X_{α}^{i−1}; ℱ_{i}(M) ⊗_{𝒪} k(ᾱ); 0]

The first arrow of the bottom row is clearly an injection. Hence

dim(ℱ_{i}(M) ⊗_{𝒪} k(ᾱ)) = ∑_{j=i}^{p} a_{j} − ∑_{j=i+1}^{p} a_{j} = a_{i}

for any point ᾱ ∈ ℙ^{r−1}. Applying Theorem 6.2.1 again, we conclude that ℱ_{i}(M) is locally free (of rank a_{i}).

Statement (2) follows immediately by applying the last diagram to both M and N. □

The following proposition examines the exactness properties of the functors ℱ_{i}. Recall from Section 5.3 that cJt(kE) is the exact category whose objects are the modules of constant Jordan type and whose exact sequences are the locally split exact sequences of such modules. We write Vec(ℙ^{r−1}) for the category of vector bundles over ℙ^{r−1} and morphisms of vector bundles.

Proposition 8.4.4. The functors ℱ_{i}: cJt(kE) → Vec(ℙ^{r−1}) are exact for 1 ≤ i ≤ p.

Proof. Let 1 ≤ i ≤ p and let

0 → M_{1} → M_{2} → M_{3} → 0

be a locally split exact sequence of modules of constant Jordan type. Then the Jordan type of the middle term is the sum of the Jordan types of the end terms. Hence

(8.4.3)  rank ℱ_{i}(M_{2}) = rank ℱ_{i}(M_{1}) + rank ℱ_{i}(M_{3}).

Consider the map ℱ_{i}(M_{2}) → ℱ_{i}(M_{3}). By Proposition 8.4.3 if ᾱ ∈ ℙ^{r−1} then

ℱ_{i}(M_{2}) ⊗_{𝒪} k(ᾱ) → ℱ_{i}(M_{3}) ⊗_{𝒪} k(ᾱ)

is surjective for all ᾱ ∈ ℙ^{r−1}. So by Theorem 6.2.1, ℱ_{i}(M_{2}) → ℱ_{i}(M_{3}) is surjective. The same argument applied to ℱ_{i}(M_{1}) → ℱ_{i}(M_{2}) shows that it is injective. Finally, equation (8.4.3) implies exactness in the middle. □

Remark 8.4.5. In Example 8.5.6 and Section 8.12, we shall see that the functors ℱ_{i} do not in general take locally split exact sequences of modules of constant Jordan type to split exact sequences of vector bundles.

Lemma 8.4.6. Let M be a kE-module. The trivial bundle M̃ has a filtration in which the filtered quotients are isomorphic to ℱ_{i,j}(M) for 0 ≤ j < i ≤ p.

Proof. We consider two filtrations on M̃, by kernels and image:

0 ⊆ Ker θ_{M} ⊆ … ⊆ Ker θ_{M}^{p−1} ⊆ M̃
0 = Im θ_{M}^{p} ⊆ Im θ_{M}^{p−1} ⊆ … ⊆ Im θ_{M} ⊆ Im θ_{M}^{0} = M̃

To simplify notation, we set 𝒦_{j} = Ker θ_{M}^{i} and ℐ_{i} = Im θ_{M}^{p−i}. Using the standard refinement procedure, we refine the kernel filtration by the image filtration:

𝒦_{j} ⊆ (𝒦_{j+1} ∩ ℐ_{1}) + 𝒦_{j} ⊆ … ⊆ (𝒦_{j+1} ∩ ℐ_{ℓ}) + 𝒦_{j} ⊆ (𝒦_{j+1} ∩ ℐ_{ℓ+1}) + 𝒦_{j} ⊆ … ⊆ 𝒦_{j+1}

For any three sheaves A, B, C with B ⊆ A, the second isomorphism theorem and the modular law imply that

\frac{A + C}{B + C} = \frac{A + (B + C)}{B + C} ≅ \frac{A}{A ∩ (B + C)} = \frac{A}{B + (A ∩ C)}.

Hence, we can identify the subquotients of the refined kernel filtration above as

\frac{(𝒦_{j+1} ∩ ℐ_{ℓ+1}) + 𝒦_{j}}{(𝒦_{j+1} ∩ ℐ_{ℓ}) + 𝒦_{j}} ≅ \frac{𝒦_{j+1} ∩ ℐ_{ℓ+1}}{(𝒦_{j+1} ∩ ℐ_{ℓ}) + (𝒦_{j} ∩ ℐ_{ℓ+1})}.

Setting i = p − ℓ + j, we get that the latter quotient is precisely ℱ_{i,j}(M) (note that when j > ℓ, the corresponding subquotient is trivial). □

Essentially the same proof gives the following.

Lemma 8.4.7. Let M be a kE-module.

(1) The subsheaf Ker (θ_{M}^{i}) ⊆ M̃ has a filtration whose filtered quotients are isomorphic to ℱ_{j,ℓ}(M) with ℓ ≤ i.

(2) The subsheaf Im (θ_{M}^{i}) ⊆ M̃ has a filtration whose filtered quotients are isomorphic to ℱ_{j,ℓ}(M) with j − ℓ > i. □

Lemma 8.4.8. For 0 ≤ j < i, we have ℱ_{i,j}(M) ≅ ℱ_{i}(M)(j).

Proof. For 0 < j < i, the map θ_{M}: M̃ → M̃(1) induces a natural isomorphism ℱ_{i,j}(M) → ℱ_{i,j−1}(M)(1). Since ℱ_{i,0} = ℱ_{i}, the result follows by induction on j. □

Theorem 8.4.9. Let M be a kE-module. The trivial sheaf M̃ has a filtration in which the filtered quotients are isomorphic to ℱ_{i}(M)(j) for 0 ≤ j < i ≤ p.

Proof. This follows from Lemmas 8.4.6 and 8.4.8. □

Lemma 8.4.10. Suppose that M has constant Jordan type [p]^{a_{p}} … [1]^{a_{1}}. Then the sheaf ℱ_{i,j}(M) is a vector bundle of rank a_{i}.

Proof. This follows from Proposition 8.4.3 and Lemma 8.4.8. □

I’d like to thank Julia Pevtsova for help with the proof of the following theorem.

Theorem 8.4.11. Let M be a kE-module. Then the following are equivalent.

(1) M has constant Jordan type.

(2) ℱ_{i}(M) is a vector bundle for each i with 1 ≤ i ≤ p.

Proof. If M has constant Jordan type then by Proposition 8.4.3 (1) ℱ_{i}(M) is a vector bundle for 1 ≤ i ≤ p. Conversely, if ℱ_{i}(M) is a vector bundle for 1 ≤ i ≤ p then also ℱ_{j,ℓ}(M) is a vector bundle for 0 ≤ ℓ < j ≤ p by Lemma 8.4.8. By Lemma 8.4.7 (2), Coker (θ_{M}^{i}) has a filtration in which the filtered subquotients are isomorphic to ℱ_{j,ℓ}(M) with j − ℓ ≤ i. So by Proposition 6.2.5 Coker (θ_{M}^{i}) is a vector bundle for 1 ≤ i ≤ p. Since tensor product is right exact, for 0 ≠ α ∈ 𝔸^{r}(k) we have

Coker (θ_{M}^{i} ⊗_{𝒪} k(ᾱ)) ≅ Coker (θ_{M}^{i}) ⊗_{𝒪} k(ᾱ) ≅ Coker (X_{α}^{i}, M)

and so X_{α}^{i} has constant rank on M for 1 ≤ i ≤ p. Thus M has constant Jordan type. □

Remark 8.4.12. It follows from the proof of Proposition 8.4.3 that the subquotient functors ℱ_{i,j} are linked as follows:

[figure: ℱ_{p,p−1}; ℱ_{p−1,p−2}; ℱ_{p,p−2}; ℱ_{p−2,p−3}; ℱ_{p−1,p−3}; …; …; …; …; …; ℱ_{1,0}; ℱ_{p−1,1}; …; ℱ_{p,1}; ℱ_{p−2,0}; ℱ_{p−1,0}; ℱ_{p,0}]

This should be compared with the corresponding diagram in Section 2.2.

The following is an immediate consequence of the definitions.

Proposition 8.4.13. If M is a homogeneous kE-module then the vector bundles ℱ_{i}(M) are homogeneous. □

8.5. Twists and Syzygies

Theorem 8.5.1. Let M be a finite dimensional kE-module and let 1 ≤ i ≤ p − 1. Then there is a natural isomorphism

ℱ_{i}(M)(−p + i) ≅ ℱ_{p−i}(ΩM).

Proof. Consider the diagram

(8.5.1)  \xymatrix{0 \ar[r] & Ω̃M \ar[r] \ar[d]^{θ_{ΩM}} & P̃_{M} \ar[r] \ar[d]^{θ_{P_{M}}} & M̃ \ar[r] \ar[d]^{θ_{M}} & 0 \\ 0 \ar[r] & Ω̃M(1) \ar[r] & P̃_{M}(1) \ar[r] & M̃(1) \ar[r] & 0,}

where P_{M} is a projective cover of M. Let

δ: Ker θ_{M} → Coker θ_{ΩM}

be the switchback map. A simple diagram chase in conjunction with the fact that θ_{P_{M}}^{p} = 0 yields that the restriction of δ to Ker θ_{M} ∩ Im θ_{M}^{i−1} lands in

\frac{Ker θ_{ΩM}^{p−i}}{Ker θ_{ΩM}^{p−i} ∩ Im θ_{ΩM}} (1).

Projecting the latter onto

ℱ_{p−i,p−i−1}(ΩM)(1) = \frac{Ker θ_{ΩM}^{p−i}}{Ker θ_{ΩM}^{p−i−1} + Ker θ_{ΩM}^{p−i} ∩ Im θ_{ΩM}} (1),

we get a map of bundles:

δ′: Ker θ_{M} ∩ Im θ_{M}^{i−1} → ℱ_{p−i,p−i−1}(ΩM)(1).

Since δ′ evidently kills Ker θ_{M} ∩ Im θ_{M}^{i}, we conclude that δ′ factors through ℱ_{i}(M). Hence, we have an induced map

δ̄: ℱ_{i}(M) → ℱ_{p−i,p−i−1}(ΩM)(1).

A simple block count shows that this is an isomorphism at each fiber. Hence by Theorem 6.2.1 this is an isomorphism of bundles. Thus using Lemma 8.4.8 (i.e., applying θ_{ΩM} a further p − i − 1 times), we have

ℱ_{i}(M) ≅ ℱ_{p−i,p−i−1}(ΩM)(1) ≅ ℱ_{p−i}(ΩM)(p − i).

Twisting by 𝒪(−p + i), we get the desired isomorphism.

Let f: M → N be a map of kE-modules. The naturality of the isomorphism

ℱ_{i}(M)(−p + i) ≅ ℱ_{p−i}(ΩM)

is equivalent to the commutativity of the diagram

(8.5.2)  \xymatrix{ℱ_{i}(M)(−p + i) \ar[d]^{≅} \ar[r]^{ℱ_{i}(f)(−p+i)} & ℱ_{i}(N)(−p + i) \ar[d]^{≅} \\ ℱ_{p−i}(ΩM) \ar[r]^{ℱ_{p−i}(Ωf)} & ℱ_{p−i}(ΩN).}

Note that although ΩM is only well defined up to projective summands, and Ωf is only well defined up to maps that factor through a projective, nonetheless ℱ_{p−i} is zero on projective modules and the map ℱ_{p−i}(Ωf) is well defined. The commutativity follows from the construction of the map δ̄ and naturality of the isomorphism of Lemma 8.4.8. □

Corollary 8.5.2. Let M be a finite dimensional kE-module and let 1 ≤ i ≤ p − 1. Then ℱ_{i}(Ω^{2}M) ≅ ℱ_{i}(M)(−p).

Proof. Apply the theorem twice. □

Corollary 8.5.3. We have ℱ_{1}(Ω^{2n}k) ≅ 𝒪(−np), and ℱ_{p−1}(Ω^{2n−1}k) ≅ 𝒪(1 − np).

Proof. This follows from the theorem and the corollary, using the isomorphism ℱ_{1}(k) ≅ 𝒪. □

Remark 8.5.4. If p = 2 then Theorem 8.5.1 and Corollary 8.5.3 reduce to the statements that ℱ_{1}(ΩM) ≅ ℱ_{1}(M)(−1) and ℱ_{1}(Ω^{n}k) ≅ 𝒪(−n).

Theorem 8.5.5. Let M^{∗} be the k-linear dual of M, as a kE-module. Then

ℱ_{i}(M^{∗}) ≅ ℱ_{i}(M)^{∨}(−i + 1).

Proof. This follows from the more obvious isomorphism

ℱ_{i,i−1}(M^{∗}) ≅ ℱ_{i,0}(M)^{∨}

together with Lemma 8.4.8. □

Example 8.5.6. There is a locally split short exact sequence of modules of constant Jordan type for ℤ/2 × ℤ/2 in characteristic two:

0 → Ω^{2}(k) → Ω(k) ⊕ Ω(k) → k → 0.

Applying ℱ_{1} to this sequence gives the short exact sequence of sheaves

0 → 𝒪(−2) → 𝒪(−1) ⊕ 𝒪(−1) → 𝒪 → 0

on ℙ^{1}. This example shows that in Proposition 8.4.4, the functors ℱ_{i} do not in general take locally split sequences of modules of constant Jordan type to split short exact sequences of vector bundles.

8.6. Chern Numbers of ℱ_{i}(M)

Lemma 8.6.1. Let M be a kE-module of constant Jordan type.

(i) As elements of ℤ[h]/(h^{r}) we have

∏_{0≤j<i≤p}(1 + jh)^{rank ℱ_{i}(M)}c (ℱ_{i}(M), \frac{h}{1 + jh}) = ∏_{0≤j<i≤p} c(ℱ_{i}(M)(j)) = 1.

(ii) As elements of ℚ[h]/(h^{r}) we have

∑_{1≤i≤p} (\frac{1 − e^{ih}}{1 − e^{h}} ) Ch(ℱ_{i}(M)) = ∑_{0≤j<i≤p} Ch(ℱ_{i}(M)(j)) = dim M.

Proof. Combining Proposition 7.3.2 with Lemma 8.4.6, we have

∏_{0≤j<i≤p} c(ℱ_{i,j}(M)) = c(M̃) = 1,
∑_{0≤j<i≤p} Ch(ℱ_{i,j}(M)) = Ch(M̃) = dim M.

Using Lemma 8.4.8 we can rewrite these as

∏_{0≤j<i≤p} c(ℱ_{i}(M)(j)) = 1,  ∑_{0≤j<i≤p} Ch(ℱ_{i}(M)(j)) = dim M.

The lemma now follows by applying the formula in Lemma 7.5.1 for the Chern polynomial and Chern character of the twists of a bundle. □

Proposition 8.6.2. Let M be a kE-module of constant Jordan type. Then for 1 ≤ i ≤ p − 1 we have

(i) c(ℱ_{p−i}(Ω(M), h)) = (1 + (−p + i)h)^{rank ℱ_{i}(M)}c (ℱ_{i}(M), \frac{h}{1 + (−p + i)h}).

(ii) Ch(ℱ_{p−i}(Ω(M))) = e^{(−p+i)h}Ch(ℱ_{i}(M)).

(iii) c(ℱ_{i}(Ω^{2}(M), h)) = (1 − ph)^{rank ℱ_{i}(M)}c (ℱ_{i}(M), \frac{h}{1 − ph}).

(iv) Ch(ℱ_{i}(Ω^{2}(M))) = e^{−ph}Ch(ℱ_{i}(M)).

(v) c(ℱ_{i}(M^{∗}), h) = (1 − (i − 1)h)^{rank ℱ_{i}(M)}c (ℱ_{i}(M), \frac{−h}{1 − (i − 1)h}).

(vi) Ch(ℱ_{i}(M^{∗}), h) = e^{−(i−1)h}Ch(ℱ_{i}(M), −h).

Proof. Parts (i)-(iv) follow by combining Lemma 7.5.1 with Theorem 8.5.1 and Corollary 8.5.2. Parts (v) and (vi) follow by combining Lemmas 7.5.2 and 7.5.1 with Theorem 8.5.5:

c(ℱ_{i}(M^{∗}), h) = c(ℱ_{i}(M)^{∨}(−i + 1), h)

= c(ℱ_{i}(M)(i − 1), −h)

= (1 − (i − 1)h)^{rank ℱ_{i}(M)}c (ℱ_{i}(M), \frac{−h}{1 − (i − 1)h}) .

and

Ch(ℱ_{i}(M^{∗}), h) = Ch(ℱ_{i}(M)^{∨}(−i + 1), h)
= Ch(ℱ_{i}(M)(i − 1), −h)
= e^{−(i−1)h}Ch(ℱ_{i}(M), −h).  □

Proposition 8.6.3. Let M be a kE-module of constant Jordan type. Then as elements of ℚ[h]/(h^{r}) we have

Ch(ℱ_{p}(Ω^{2}(M))) = e^{−ph}Ch(ℱ_{p}(M)) + (\frac{1 − e^{h}}{1 − e^{ph}} ) (dim Ω^{2}(M) − e^{−ph} dim M).

Proof. Applying Lemma 8.6.1 to Ω^{2}(M) and M, together with Lemma 7.5.1 and Corollary 8.5.2 we have

(\frac{1 − e^{ph}}{1 − e^{h}} ) Ch(ℱ_{p}(Ω^{2}(M))) = ∑_{j=0}^{p−1} Ch(ℱ_{p}(Ω^{2}(M)(j))

= dim Ω^{2}(M) − ∑_{0≤j<i<p} Ch(ℱ_{i}(Ω^{2}(M))(j))

= dim Ω^{2}(M) − ∑_{0≤j<i<p} Ch(ℱ_{i}(M)(j − p))

= dim Ω^{2}(M) − e^{−ph} ∑_{0≤j<i<p} Ch(ℱ_{i}(M)(j))

= dim Ω^{2}(M) − e^{−ph} (dim M − ∑_{j=0}^{p−1} Ch(ℱ_{p}(M)(j)))

= dim Ω^{2}(M) − e^{−ph} (dim M − (\frac{1 − e^{ph}}{1 − e^{h}} ) Ch(ℱ_{p}(M))) .

Now multiply both sides by \frac{1 − e^{h}}{1 − e^{ph}}. □

8.7. The Construction: p = 2

In this section and the next we give the construction used in the proof of Theorem 8.1.1 in the two cases p = 2 and p odd. Then in Section 8.9 we give the details of the proof using this construction. Since H^{1}(E, k) is the vector space dual of J(kE)/J^{2}(kE), there are elements y_{1}, …, y_{r} forming a vector space basis for H^{1}(E, k) and corresponding to the linear functions Y_{1}, …, Y_{r} on J(kE)/J^{2}(kE) introduced in Section 8.2. Because of the difference in structure of the cohomology ring, we divide the discussion into two cases, according as p = 2 or p is odd. In this section we deal with the case p = 2, and in the next section we explain the modifications needed for the slightly more difficult case where p is odd.

So now we assume p = 2. In this case the cohomology ring H^{∗}(E, k) is the polynomial algebra k[y_{1}, …, y_{r}]. We define a k-algebra homomorphism

ρ: H^{∗}(E, k) = k[y_{1}, …, y_{r}] → k[Y_{1}, …, Y_{r}]

by ρ(y_{i}) = Y_{i}.

Lemma 8.7.1. If ζ ∈ H^{n}(E, k) is represented by a cocycle ζ̂: Ω^{n+j}k → Ω^{j}k (with j ∈ ℤ) then the diagram

\xymatrix{𝒪(−n − j) \ar[d]^{≅} \ar[r]^{ρ(ζ)} & 𝒪(−j) \ar[d]^{≅} \\ ℱ_{1}(Ω^{n+j}k) \ar[r]^{ℱ_{1}(ζ̂)} & ℱ_{1}(Ω^{j}k)}

commutes.

Proof. Consider ζ̂: Ω^{n}k → k. The commutative diagram (8.5.2) applied to ζ̂ and iterated j times becomes

\xymatrix{ℱ_{1}(Ω^{n}k)(−j) \ar[d]^{≃} \ar[r]^{ℱ_{1}(ζ̂)(−j)} & ℱ_{1}(k)(−j) \ar[d]^{≃} \\ ℱ_{1}(Ω^{n+j}k) \ar[r]^{ℱ_{1}(Ω^{j} ζ̂)} & ℱ_{1}(Ω^{j}k).}

Hence, it suffices to assume that j = 0. Additivity of the functor ℱ_{1} allows us to assume that ζ is a monomial on generators y_{1}, …, y_{r}. Finally, since multiplication in cohomology corresponds to composition of the corresponding maps on Heller shifts of k, it suffices to prove our statement for a generator ζ = y_{i} of degree one.

In the case j = 0, ζ = y_{i}, we need to show that the following diagram commutes (this is the diagram above twisted by 𝒪(1)):

(8.7.1)  [figure: ℱ_{1}(k); ≃; Y_{i}; ℱ_{1}(k)(1); ℱ_{1}(Ωk)(1); ℱ_{1}(ŷ_{i})(1); ℱ_{1}(k)(1).]

Let E_{i} be the subgroup of index two in E such that y_{i} is inflated from E/E_{i} to E, namely the subgroup generated by all of g_{1}, …, g_{r} except g_{i}. Then y_{i} represents the class of the extension

0 → k → M_{i} → k → 0

where M_{i} is the permutation module on the cosets of E_{i}. This is a module of length two on which X_{1}, …, X_{r} act as zero except for X_{i}, which acts as a Jordan block of length two. We have a commutative diagram of kE-modules

(8.7.2)  [figure: 0; Ωk; y_{i}; P_{0}; k; 0; 0; k; M_{i}; k; 0.]

The left vertical isomorphism \overline{δ}: ℱ_{1}(k) →^{∼} ℱ_{1}(Ωk)(1) of the diagram (8.7.1) is given by the switchback map for the short exact sequence

0 → Ωk → P_{0} → k → 0

as in Diagram (8.5.1). Applying θ to the commutative diagram on free 𝒪-modules induced by the module diagram (8.7.2), we get a commutative diagram

[figure: 0; Ω̃k; ỹ_{i}; P̃_{0}; 𝒪; 0; 0; Ω̃k(1); ỹ_{i}(1); P̃_{0}(1); 𝒪(1); 0; 0; 𝒪; M̃_{i}; 𝒪; 0; 0; 𝒪(1); M̃_{i}(1); 𝒪(1); 0]

where all horizontal arrows going from the back to the front are given by the operator θ on the corresponding module. In the bottom layer of this diagram,

\xymatrix{0 \ar[r] & 𝒪 \ar[r] \ar[d]^{θ_{k}} & M̃_{i} \ar[r] \ar[d]^{θ_{M_{i}}} & 𝒪 \ar[r] \ar[d]^{θ_{k}} & 0 \\ 0 \ar[r] & 𝒪(1) \ar[r] & M̃_{i}(1) \ar[r] & 𝒪(1) \ar[r] & 0}

the left- and right-hand maps θ_{k} are zero.

The map ỹ_{i}: Ω̃k → 𝒪 is induced by y_{i}: Ωk → k. To compute the composite ℱ_{1}(ŷ_{i})(1) ∘ \overline{δ} we first do the switchback map of the top layer and then push the result down via ỹ_{i}(1). Since the diagram is commutative, we can first push down via the identity map of the right vertical back arrow and then do the switchback of the bottom layer. Hence, the composite ℱ_{1}(ŷ_{i})(1) ∘ \overline{δ} is given by the switchback map \overline{δ}: 𝒪 → 𝒪(1) of the bottom layer. The action of θ_{M_{i}} is given by the matrix

(\begin{matrix}0 & 0 \\ Y_{i} & 0 \end{matrix})

(see Remark 8.2.1), and so the switchback map is multiplication by Y_{i}. □

8.8. The Construction: p Odd

In this section we turn to construction used in the proof of Theorem 8.1.1 in the case where p is odd. We write β: H^{1}(E, k) → H^{2}(E, k) for the Bockstein map, and we set x_{i} = β(y_{i}). In terms of Massey products, this is given by x_{i} = −(y_{i}, y_{i}, …, y_{i}) (p terms).

The cohomology ring is a tensor product of an exterior algebra with a polynomial algebra:

H^{∗}(E, k) ≅ Λ(y_{1}, …, y_{r}) ⊗_{k} k[x_{1}, …, x_{r}],

see Section 1.8. We define a k-algebra homomorphism

ρ: k[x_{1}, …, x_{r}] → k[Y_{1}, …, Y_{r}]

by ρ(x_{i}) = Y_{i}^{p}.

Lemma 8.8.1. Let ζ be a degree n polynomial in k[x_{1}, …, x_{r}], regarded as an element of H^{2n}(E, k). If ζ is represented by a cocycle ζ̂: Ω^{2(n+j)}k → Ω^{2j}k then the diagram

\xymatrix{𝒪(−p(n + j)) \ar[r]^{ρ(ζ)} \ar[d]^{≅} & 𝒪(−pj) \ar[d]^{≅} \\ ℱ_{1}(Ω^{2(n+j)}k) \ar[r]^{ℱ_{1}(ζ̂)} & ℱ_{1}(Ω^{2j}k)}

commutes.

Proof. The proof is similar to the proof in the case p = 2, but more complicated. Again it suffices to treat the case where ζ = x_{i} and j = 0. In other words, we need to compute the composite ℱ_{1}(x̂_{i}) ∘ f in the diagram

[figure: ℱ_{1}(k); ≃; f; Y_{i}^{p}; ℱ_{1}(k)(p); ℱ_{1}(Ω^{2}k)(p); ℱ_{1}(x̂_{i}); ℱ_{1}(k)(p).]

where f: ℱ_{1}(k) → ℱ_{1}(Ω^{2}k)(p) is the isomorphism of Corollary 8.5.2. Let

\xymatrix{0 \ar[r] & Ω^{2}k \ar[r] & P_{1} \ar[r] & P_{0} \ar[r] & k \ar[r] & 0}

be a truncated projective resolution of k. Tracing through the proof of Theorem 8.5.1, we see that f: ℱ_{1}(k) → ℱ_{1}(Ω^{2}k)(p) is a composite of three maps:

(1) the switchback of the top two rows of the diagram (8.8.1) below which gives the isomorphism ℱ(k) → ℱ_{p−1,p−2}(Ωk)(1),

(2) followed by the isomorphism

θ_{Ωk}^{p−2} : ℱ_{p−1,p−2}(Ωk)(1) → ℱ_{p−1}(Ωk)(p − 1)

of Lemma 8.4.8,

(3) followed by another switchback map, now for the bottom two rows of diagram (8.8.1), which gives the isomorphism ℱ_{p−1}(Ωk)(p − 1) ≃ ℱ_{1}(Ω^{2}k)(p).

(8.8.1) \xymatrix{ &  & 0 \ar[r] & Ω̃k \ar[r] \ar[d] & P̃_{0} \ar[d] \ar[r] & 𝒪 \ar[r] \ar[d] & 0 \\  &  & 0 \ar[r] & Ω̃k(1) \ar[d]^{θ_{Ωk}^{p−2}} \ar[r] & P̃_{0}(1) \ar[r] & 𝒪(1) \ar[r] & 0 \\ 0 \ar[r] & Ω̃^{2}k(p − 1) \ar[r] \ar[d] & P̃_{1}(p − 1) \ar[r] \ar[d] & Ω̃k(p − 1) \ar[r] \ar[d] & 0 \\ 0 \ar[r] & Ω̃^{2}k(p) \ar[r] & P̃_{1}(p) \ar[r] & Ω̃k(p) \ar[r] & 0}

Let E_{i} be the subgroup of index p such that x_{i} is inflated from E/E_{i}, namely the subgroup generated by all of g_{1}, …, g_{r} except for g_{i}. We let M_{i} be the permutation module on the cosets of E_{i}. This is a module of length p on which X_{1}, …, X_{r} act as zero except for X_{i}, which acts as a Jordan block of length p. Then x_{i} represents the class of the 2-fold extension

\xymatrix{0 \ar[r] & k \ar[r] & M_{i} \ar[r] & M_{i} \ar[r] & k \ar[r] & 0}

where the middle map is multiplication by X_{i}. We construct a diagram analogous to (8.8.1) for this extension:

(8.8.2) \xymatrix{ &  & 0 \ar[r] & Ñ_{i} \ar[r] \ar[d] & M̃_{i} \ar[d] \ar[r] & 𝒪 \ar[r] \ar[d] & 0 \\  &  & 0 \ar[r] & Ñ_{i}(1) \ar[d]^{θ_{N_{i}}^{p−2}} \ar[r] & M̃_{i} \ar[r] & 𝒪(1) \ar[r] & 0 \\ 0 \ar[r] & 𝒪(p − 1) \ar[r] \ar[d] & M̃_{i}(p − 1) \ar[r] \ar[d] & Ñ_{i}(p − 1) \ar[r] \ar[d] & 0 \\ 0 \ar[r] & 𝒪(p) \ar[r] & M̃_{i}(p) \ar[r] & 𝒪(p) \ar[r] & 0.}

Here, N_{i} = Im {X_{i}: M_{i} → M_{i}}. Just as in the proof of Lemma 8.8.1, the module diagram

[figure: 0; Ω^{2}k; x_{i}; P_{1}; P_{0}; k; 0; 0; k; M_{i}; M_{i}; k; 0]

induces a commutative diagram of vector bundles with (8.8.1) on top and (8.8.2) at the bottom. Arguing as in the proof of Lemma 8.7.1, we compute the composite ℱ_{1}(x̂_{i}) ∘ f by first mapping the rightmost 𝒪 of diagram (8.8.1) identically to the rightmost 𝒪 of Diagram (8.8.2), and then applying our composite of a switchback, followed by θ_{N_{i}}^{p−2}, followed by another switchback in the diagram (8.8.2). The maps θ_{M_{i}}: M̃_{i} → M̃_{i}(1) and θ_{N_{i}}: Ñ_{i} → Ñ_{i}(1) are simply multiplication by Y_{i}. Since the leftmost and rightmost vertical arrows in (8.8.2) are zero, to compute the composite of the three maps involved in diagram (8.8.2), we have to multiply first by Y_{i}, then by Y_{i}^{p−2}, then by Y_{i} again. Hence, ℱ_{1}(x̂_{i}) ∘ f = Y_{i}^{p}. □

8.9. Proof of the Realisation Theorem

We are ready to prove the main theorem of this chapter, Theorem 8.1.1. We begin with a couple of lemmas. The following lemma will be applied to modules over K[X_{α}]/(X_{α}^{p}), where projective and injective modules coincide.

Lemma 8.9.1. Let ϕ = (\begin{matrix}a & b \\ c & d \end{matrix}) : A ⊕ P → B ⊕ Q be an injective map of modules, with

a: A → B a split injection and P an injective module. Then ϕ is a split injection.

Proof. Let B = B_{1} ⊕ B_{2}, where B_{1} = Im a, and let

a = (\begin{matrix}a_{1} \\ 0 \end{matrix}) : A → B_{1} ⊕ B_{2}, b = (\begin{matrix}b_{1} \\ b_{2} \end{matrix}) : P → B_{1} ⊕ B_{2}.

With this notation, we have

ϕ = (\begin{matrix}a_{1} & b_{1} \\ 0 & b_{2} \\ c & d \end{matrix}) : A ⊕ P → B_{1} ⊕ B_{2} ⊕ Q.

Since a_{1} is an isomorphism, there exist maps u : B_{1} → Q, v : P → A such that c = ua_{1} and b_{1} = a_{1}v. Hence,

(\begin{matrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ −u & 0 & 1 \end{matrix}) (\begin{matrix}a_{1} & b_{1} \\ 0 & b_{2} \\ c & d \end{matrix}) (\begin{matrix}1 & −v \\ 0 & 1 \end{matrix}) = (\begin{matrix}a_{1} & 0 \\ 0 & b_{2} \\ 0 & d − ub_{1} \end{matrix}) : A ⊕ P → B_{1} ⊕ B_{2} ⊕ Q.

By assumption, this map is injective. Hence, the map

(\begin{matrix}b_{2} \\ d − ub_{1} \end{matrix}) : P → B_{2} ⊕ Q

is injective and therefore split. □

Lemma 8.9.2. Let 0 → A → B → C → 0 be a short exact sequence of kE-modules, such that A and B have trivial stable constant Jordan type and ℱ_{1}(A) → ℱ_{1}(B) is injective. Then C is of trivial stable constant Jordan type and ℱ_{1}(C) is the cokernel of ℱ_{1}(A) → ℱ_{1}(B).

Proof. Applying Lemma 8.9.1 to the restriction of the sequence to K[X_{α}]/(X_{α}^{p}) for each 0 ≠ α ∈ 𝔸^{r}(k), we conclude that this sequence is locally split. Hence, C is a module of trivial stable constant Jordan type. By Proposition 8.4.4, applying ℱ_{1} gives an exact sequence

0 → ℱ_{1}(A) → ℱ_{1}(B) → ℱ_{1}(C) → 0.  □

Theorem 8.9.3. Given any vector bundle ℱ of rank s on ℙ^{r−1}, there exists a finitely generated kE-module M of stable constant Jordan type [1]^{s} such that

(i) if p = 2, then ℱ_{1}(M) ≅ ℱ.

(ii) if p is odd, then ℱ_{1}(M) ≅ F^{∗}(ℱ), the inverse image of ℱ along the Frobenius morphism F: ℙ^{r−1} → ℙ^{r−1}.

Proof. Given a vector bundle ℱ on ℙ^{r−1}, using Corollary 6.12.5 of the Hilbert syzygy theorem we can form a resolution by sums of twists of the structure sheaf:

(8.9.1)  0 → ⨁_{j=1}^{m_{r}} 𝒪(a_{r,j}) → ⋯ → ⨁_{j=1}^{m_{1}} 𝒪(a_{1,j}) → ⨁_{j=1}^{m_{0}} 𝒪(a_{0,j}) → ℱ → 0.

Each of the maps in this resolution is a matrix whose entries are homogeneous polynomials in Y_{1}, …, Y_{r} and each kernel is again a vector bundle. Replacing each Y_{i} with y_{i} if p = 2 and with x_{i} if p is odd gives matrices of cohomology elements which we may use to form a sequence of modules and homomorphisms, which takes the form

(8.9.2)  0 → ⨁ Ω^{−εa_{r,j}}(k) → ⋯ → ⨁ Ω^{−εa_{1,j}}(k) → ⨁ Ω^{−εa_{0,j}}(k)

where ε = 1 if p = 2 and ε = 2 if p is odd. We discussed these maps x_{i} and y_{i} in Section 1.8, and showed that they may be constructed in such a way that they strictly commute. So this sequence is a complex in the module category mod(kE). It therefore determines an object in the derived category D^{b}(kE). Using the functor ℛ: D^{b}(kE) → stmod(kE) (see Section 1.6) we get a module M in stmod(kE). The claim is that M has stable constant Jordan type [1]^{s}, and ℱ_{1}(M) is isomorphic to ℱ if p = 2 and to F^{∗}(ℱ) if p is odd.

If p = 2 it follows from Lemma 8.7.1 that applying ℱ_{1} to (8.9.2) gives all but the augmentation part of the sequence (8.9.1). If p is odd, it follows from Lemma 8.8.1 that it gives a modification of this sequence where each of the original matrices has been altered by replacing the variables Y_{i} by their p^{th} powers. This is the inverse image of the original resolution along the Frobenius map F: ℙ^{r−1} → ℙ^{r−1}, and so it is a resolution of the inverse image bundle F^{∗}(ℱ).

Let M_{i} be the image in the stable category of the complex obtained by removing the terms of degrees from 0 to i − 1 from the complex (8.9.2). So for example M_{r} is just the left-hand term in this complex and M_{0} is equal to M. Since the functor ℛ preserves triangles, we obtain triangles

M_{i+1} → ⨁ Ω^{−εa_{i,j}}(k) → M_{i}

in stmod(kE). Thus we have constructed a diagram

[figure: ⨁ Ω^{−εa_{r,j}}(k); ⨁ Ω^{−εa_{r−1,j}}(k); ⨁ Ω^{−εa_{r−2,j}}(k); · · · · · ·; ⨁ Ω^{−εa_{0,j}}(k); M_{r−1}; M_{r−2}; M_{1}; M_{0}]

Applying ℱ_{1} to this diagram, we see by downwards induction on i and Lemma 8.9.2 that ℱ_{1}(M_{i}) is the ith kernel in (8.9.1) if p = 2, and its Frobenius twist if p is odd, so that

ℱ_{1}(M_{i+1}) → ℱ_{1}(⨁ Ω^{−εa_{i,j}}(k))

is injective with cokernel ℱ_{1}(M_{i}). Thus all the M_{i} are modules of trivial stable constant Jordan type. Finally, applying this with i = 0, we see that ℱ_{1}(M) ≅ ℱ if p = 2 and ℱ_{1}(M) ≅ F^{∗}(ℱ) if p is odd. □

Remark 8.9.4. We shall return to this theorem in Chapter 11, where we shall interpret the construction of Section 8.8 in terms of the Orlov correspondence.

8.10. Functoriality

Let ℱ and ℱ′ be vector bundles on ℙ^{r−1} and f: ℱ → ℱ′ be a sheaf homomorphism. Let M and M′ be the modules constructed in the proof of Theorem 8.9.3 realising ℱ and ℱ′ if p = 2 and realising F^{∗}(ℱ) and F^{∗}(ℱ′) if p is odd.

By Corollary 6.12.5, we obtain a morphism of resolutions

(8.10.1) \xymatrix{0 \ar[r] & ⨁_{j=1}^{m_{r}} 𝒪(a_{r,j}) \ar[r] \ar[d] & ⋯ \ar[r] & ⨁_{j=1}^{m_{1}} 𝒪(a_{1,j}) \ar[r] \ar[d] & ⨁_{j=1}^{m_{0}} 𝒪(a_{0,j}) \ar[r] \ar[d] & ℱ \ar[r] \ar[d]^{f} & 0. \\ 0 \ar[r] & ⨁_{j=1}^{m_{r}} 𝒪(a′_{r,j}) \ar[r] & ⋯ \ar[r] & ⨁_{j=1}^{m_{1}} 𝒪(a′_{1,j}) \ar[r] & ⨁_{j=1}^{m_{0}} 𝒪(a′_{0,j}) \ar[r] & ℱ′ \ar[r] & 0.}

Each of the vertical maps in this diagram is a matrix whose entries are homogeneous polynomials in Y_{1}, …, Y_{r}. As in the proof of Theorem 8.9.3, we replace Y_{i} with y_{i} (p = 2) or x_{i} (p odd) to obtain the vertical maps in the following commutative diagram.

(8.10.2)  \xymatrix{0 \ar[r] & ⨁ Ω^{−εa_{r,j}}(k) \ar[r] \ar[d] & … \ar[r] & ⨁ Ω^{−εa_{1,j}}(k) \ar[r] \ar[d] & ⨁ Ω^{−εa_{0,j}}(k) \ar[d] \\ 0 \ar[r] & ⨁ Ω^{−εa′_{r,j}}(k) \ar[r] & … \ar[r] & ⨁ Ω^{−εa′_{1,j}}(k) \ar[r] & ⨁ Ω^{−εa′_{0,j}}(k)}

This gives us a morphism in D^{b}(kE). Applying the functor ℛ: D^{b}(kE) → stmod(kE), we obtain a module homomorphism from M to M′.

A chain homotopy between maps of complexes (8.10.1) lying over f: ℱ → ℱ′ is again given by matrices of polynomials, and therefore gives rise to a chain homotopy between maps in the Diagram (8.10.2). Therefore different choices of chain maps in (8.10.1) induce the same map in D^{b}(kE) and hence the same map in stmod(kE).

It follows that the construction in Theorem 8.9.3 gives a functor

𝒢: Vec(ℙ^{r−1}) → stmod(kE)

whose image consists of modules of constant Jordan type.

8.11. Tensor Products

Similar techniques to those used in Section 8.5 prove the following theorems.

Theorem 8.11.1. Let M and N be kE-modules of constant Jordan type. Then

ℱ_{1}(M ⊗_{k} N) ≅ ⨁_{i=1}^{p−1}(ℱ_{i}(M) ⊗_{𝒪} ℱ_{i}(N))(i − 1),

ℱ_{2}(M ⊗_{k} N) ≅ ⨁_{i=1}^{p−2}(ℱ_{i}(M) ⊗_{𝒪} ℱ_{i+1}(N) ⊕ ℱ_{i+1}(M) ⊗_{𝒪} ℱ_{i}(N))(i − 1)

and more generally for 1 ≤ ℓ ≤ p − 1 we have

ℱ_{ℓ}(M ⊗_{k} N) ≅ ⨁_{i=1}^{p−ℓ} ⨁_{j=0}^{ℓ−1}(ℱ_{i+j}(M) ⊗_{𝒪} ℱ_{i+ℓ−1−j}(N))(i − 1)

cf. (2.3.1). □

It is harder to write down a formula for ℱ_{p}(M ⊗_{k} N), and we do not attempt it here. Symmetric and exterior powers may be treated in the same way. For example we have the following theorem.

Theorem 8.11.2. Let p be odd, and let M be a kE-module of constant Jordan type. Then

ℱ_{1}(Λ^{2}(M)) ≅ ⨁_{i=1}^{p−1} Λ^{2}(ℱ_{i}(M))(i − 1),

ℱ_{1}(S^{2}(M)) ≅ ⨁_{i=1}^{p−1} S^{2}(ℱ_{i}(M))(i − 1),

ℱ_{2}(Λ^{2}(M)) ≅ ℱ_{2}(S^{2}(M)) ≅ ⨁_{i=1}^{p−2}(ℱ_{i}(M) ⊗_{𝒪} ℱ_{i+1}(M))(i − 1)

and more generally for 1 ≤ ℓ ≤ p − 1 we have

ℱ_{ℓ}(Λ^{2}(M)) ≅ \begin{cases}⨁_{i=1}^{p−ℓ}(Λ^{2}(ℱ_{i}(M)) ⊕ ⨁_{j=0}^{(ℓ−1)/2}(ℱ_{i+j}(M) ⊗_{𝒪} ℱ_{i+ℓ−1−j}(M)))(i − 1) & ℓ odd \\ ⨁_{i=1}^{p−ℓ} ⨁_{j=0}^{(ℓ−1)/2}(ℱ_{i+j}(M) ⊗_{𝒪} ℱ_{i+ℓ−1−j}(M))(i − 1) & ℓ even. \end{cases}

The formula for ℱ_{ℓ}(S^{2}(M)) is the same with Λ^{2} replaced by S^{2} on the right-hand side for ℓ odd.

Higher symmetric and exterior powers, and more general Schur functors are much more complicated.

8.12. Negative Tate Cohomology

In this section we examine in detail the locally split sequences of modules of constant Jordan type coming from negative Tate cohomology that were briefly discussed in Example 5.3.5, and the short exact sequences of vector bundles coming from them (Proposition 8.4.4).

Proposition 8.12.1. Let E be an elementary abelian p-group, and let E′ be a shifted subgroup of strictly smaller rank. If ζ ∈ Ĥ^{−n}(E, k) with n > 0 then the restriction of ζ to Ĥ^{−n}(E′, k) is equal to zero.

Proof. By a change of basis, we may assume that E′ is a subgroup of E. Since E is elementary abelian, there exists another subgroup E′′ such that E = E′ × E′′. It follows from the Künneth theorem for group cohomology that if η′ ∈ H^{n−1}(E′, k) then there exists η ∈ H^{n−1}(E, k) such that res_{E,E′}(η) = η′. Taking transfer we get

Tr_{E′,E}(η′) = Tr_{E′,E}res_{E,E′}(η) = |E : E′|.η = 0.

So transfer from E′ to E in cohomology is zero. Since

res_{E,E′}: Ĥ^{−n}(E, k) → Ĥ^{−n}(E′, k)

is Tate dual to this transfer map (see Section 1.5), it follows that it is also the zero map. □

Now suppose that E has rank r ≥ 2. Let j ∈ ℤ and let n > 0. If

0 ≠ ζ ∈ Ĥ^{−n}(E, k) ≅ Ext_{kE}^{1}(Ω^{j−n−1}(k), Ω^{j}(k))

then we have a corresponding short exact sequence

(8.12.1)  0 → Ω^{j}(k) → M → Ω^{j−n−1}(k) → 0.

Let us assume that if p > 2 then j is even and n is odd, so that both ends of this short exact sequence have stable constant Jordan type [1]. By Proposition 8.12.1, this sequence splits on restriction to every cyclic shifted subgroup of E. So it is a locally split short exact sequence of modules of constant Jordan type (see Definition 5.3.3), and M has stable constant Jordan type [1]^{2}.

Applying the functor ℱ_{1}, we obtain a short exact sequence of vector bundles

(8.12.2)  0 → ℱ_{1}(Ω^{j}(k)) → ℱ_{1}(M) → ℱ_{1}(Ω^{j−n−1}(k)) → 0.

Theorem 8.12.2. If r ≥ 3 then the sequence (8.12.2) splits.

Proof. By Corollary 8.5.2 and the remark following it, the vector bundles at the ends of this sequence are twists of the structure sheaf. If r ≥ 3 then by Corollary 6.9.2 for any values of a and b we have

Ext_{ℙ^{r−1}}^{1}(𝒪(a), 𝒪(b)) ≅ H^{1}(ℙ^{r−1}, 𝒪(b − a)) = 0.

It follows that the sequence splits. □

For r = 2 the situation is more complicated. If p = 2 then the sequence (8.12.2) never splits. The sequence (8.12.1) takes the form

(8.12.3)  0 → Ω^{j}(k) \underrightarrow{(\begin{matrix}η_{2} \\ −η_{1} \end{matrix} )} Ω^{j−a}(k) ⊕ Ω^{j−b}(k) ⊕ (projective) ⟶^{(η_{1},η_{2})} Ω^{j−n−1}(k) → 0

where a + b = n + 1. The ideal generated by η_{1} and η_{2} is the annihilator of ζ in H^{∗}(E, k). In the language of [47], the element ζ is the last survivor for the sequence η_{1}, η_{2}.

Finally we discuss the case r = 2, p odd. If ζ ∈ Ĥ^{−n}(E, k) has zero Tate pairing with every nilpotent element of H^{n−1}(E, k) then the sequence (8.12.2) splits. On the other hand, if there is a nilpotent element with which ζ has non-zero pairing then the annihilator of ζ is again generated by a regular sequence η_{1}, η_{2}, and the situation is as described in the p = 2 case. The extension class representing η takes the form (8.12.3) with a and b even.

As a concrete example, let r = 2 and p be odd so that

H^{∗}(E, k) = Λ(y_{1}, y_{2}) ⊗_{k} k[x_{1}, x_{2}]

and let ζ be an element which has a non-zero pairing with y_{1}y_{2}. Then η_{1}, η_{2} are x_{1}, x_{2} and the sequence takes the form

0 → Ω^{4}(k) \underrightarrow{(\begin{matrix}x_{2} \\ −x_{1} \end{matrix} )} Ω^{2}(k) ⊕ Ω^{2}(k) ⟶^{(x_{1},x_{2})} k → 0.

The sequence (8.12.2) takes the form

0 → 𝒪(−2p) → 𝒪(−p) ⊕ 𝒪(−p) → 𝒪 → 0.

8.13. The BGG Correspondence

The construction of the operator θ given in Section 8.2 is based on the Bernstein–Gelfand– Gelfand (or BGG) correspondence. Since there are a number of different versions of this correspondence, in this section we give a brief summary and put the construction in context. For the purposes of this book, it is not necessary to understand this section, but the reader may find it helpful. Further information on the BGG correspondence may be found in the papers [22, 51, 55, 91, 111, 121], and the appendix of [202].

The BGG correspondence for exterior algebras is as follows. Let Λ = Λ(X_{1}, …, X_{r}) be an exterior algebra on generators X_{1}, …, X_{r} in degree 1, and let S = k[Y_{1}, …, Y_{r}] be a polynomial algebra on generators Y_{1}, …, Y_{r} of degree 1. Then there is an equivalence of triangulated categories

(8.13.1)  D_{gr}^{b}(Λ) ≃ D_{gr}^{b}(S).

between the bounded derived categories of graded Λ-modules and graded S-modules. Objects in these categories have two gradings: an internal grading coming from the gradings on Λ and S, and a homological grading in the complexes. In this equivalence, the internal shift of degree 1 in D_{gr}^{b}(Λ) corresponds to the composite of the internal shift of degree −1 and a homological shift of degree 1 in D_{gr}^{b}(S).

Inside D_{gr}^{b}(Λ), perf_{gr}(Λ) is the thick subcategory of perfect complexes, i.e., objects isomorphic to bounded complexes of finitely generated projective modules. The quotient D_{gr}^{b}(Λ)/perf_{gr}(Λ) is equivalent to the stable module category stmod_{gr}(Λ) of finitely generated graded Λ-modules, see Section 1.6.

Inside D_{gr}^{b}(S), let fl_{gr}(S) be the thick subcategory of objects isomorphic to bounded complexes of finite length graded S-modules. Then D_{gr}^{b}(S)/fl_{gr}(S) is equivalent to D^{b}(Coh(ℙ^{r−1})), the bounded derived category of coherent sheaves on projective space, see Theorem 3.9.12.

Under the equivalence (8.13.1), perf_{gr}(Λ) corresponds to fl_{gr}(S), and so we obtain the BGG equivalence of categories

(8.13.2)  stmod_{gr}(Λ) ≃ D^{b}(Coh(ℙ^{r−1})).

There is a modified version of BGG, which is essentially the construction described in [51], which goes as follows. Let D_{dg}^{b}(S) be the bounded derived category of differential graded S-modules. This may be described as follows. We regard S as a differential graded algebra with zero differential. An object in D_{dg}^{b}(S) is a differential graded S-module, where the differential has degree 1 and where the underlying S-module is finitely generated. The arrows are obtained by taking homotopy classes of maps of differential graded modules, and inverting quasi-isomorphisms.

Let D^{b}(Λ) be the bounded derived category of ungraded Λ-modules. Then there is an equivalence of triangulated categories

(8.13.3)  D^{b}(Λ) ≃ D_{dg}^{b}(S).

Again this takes the thick subcategory perf(Λ) of D^{b}(Λ) to the thick subcategory fl_{dg}(S) of D_{dg}^{b}(S) consisting of objects isomorphic to differential graded modules of finite length. So we obtain a BGG correspondence

(8.13.4)  stmod(Λ) ≃ D_{dg}^{b}(S)/fl_{dg}(S).

There is an obvious functor

Tot: D_{gr}^{b}(S) → D_{dg}^{b}(S),

which takes a complex of graded S-modules and applies to total complex construction to obtain differential graded S-modules. Under the BGG correspondence, this corresponds to the forgetful functor

stmod_{gr}(Λ) → stmod(Λ)

which forgets the grading.

Finally, there is a functor from D_{dg}^{b}(S) to finitely generated graded S-modules, which takes a differential graded S-module to its cohomology. This passes down to a functor

H^{∗}: D_{dg}^{b}(S)/fl_{dg}(S) → Coh(ℙ^{r−1}).

Composing this with the equivalence (8.13.4) gives the functor ℱ_{1} : stmod(Λ) → Coh(ℙ^{r−1}) of Section 8.4.

Putting all these together, we obtain the following diagram of categories and functors:

[figure: D_{gr}^{b}(Λ); ≃; D_{gr}^{b}(S); Tot; D^{b}(Λ); ≃; D_{dg}^{b}(S); stmod_{gr}(Λ); ≃; D^{b}(Coh(ℙ^{r−1})); Tot; stmod(Λ); ≃; ℱ_{1}; D_{dg}^{b}(S)/fl_{dg}(S); H^{∗}; Coh(ℙ^{r−1})]

The appendix to the second and third editions of [202] gives an expanded version of the original paper. The version of the correspondence given there is particularly well suited to the study of modules of constant Jordan type for exterior algebras.

Consider a graded Λ-module M with the property that the action of

X_{α} = λ_{1}X_{1} + ⋯ + λ_{r}X_{r}

on M is exact except in degree zero for every 0 ≠ α = (λ_{1}, …, λ_{r}) ∈ 𝔸^{r}(k). The dimension of the homology of M in degree zero for the action of X_{α} is then the alternating sum of the dimensions of the graded pieces, ∑_{i}(−1)^{i} dim_{k}(M_{i}), and is hence independent of α. So such a module has constant Jordan type, and ℱ_{1}(M) is a vector bundle.

Theorem 8.13.1. Every vector bundle on ℙ^{r−1} arises as ℱ_{1}(M) for a graded Λ-module M satisfying the above condition. Furthermore, two such graded modules M and M′ satisfy ℱ_{1}(M) ≅ ℱ_{1}(M′) if and only if there are free modules F and F′ such that M ⊕ F ≅ M′ ⊕ F′.

We shall see some explicit examples of this theorem in the next chapter.

9 Examples

In this chapter we look at some particular examples of vector bundles over projective space, and show how to realise them as ℱ_{i}(M) with M a module of constant Jordan type over an elementary abelian p-group and over exterior algebras. These examples include the tangent and cotangent bundles, the null correlation bundle, some bundles of Schwarzenberger, the Tango bundle, and the Horrocks–Mumford bundle. Some of these constructions will be used in Chapter 10.

The Schwarzenberger example shows that given a set of points {x_{1}, …, x_{m}} in ℙ^{2}, there exists a module of stable constant Jordan type [1]^{2} for (ℤ/2)^{3} whose restriction to a shifted subgroup of type (ℤ/2)^{2} is Ω^{n}k ⊕ Ω^{−n}k ⊕ (projective), where n is the number of x_{i} that lie on the corresponding line ℙ^{1} ⊆ ℙ^{2}.

9.1. Modules for (ℤ/2)^{2}

Let k be an algebraically closed field of characteristic two, E = ⟨g_{1}, g_{2}⟩ ≅ (ℤ/2)^{2} and X_{1} = g_{1} − 1, X_{2} = g_{2} − 1 in kE. Thus kE = k[X_{1}, X_{2}]/(X_{1}^{2}, X_{2}^{2}). Recall from Section 1.11 that we draw a diagram for kE as a kE-module as follows.

[figure: •^{1}; •^{X_{1}}; •^{X_{2}}; •_{X_{1}X_{2}}]

We wish to understand the structure of the sheaf k̃E = kE ⊗_{k} 𝒪 on ℙ^{1} = Proj k[Y_{1}, Y_{2}] together with the operator θ = X_{1} ⊗ Y_{1} + X_{2} ⊗ Y_{2} (see Section 8.2). We begin by examining the radical layers of kE, on the grounds that θ takes us down one radical layer. So we think of k̃E as

\xymatrix{{ \atop kE/RadkE} \ar[d]^{θ} \\ { \atop RadkE/Rad^{2}kE} \ar[d]^{θ} \\ { \atop Rad^{2}kE}}  =  \xymatrix{𝒪 \ar[d]^{θ} \\ 𝒪 ⊕ 𝒪 \ar[d]^{θ} \\ 𝒪.}

Since θ is really a map from k̃E to k̃E(1), the image of θ in the second radical layer is a copy of 𝒪(−1) ⊂ 𝒪 ⊕ 𝒪 after untwisting. So we should use the Euler sequence

0 → 𝒪(−1) → 𝒪 ⊕ 𝒪 → 𝒪(1) → 0

on ℙ^{1} to rewrite the second radical layer 𝒪 ⊕ 𝒪 as an extension

\begin{matrix}𝒪(1) \\ 𝒪(−1) \end{matrix}

where the subsheaf 𝒪(−1) is the image of θ, and is spanned by the element X_{1} ⊗Y_{1} ⊕X_{2} ⊗Y_{2} when regarded as an element of

RadkE/Rad^{2}kE ⊗_{k} k[Y_{1}, Y_{2}].

Now θ^{2} = 0, so 𝒪(−1) is also the kernel of θ as a map from the second layer to the third layer. Thus we can write down the following diagram for k̃E:

(9.1.1)  [figure: 𝒪; 𝒪(1); 𝒪(−1); 𝒪]

Here, the arrows going down right give the action of the operator θ, while the lines going down left represent extensions of vector bundles. The radical layers are diagonal slices from top right to bottom left.

Now we turn to the syzygies of the trivial module. We have the following:

Ω(k) : [figure: 𝒪(1); 𝒪(−1); 𝒪] Ω^{2}(k) : [figure: 𝒪(2); 𝒪; 𝒪(1); 𝒪(−2); 𝒪(−1)]

Ω^{3}(k) : [figure: 𝒪(3); 𝒪(1); 𝒪(2); 𝒪(−1); 𝒪; 𝒪(−3); 𝒪(−2)]

and so on. Dually, we have Ω^{−1}(k) : [figure: 𝒪; 𝒪(1); 𝒪(−1)] Ω^{−2}(k) : [figure: 𝒪(1); 𝒪(2); 𝒪(−1); 𝒪; 𝒪(−2)] and so on.

9.2. Modules for (ℤ/p)^{2}

The diagram corresponding to (9.1.1) for k(ℤ/3)^{2} with k an algebraically closed field of characteristic three is as follows.

k(ℤ/3)^{2} : [figure: 𝒪; 𝒪(1); 𝒪(2); 𝒪(−1); 𝒪; 𝒪(1); 𝒪(−2); 𝒪(−1); 𝒪] The diagrams for larger primes follow the same pattern. For example for p = 5 it is as follows:

k(ℤ/5)^{2} : [figure: 𝒪; 𝒪(1); 𝒪(2); 𝒪(3); 𝒪(4); 𝒪(−1); 𝒪; 𝒪(1); 𝒪(2); 𝒪(3); 𝒪(−2); 𝒪(−1); 𝒪; 𝒪(1); 𝒪(2); 𝒪(−3); 𝒪(−2); 𝒪(−1); 𝒪; 𝒪(1); 𝒪(−4); 𝒪(−3); 𝒪(−2); 𝒪(−1); 𝒪]

Again the radical layers are the diagonal slices going from top right to bottom left.

In the case p = 3, the syzygies of the trivial module are as follows:

Ω(k) : [figure: 𝒪(1); 𝒪(2); 𝒪(−1); 𝒪; 𝒪(1); 𝒪(−2); 𝒪(−1); 𝒪]

Ω^{2}(k) : [figure: 𝒪(3); 𝒪; 𝒪(1); 𝒪(2); 𝒪(−1); 𝒪; 𝒪(1); 𝒪(−3); 𝒪(−2); 𝒪(−1)]

The next ones are more difficult:

Ω^{3}(k) : [figure: 𝒪(4); 𝒪(5); 𝒪(2); 𝒪(3); 𝒪(4); 𝒪(1); 𝒪(2); 𝒪(3); 𝒪(−2); 𝒪(−1); 𝒪; 𝒪(−4); 𝒪(−3); 𝒪(−2); 𝒪(−5); 𝒪(−4); 𝒪(−3)]

Ω^{4}(k) : [figure: 𝒪(6); 𝒪(3); 𝒪(4); 𝒪(5); 𝒪(2); 𝒪(3); 𝒪(4); 𝒪; 𝒪(1); 𝒪(2); 𝒪(−3); 𝒪(−2); 𝒪(−1); 𝒪(−4); 𝒪(−3); 𝒪(−2); 𝒪(−6); 𝒪(−5); 𝒪(−4)]

9.3. Larger Rank

We can make similar diagrams to a certain extent for larger rank. For example for p = 2 and rank 3 the group algebra has the following diagram:

k(ℤ/2)^{3} : [figure: 𝒪; 𝒯(−1); 𝒪(1); 𝒪(−1); 𝒯(−2); 𝒪]

More generally, for p = 2 and larger rank the group algebra has the following diagram:

k(ℤ/2)^{r}: [figure: 𝒪; 𝒯(−1); Λ^{2}𝒯(−2); …; Ω^{2}(3); Ω(2); 𝒪(1); 𝒪(−1); 𝒯(−2); Λ^{2}𝒯(−3); …; Ω^{2}(2); Ω(1); 𝒪]

(recall from Section 6.10 that Ω^{r−1−m}(r − m) ≅ Λ^{m}𝒯(−m)). This is in some sense an embodiment of the Koszul complex. The radical layers of the group algebra are diagonal slices in the diagram. The diagrams for Ω(k) and Ω^{−1}(k) for p = 2 are obtained from this diagram by removing the copy of 𝒪 from the top left, respectively the bottom right.

Theorem 9.3.1. Let E = (ℤ/2)^{r} and k an algebraically closed field of characteristic two. Then

ℱ_{2}(kE) ≅ ⨁_{j=0}^{r−1} Ω^{j}(j),  ℱ_{2}(Ω(k)) ≅ ⨁_{j=0}^{r−2} Ω^{j}(j),  ℱ_{2}(Ω^{−1}(k)) ≅ ⨁_{j=1}^{r−1} Ω^{j}(j).

For p = 3 and rank 3 the group algebra k(ℤ/3)^{3} has the following diagram:

[figure: 𝒪; 𝒯(−1); S^{2}𝒯(−2); Ω(3); 𝒪(2); 𝒪(−1); 𝒯(−2); S^{2}𝒯(−3); Ω(2); 𝒪(1); 𝒪(−2); 𝒯(−3); S^{2}𝒯(−4); Ω(1); 𝒪,]

where S^{2}𝒯 denotes the symmetric square of the tangent bundle, which is a simple bundle of rank three on ℙ^{2}. Note that we have

S^{2}𝒯(−3) ≅ S^{2}Ω(3).

For p = 3 and rank 4 the group algebra k(ℤ/3)^{4} has the following diagram:

[figure: 𝒪; 𝒯(−1); S^{2}𝒯(−2); L^{[2][1]}𝒯(−3); S^{2}Ω(4); Ω(3); 𝒪(2); 𝒪(−1); 𝒯(−2); S^{2}𝒯(−3); L^{[2][1]}𝒯(−4); S^{2}Ω(3); Ω(2); 𝒪(1); 𝒪(−2); 𝒯(−3); S^{2}𝒯(−4); L^{[2][1]}𝒯(−5); S^{2}Ω(2); Ω(1); 𝒪.]

Here, L^{[2][1]} is the simple functor in characteristic three obtained by taking the quotient of the symmetric cube by the Frobenius cube:

0 → F^{3}(V) → S^{3}(V) → L^{[2][1]}(V) → 0.

We have

L^{[2][1]}𝒯(−4) ≅ L^{[2][1]}Ω(4).

It is easy to see how these patterns continue in the light of the description of the radical layers of kE given in Section 1.17.

In the same vein, for E of arbitrary rank and for any prime we have

Soc^{2}(kE) : [figure: 𝒪(1); Ω(1); 𝒪] kE/Rad^{2}(kE) : [figure: 𝒪; 𝒯(−1); 𝒪(−1)] ,

see Section 9.5.

9.4. Nilvarieties

The three dimensional nilvariety of constant Jordan type given in Example 4.1.6, where the operator θ has matrix

(\begin{matrix}0 & Y_{1} & 0 \\ Y_{2} & 0 & Y_{1} \\ 0 & −Y_{2} & 0 \end{matrix}) ,

may be thought of as a counterexample to the analogue of Dade’s Lemma 1.9.5 for nilvarieties, because the dimension is not divisible by 9.

This example can be drawn in the fashion of the previous section as follows:

[figure: 𝒪(1); 𝒪; 𝒪(−1)]

while the first example of dimension five described in Example 4.1.7 gives

[figure: 𝒪(2); 𝒪(1); 𝒪; 𝒪(−1); 𝒪(−2)]

A closely related example is the nilvariety where the operator θ has matrix

(\begin{matrix}0 & 0 & Y_{1} & 0 \\ Y_{2} & 0 & 0 & Y_{1} \\ 0 & 0 & 0 & 0 \\ 0 & 0 & −Y_{2} & 0 \end{matrix}) .

This may be thought of as a nilvariety with diagram

\xymatrix{𝒪 \ar[dr] &  & 𝒪(1) \ar[dr] \\  & 𝒪(−1) &  & 𝒪}

and forming a counterexample to the analogue of Dade’s Endotrivial Theorem 5.4.2 for nilvarieties.

9.5. The Tangent and Cotangent Bundles

Example 9.5.1. Let M = kE/J^{2}(kE). Then M has constant Jordan type [2] [1]^{r−1}. In the short exact sequence of vector bundles

0 → { \atop M/RadM} ⟶^{θ} { \atop RadM}(1) → ℱ_{1}(M)(1) → 0

the map θ (induced by θ_{M}) is equal to the map defining the tangent bundle 𝒯 of ℙ^{r−1}:

0 → 𝒪 → 𝒪(1)^{r} → 𝒯 → 0.

It follows that ℱ_{1}(M) ≅ 𝒯(−1). On the other hand we have ℱ_{2,1}(M) ≅ 𝒪, and hence ℱ_{2}(M) ≅ 𝒪(−1).

The module J^{2}(kE) is isomorphic to Ω(kE/J^{2}(kE)). Applying Theorem 8.5.1, we find that

ℱ_{p−1}(J^{2}(kE)) ≅ 𝒯(−p), ℱ_{p−2}(J^{2}(kE)) ≅ 𝒪(−p + 1).

Example 9.5.2. Dually, let M = Soc^{2}(kE), also of constant Jordan type [2] [1]^{r−1}. Then in the short exact sequence

0 → ℱ_{1}(M)(−1) → { \atop M/RadM}(−1) ⟶^{θ} { \atop RadM} → 0

the map θ is equal to the map defining the cotangent bundle Ω on ℙ^{r−1}:

0 → Ω → 𝒪(−1)^{r} → 𝒪 → 0.

It follows that ℱ_{1}(M) ≅ Ω(1). On the other hand, we have ℱ_{2}(M) ≅ 𝒪.

The module kE/Soc^{2}(kE) is isomorphic to Ω^{−1}(Soc^{2}(kE)). Again applying Theorem 8.5.1, we find that

ℱ_{p−1}(kE/Soc^{2}(kE)) ≅ Ω(2), ℱ_{p−2}(kE/Soc^{2}(kE)) ≅ 𝒪(2).

Example 9.5.3. Suppose that p ≥ 3, and let M = Rad(kE)/Rad^{3}(kE), of contant Jordan type [2]^{r}[1]^{r(r−1)/2}. Then the short exact sequence

0 → ℱ_{2,1}(M) ⟶^{θ} { \atop RadM}(1) → ℱ_{1}(M)(1) → 0

can be viewed as follows. Consider the two-term complex 𝒪 → 𝒪(1)^{r} where 𝒪 is the term of degree zero and the homology is the tangent bundle 𝒯 in degree one. Since we are in odd characteristic, the tensor square of this complex decomposes as the direct sum of the exterior square and the symmetric square. The exterior square is a complex of the form 0 → 𝒪(1)^{r} → 𝒪(2)^{r(r+1)/2} whose homology in degree two is the symmetric square S^{2}(𝒯). Untwisting, we get a short exact sequence

0 → 𝒪^{r} → 𝒪(1)^{r(r+1)/2} → S^{2}(𝒯)(−1) → 0.

The first map in this sequence is the map corresponding to θ, and so

ℱ_{1}(M) ≅ S^{2}(𝒯)(−2).

Example 9.5.4. Suppose again that p ≥ 3 and consider the kE-module of Loewy length three with underlying vector space V^{∗} ⊕ (V ⊗_{k} V^{∗}) ⊕ V where V = J(kE)/J^{2}(kE), and action given by

X: (\begin{matrix}ϕ \\ X′ ⊗ ϕ′ \\ X′′ \end{matrix}) ↦ (\begin{matrix}0 \\ X ⊗ ϕ \\ ϕ′(X)X′ + ϕ′(X′)X \end{matrix}) .

Here, X, X′, X′′ are linear combinations of X_{1}, …, X_{r} and ϕ, ϕ′ are linear forms. It is easy to check that these matrices commute and cube to zero, and hence give a kE-module M. We have

V^{∗} ⊗ 𝒪(−1) ⟶^{θ} V ⊗ V^{∗} ⊗ 𝒪 → V ⊗ 𝒪(1)

and ℱ_{1}(M) = Ker θ/Ker θ ∩ Im θ as a subquotient of the middle term. This is the vector bundle of rank (r − 1)^{2} given by Hom(𝒯, 𝒯) ≅ 𝒯 ⊗ Ω. The Jordan type of this module is [3] [2]^{2(r−1)}[1]^{(r−1)^{2}}.

On the other hand, if we want anticommuting matrices, we let

X: (\begin{matrix}ϕ \\ X′ ⊗ ϕ′ \\ X′′ \end{matrix}) ↦ (\begin{matrix}0 \\ X ⊗ ϕ \\ ϕ′(X)X′ − ϕ′(X′)X \end{matrix}) .

This then gives a module M for the exterior algebra Λ(X_{1}, …, X_{r}), this time with ℱ_{1}(M) ≅ Hom(𝒯, 𝒯)/(multiples of the identity) ≅ (𝒯⊗Ω)/𝒪. This time the Jordan type is [2]^{2r}[1]^{r^{2}−2r}.

9.6. The Null Correlation Bundle, p = 2

In Section 6.7, we described the null correlation bundle as an example of an indecomposable vector bundle of rank (r − 2) on ℙ^{r−1} for r even.

Let p = 2 and r = 2m ≥ 4 be even, E = (ℤ/2)^{r}, and let M be the (r + 2)-dimensional kE-module with basis elements u, v_{1}, …, v_{r}, w, with action given by: X_{i}u = v_{i}, X_{i}v_{j} = w if |i − j| = m, X_{i}v_{j} = 0 otherwise. Then ℱ_{1}(M) is the null correlation bundle of rank r − 2 defined by the symplectic form

⟨∑ a_{i}X_{i}, ∑ b_{i}X_{i}⟩ = ∑_{|i−j|=m} a_{i}b_{j}.

The monad (6.7.1) for the null correlation bundle,

0 → 𝒪(−1) → 𝒪^{r} → 𝒪(1) → 0,

can be written in the form

0 → ˜⟨u⟩(−1) ⟶^{θ} { \atop ⟨v_{1}, …, v_{r}⟩} ⟶^{θ} ˜⟨w⟩(1) → 0.

It follows that for this module M, ℱ_{1}(M) is isomorphic to the cohomology of this monad, namely the null correlation bundle.

The case r = 4 of the module M was described in Example 5.2.11. In general, the matrix for the operator θ is given by

( Y_{1} ; ⋮ ; Y_{r} ; Y_{r} ⋯ Y_{1} )

where blank entries are taken to be zero as usual.

9.7. The Null Correlation Bundle, p Odd

Realising the null correlation bundle when p is odd is an instructive exercise in the techniques of Section 8.8; as in that section, we shall content ourselves with realising the Frobenius twist of the null correlation bundle.

Let p be an odd prime, r = 2m ≥ 4 be even, and E = (ℤ/p)^{r}. Again we use the same symplectic form, but instead of using extensions of k by k as elements of H^{1}(E, k) we are obliged to use elements of H^{2}(E, k) as explained in Section 8.8. Therefore we take the sequence of modules and maps

Ω^{2}k → k^{r} → Ω^{−2}k

where the first map is given by the column vector of elements (x_{1}, …, x_{m}, x_{m+1}, …, x_{r})^{t} in H^{2}(E, k) and the second is given by the row vector (x_{m+1}, …, x_{r}, −x_{1}, …, −x_{m}). Since the elements x_{i} ∈ H^{2}(E, k) commute, the composite of these maps represents the zero element of H^{4}(E, k) and so it factors through a projective module.

This allows us to perform the following construction in the stable module category: complete the map Ω^{2}k → k^{r} to a triangle

Ω^{2}k → k^{r} → X

in the stable module category stmod(kE). Then there exists a map X → Ω^{−2}k whose composite with k^{r} → X is the given map k^{r} → Ω^{−2}k. Complete this to a triangle

Y → X → Ω^{−2}k,

in stmod(kE), and then Y will be the required module of stable constant Jordan type [1]^{r−2}. We illustrate this with the following diagram in stmod(kE):

\xymatrix{ &  &  & Ω^{−2}k. \\ Ω^{2}k \ar[r] & k^{r} \ar[urr] \ar[r] & X \ar[ur] \\  & Y \ar[ur]}

Note that as in Section 8.8, ℱ_{1}(Y) is not isomorphic to the null correlation bundle but rather to the Frobenius twist of the null correlation bundle.

Let us make this construction more explicit in mod(kE). Recall that the beginning of the minimal resolution of k as a kE-module is as follows:

\xymatrix{0 \ar[r] & Ω^{2}k \ar[r] & P_{1} \ar[rr] \ar@{-->}[dr] &  & P_{0} \ar[r] & k \ar[r] & 0. \\  &  &  & Ωk \ar@{-->}[ur] \ar[dr] \\  &  & 0 \ar[ur] &  & 0}

We turn Ω^{2}k → k^{r} into an injective map by adding P_{1} to k^{r}. So X is an extension of Ωk by k^{r} as in the following diagram:

[figure: 0; 0; k^{r}; k^{r}; 0; Ω^{2}k; k^{r} ⊕ P_{1}; X; 0; 0; Ω^{2}k; P_{1}; Ωk; 0; 0; 0]

Now the composite Ω^{2}k → k^{r} → Ω^{−2}k factors through some projective module, and therefore through the projective cover P_{−1} of Ω^{−2}k, which happens to be isomorphic to P_{1} as both are direct sums of r copies of the group algebra. So there exists a surjective map k^{r} ⊕P_{1} → Ω^{−2}k such that the composite

Ω^{2}k → k^{r} ⊕ P_{1} → Ω^{−2}k

is the zero map in mod(kE). The required module Y is the cohomology of this complex in the middle term. So it fits into a commutative diagram as follows:

[figure: 0; 0; 0; Ω^{−1}k; P_{1}; Ω^{−2}k; 0; 0; Y; X ⊕ P_{1}; Ω^{−2}k; 0; X; X; 0; 0]

From these diagrams we can see that the module Y has a filtration with filtered quotients (from the top) Ωk, k^{r}, Ω^{−1}k. The extensions of Ωk by k^{r} and of k^{r} by Ω^{−1}k are given by the column vector (x_{1}, …, x_{m}, x_{m+1}, …, x_{r})^{t} and the row vector (x_{m+1}, …, x_{r}, −x_{1}, …, −x_{m}), respectively. The reason the threefold extension exists is that the cup product of these two vectors is zero in H^{4}(E, k).

Proposition 9.7.1. In odd characteristic, for the module Y in the above construction we have ℱ_{1}(Y) ≅ F^{∗}(ℱ_{N}), the inverse image of the null correlation bundle along the Frobenius map on ℙ^{r−1}.

Remark 9.7.2. The constructions given in this section and the last work with any monad of the form

0 → 𝒪(−1)^{⊕a} → 𝒪^{⊕b} → 𝒪(1)^{⊕c} → 0.

In the next two sections we discuss more examples of monads of this form, giving rise to instanton bundles and Schwarzenberger’s bundles.

9.8. Instanton Bundles

There is a generalisation of the null correlation bundle called an instanton bundle on ℙ^{r−1} when r is even. It is a bundle ℱ_{k} appearing as the cohomology of a monad of the form

0 → 𝒪(−1)^{⊕k} ⟶^{β} 𝒪^{⊕(2k+r−2)} ⟶^{α} 𝒪(1)^{⊕k} → 0

with k ≥ 1. The positive integer k is called the quantum number (or instanton number or charge) of the bundle. The usual definition of an instanton bundle is technical, and we shall not give it here, instead referring the reader to Okonek and Spindler [203]. It is proved in Proposition 2.4 of Hoffmann [157] that their definition is equivalent to the one given above together with the condition of trivial splitting type; the latter is an open condition and is omitted by some authors. There is a considerable literature on instanton bundles, see for example Ancona and Ottaviani [9, 10], Barth [34], Costa and Ottaviani [94, 95], Hartshorne [148], Hoffmann [157], Miro-Roig and Orus-Lacort [194], Okonek and Spindler [203], Rao [213] and Vedernikov [240]. In particular, in [95] it is shown that the moduli space of instanton bundles on ℙ^{r−1} with instanton number k is an affine variety. The paper of Atiyah, Drinfeld, Hitchin and Manin [15] explains the connection of complex instanton bundles over ℙ^{3} with solutions of the Yang–Mills equations.

Instead of giving the definition, we shall give examples of instanton bundles for each even r ≥ 4 and each k ≥ 1, and describe some properties. The examples with k = 1 are just the null correlation bundles.

Example 9.8.1. Let α be the k × (2k + r − 2) matrix whose (i, j) entry is Y_{j−2i−1} if 1 ≤ j − 2i − 1 ≤ r and zero otherwise, and let β be the (2k + r − 2) × k matrix whose (i, j) entry is (−1)^{i+1}Y_{r+2+i−2j} if 1 ≤ r + 2 + i − 2j ≤ r and zero otherwise.

It is easy to check that these matrices have zero product, and that α is surjective and β is injective at all points of projective space. With these choices for α and β, we obtain an instanton bundle.

For example, if r = 6 and k = 3, the maps α and β are given by

α = ( \begin{matrix}Y_{1} & Y_{2} & Y_{3} & Y_{4} & Y_{5} & Y_{6} \\ & & Y_{1} & Y_{2} & Y_{3} & Y_{4} & Y_{5} & Y_{6} \\ & & & & Y_{1} & Y_{2} & Y_{3} & Y_{4} & Y_{5} & Y_{6} \end{matrix} ) β = ( \begin{matrix}Y_{6} \\ −Y_{5} \\ Y_{4} & Y_{6} \\ −Y_{3} & −Y_{5} \\ Y_{2} & Y_{4} & Y_{6} \\ −Y_{1} & −Y_{3} & −Y_{5} \\ & Y_{2} & Y_{4} \\ & −Y_{1} & −Y_{3} \\ & & Y_{2} \\ & & −Y_{1} \end{matrix} ) .

Proposition 9.8.2. The Chern polynomial of an instanton bundle ℱ_{k} on ℙ^{r−1} (r even) is equal to

c(ℱ_{k}) = (1 − h^{2})^{−k} = 1 + kh^{2} + \binom{k + 1}{2}h^{4} + \binom{k + 2}{3}h^{6} + ⋯

and the Chern character is given by

Ch(ℱ_{k}) = (2k + r − 2) − k(e^{h} + e^{−h}).

Proof. This follows from Lemma 7.3.5. □

Again for p = 2 we can take the monad describing an instanton bundle,

0 → 𝒪(−1)^{⊕k} → 𝒪^{⊕(2k+r−2)} → 𝒪(1)^{⊕k} → 0,

and apply the constructions of the last two sections to it. The modules M_{k} obtained this way for p = 2 have constant Jordan type [2]^{k}[1]^{r−2}, and ℱ_{1}(M_{k}) ≅ ℱ_{k}, the instanton bundle with quantum number k.

For example if r = 6 and k = 3, using the instanton bundle described in Example 9.8.1, the operator θ for this module is represented by the matrix

( \begin{matrix}Y_{6} \\ Y_{5} \\ Y_{4} & Y_{6} \\ Y_{3} & Y_{5} \\ Y_{2} & Y_{4} & Y_{6} \\ Y_{1} & Y_{3} & Y_{5} \\ & Y_{2} & Y_{4} \\ & Y_{1} & Y_{3} \\ & & Y_{2} \\ & & Y_{1} \\ & & & Y_{1} & Y_{2} & Y_{3} & Y_{4} & Y_{5} & Y_{6} \\ & & & & & Y_{1} & Y_{2} & Y_{3} & Y_{4} & Y_{5} & Y_{6} \\ & & & & & & & Y_{1} & Y_{2} & Y_{3} & Y_{4} & Y_{5} & Y_{6} \end{matrix}  )

A diagram for this module looks as follows (we have not used any notation to distinguish the actions of X_{1}, …, X_{6}, but it is easy to see which is which from the above matrix).

[figure: •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •]

For p odd, again we need to use the more complicated construction described in Section 9.7 (and of course put the signs back into the matrices). The modules we obtain this way have ℱ_{1}(M) isomorphic to F^{∗}(ℱ_{k}), the inverse image of the instanton bundle along the Frobenius map.

9.9. Schwarzenberger’s Bundles

The next example concerns the bundles ℱ(x_{1}, …, x_{m}). These were constructed by Schwarzenberger, and appear in the book of Okonek, Schneider and Spindler [202, Theorem 2.2.5]; see also [13]. Let k be an algebraically closed field of characteristic 2 and let E be an elementary abelian group of order 8. Let {x_{1}, …, x_{m}} be a finite set of points in ℙ^{2}. Then the following is a construction of a kE-module M of dimension 4m + 2 of constant Jordan type [2]^{2m}[1]^{2} with the following property. Given a shifted subgroup E′ of E of rank two, the restriction of M to E′ is Ω^{n}k ⊕Ω^{−n}k ⊕(projective), where n is the number of points from the set {x_{1}, …, x_{m}} which lie on the corresponding line ℙ^{1} ⊆ ℙ^{2}. The construction relies on the fact that there’s an easier construction of the corresponding bundle than the one given in [202]. Namely, there’s a monad of the form

𝒪(−1)^{m} → 𝒪^{2m+2} → 𝒪(1)^{m}

whose homology is ℱ(x_{1}, …, x_{m}). The maps in this monad are given as follows. The lefthand map is

(\begin{matrix}w_{1} & w_{2} & … & w_{m} \\ v_{1} & 0 & & 0 \\ −u_{1} & 0 & … & 0 \\ 0 & v_{2} & & 0 \\ 0 & −u_{2} & & 0 \\ & & … \\ 0 & 0 & & v_{m} \\ 0 & 0 & & −u_{m} \\ 0 & 0 & & 0 \end{matrix})

while the right-hand map is

(\begin{matrix}0 & u_{1} & v_{1} & 0 & 0 & … & 0 & 0 & w_{1} \\ 0 & 0 & 0 & u_{2} & v_{2} & & 0 & 0 & w_{2} \\ & & & & & … \\ & & & & & … \\ 0 & 0 & 0 & 0 & 0 & & u_{m} & v_{m} & w_{m} \end{matrix}) .

Here, for each i with 1 ≤ i ≤ m, the linear functions u_{i}, v_{i}, w_{i} are chosen to be any basis for the space spanned by Y_{1}, Y_{2}, Y_{3} with the property that u_{i} and v_{i} vanish on x_{i} but w_{i} does not.

Example 9.9.1. Suppose that x_{1} = (1 : 0 : 0), x_{2} = (0 : 1 : 0), x_{3} = (0 : 0 : 1), x_{4} = (1 : 1 : 1) and x_{5} = (1 : a : b). Then one choice of bases gives the matrices

(\begin{matrix}Y_{1} & Y_{2} & Y_{3} & Y_{1} & Y_{1} \\ Y_{2} & 0 & 0 & 0 & 0 \\ −Y_{3} & 0 & 0 & 0 & 0 \\ 0 & Y_{3} & 0 & 0 & 0 \\ 0 & −Y_{1} & 0 & 0 & 0 \\ 0 & 0 & Y_{1} & 0 & 0 \\ 0 & 0 & −Y_{1} & 0 & 0 \\ 0 & 0 & 0 & Y_{1} − Y_{2} & 0 \\ 0 & 0 & 0 & −Y_{1} + Y_{3} & 0 \\ 0 & 0 & 0 & 0 & aY_{1} − Y_{2} \\ 0 & 0 & 0 & 0 & −bY_{1} + Y_{3} \\ 0 & 0 & 0 & 0 & 0 \end{matrix}) and (\begin{matrix}0 & Y_{3} & Y_{2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & Y_{1} \\ 0 & 0 & 0 & Y_{1} & Y_{3} & 0 & 0 & 0 & 0 & 0 & 0 & Y_{2} \\ 0 & 0 & 0 & 0 & 0 & Y_{2} & Y_{1} & 0 & 0 & 0 & 0 & Y_{3} \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & Y_{1} − Y_{3} & Y_{1} − Y_{2} & 0 & 0 & Y_{1} \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & bY_{1} − Y_{3} & aY_{1} − Y_{2} & Y_{1} \end{matrix}) .

So we form a kE-module M with J^{3}(M) = 0 and Loewy layers of dimensions m, 2m + 2 and m, with these maps as the data for θ. In other words, the monad above is

{ \atop M/J(M)}(−1) ⟶^{θ} { \atop J(M)/J^{2}(M)} ⟶^{θ} { \atop J^{2}(M)}(1).

It is easy to see that ℱ_{1}(M) is isomorphic to the given Schwarzenberger bundle, and that the restriction of M to E′ has the required isomorphism type. The matrix for θ is as follows:

( \begin{matrix}w_{1} & w_{2} & … & w_{m} \\ v_{1} & 0 & & 0 \\ −u_{1} & 0 & … & 0 \\ 0 & v_{2} & & 0 \\ 0 & −u_{2} & & 0 \\ & & … \\ 0 & 0 & & v_{m} \\ 0 & 0 & & −u_{m} \\ 0 & 0 & & 0 \\ & & & & 0 & u_{1} & v_{1} & 0 & 0 & … & 0 & 0 & w_{1} \\ & & & & 0 & 0 & 0 & u_{2} & v_{2} & & 0 & 0 & w_{2} \\ & & & & & & & & & … \\ & & & & & & & & & … \\ & & & & 0 & 0 & 0 & 0 & 0 & & u_{m} & v_{m} & w_{m} \end{matrix}  ) .

9.10. The Examples of Tango

Recall from §6.8 that Tango produced examples of indecomposable bundles ℱ of rank r − 2 on ℙ^{r−1} = ℙ(V) fitting into exact sequences

0 → 𝒪(−2) → 𝒪(−1)^{r} → 𝒪^{2r−3} → ℱ → 0,

one for each subspace W of codimension 2r − 3 in Λ^{2}V containing no non-trivial wedges. For p = 2 it is easy to construct modules of constant Jordan type for (ℤ/2)^{r} realising these bundles. Namely, thinking of V as J(kE)/J^{2}(kE), we can regard W as a subspace of J^{2}(kE)/J^{3}(kE) = Λ^{2}V. Let Ŵ be the inverse image of W in J^{2}(kE). Then M = kE/Ŵ is a kE-module of dimension 3r − 2 of constant Jordan type [2]^{r}[1]^{r−2} satisfying

ℱ_{1}(M) ≅ ℱ.

The sequence above defining ℱ can be written as

0 → { \atop M/Rad(M)}(−2) → { \atop Rad(M)/Rad^{2}(M)}(−1) → { \atop Rad^{2}(M)} → ℱ_{1}(M) → 0.

One can visualise the module M as follows:

Λ^{2}V/W Vk

9.11. The Horrocks–Mumford Bundle

The Horrocks–Mumford bundle is an indecomposable vector bundle of rank two on ℙ^{4} in any characteristic. In this section we construct this bundle and examine its automorphisms. Literature includes the original paper of Horrocks and Mumford [161] as well as the papers of Aure [20], Barth [33], Barth, Hulek and Moore [35], Barth and Moore [36], Boralevi [59], Borcea [61], Decker [99, 100], Decker and Schreyer [102, 103], Dowbor and Meltzer [106], Hulek [163, 164], Hulek and Vandeven [165], Kaji [172], Lee [186], as well as Example 7.3 of Eisenbud, Fløystad and Schreyer [111] and Section 2.3.3 of Okonek, Schneider and Spindler [202].

Let V be a k-vector space of dimension five with basis v_{0}, …, v_{4}. We read these subscripts modulo five, so that for example v_{5} = v_{0}. Let v_{0}^{∗}, …, v_{4}^{∗} be the dual basis of V^{∗}, and identify V^{∗} with Λ^{4}V via

v_{i}^{∗} = v_{i+1} ∧ v_{i+2} ∧ v_{i+3} ∧ v_{i+4}.

Similarly, Λ^{2}V^{∗} is identified with Λ^{3}V via

(9.11.1)  v_{i}^{∗} ∧ v_{i+1}^{∗} = v_{i+2} ∧ v_{i+3} ∧ v_{i+4}
(9.11.2)  v_{i}^{∗} ∧ v_{i+2}^{∗} = −v_{i+1} ∧ v_{i+3} ∧ v_{i+4}.

These identifications amount to fixing a determinant form on V.

Let V′ be another k-vector space of dimension five with basis v′_{0}, …, v′_{4} and define maps

f_{+} : V′ → Λ^{2}V  f_{+}(v′_{i}) = v_{i+2} ∧ v_{i−2}
f_{−}: V′ → Λ^{2}V  f_{−}(v′_{i}) = v_{i+1} ∧ v_{i−1}.

Let f_{+}^{∗}, f_{−}^{∗}: Λ^{2}V^{∗} → (V′)^{∗} be the transposed maps.

Lemma 9.11.1. If v′ ∈ V′, v, v_{+}, v_{−} ∈ V satisfy f_{+}(v′) = v ∧ v_{+} and f_{−}(v′) = v ∧ v_{−} then v′ = 0.

Proof. We have

f_{+}(v′) ∧ f_{−}(v′) = v ∧ v_{+} ∧ v ∧ v_{−} = 0.

If v′ = ∑_{i} a_{i}v′_{i} then we have

0 = f_{+}(v′) ∧ f_{−}(v′) = ∑_{i} a_{i}^{2}v_{i+2} ∧ v_{i−2} ∧ v_{i+1} ∧ v_{i−1} = ∑_{i} a_{i}^{2}v_{i}^{∗}

and hence each a_{i} = 0. □

Now take the Koszul complex on ℙ^{4} = ℙ(V):

\xymatrix{0 \ar[r] & 𝒪 → V ⊗ 𝒪(1) \ar[rr] \ar[dr] &  & Λ^{2}V ⊗ 𝒪(2) \ar[rr] \ar[dr] &  & Λ^{2}V^{∗} ⊗ 𝒪(3) \ar[rr] \ar[dr] &  & V^{∗} ⊗ 𝒪(4) \ar[r] & 𝒪(5) \ar[r] & 0 \\  &  & 𝒯 \ar[ur] &  & Λ^{2}𝒯 \ar@{-->}[ur] &  & Λ^{3}𝒯 \ar[ur]}

Let W be a two dimensional vector space with basis w_{1}, w_{2}, and and define

p: V′ ⊗ 𝒪(2) \underrightarrow{(\begin{matrix}f_{+} \\ f_{−} \end{matrix} )} W ⊗ Λ^{2}V ⊗ 𝒪(2) → W ⊗ Λ^{2}𝒯
q: W ⊗ Λ^{2}𝒯 → W ⊗ Λ^{2}V^{∗} ⊗ 𝒪(3) ⟶^{(−f_{−}^{∗} f_{+}^{∗})} (V′)^{∗} ⊗ 𝒪(3).

Here, we are just regarding W ⊗ Λ^{2}V as a direct sum of two copies of Λ^{2}V, and the first map in the definition of p then has components f_{+} and f_{−}, tensored with the identity map on 𝒪(2). The point of using the space W is that automorphisms of the Horrocks–Mumford bundle will make use of linear transformations on W. Similar remarks apply to the second map in the definition of q.

Lemma 9.11.2. We have q ∘ p = 0.

Proof. First note that the composite of the first map in the definition of q with the second map in the definition of p is equal to the differential in the Koszul complex. Let ϕ be a function of some degree in a graded module defining 𝒪(2), and we evaluate q ∘ p on an element of the form v′_{i} ⊗ ϕ. So we first apply ( \begin{matrix}f_{+} \\ f_{−} \end{matrix} ) ⊗ 1 to get

(\begin{matrix}v_{i+2} ∧ v_{i−2} \\ v_{i+1} ∧ v_{i−1} \end{matrix}) ⊗ ϕ.

Next we apply the Koszul differential to obtain

(\begin{matrix}v_{i−1} ∧ v_{i+2} ∧ v_{i−2} ⊗ Y_{i−1}ϕ + v_{i} ∧ v_{i+2} ∧ v_{i−2} ⊗ Y_{i}ϕ + v_{i+1} ∧ v_{i+2} ∧ v_{i−2} ⊗ Y_{i+1}ϕ \\ v_{i−2} ∧ v_{i+1} ∧ v_{i−1} ⊗ Y_{i−2}ϕ + v_{i} ∧ v_{i+1} ∧ v_{i−1} ⊗ Y_{i}ϕ + v_{i+2} ∧ v_{i+1} ∧ v_{i−1} ⊗ Y_{i+2}ϕ \end{matrix}) .

Applying the identification (9.11.1) of Λ^{2}V^{∗} with Λ^{3}V, we rewrite this as

(\begin{matrix}v_{i}^{∗} ∧ v_{i+1}^{∗} ⊗ Y_{i−1}ϕ + v_{i+1}^{∗} ∧ v_{i−1}^{∗} ⊗ Y_{i}ϕ + v_{i−1}^{∗} ∧ v_{i}^{∗} ⊗ Y_{i+1}ϕ \\ v_{i}^{∗} ∧ v_{i+2}^{∗} ⊗ Y_{i−2}ϕ + v_{i+2}^{∗} ∧ v_{i−2}^{∗} ⊗ Y_{i}ϕ + v_{i−2}^{∗} ∧ v_{i}^{∗} ⊗ Y_{i+2}ϕ \end{matrix}) .

Finally, applying (−f_{−}^{∗}, f_{+}^{∗}) ⊗ 1 we get

0 − (v′_{i})^{∗} ⊗ Y_{i}ϕ + 0 + 0 + (v′_{i})^{∗} ⊗ Y_{i}ϕ + 0 = 0.  □

Lemma 9.11.3. We have

(1) p is a locally split injection, and (2) q is a locally split surjection.

Proof. (1) By Theorem 6.2.1 it suffices to prove that p(−2) and hence p is injective on fibres at every point in ℙ^{4} = ℙ(V). Choose a representative 0 ≠ v ∈ V of v̄ ∈ ℙ^{4}. If v′ is in the kernel of p at v̄ then ( \begin{matrix}f_{+} \\ f_{−} \end{matrix} ) (v′) is in the kernel of the differential in the Koszul complex and so there exist vectors v_{+} and v_{−} such that f_{+}(v′) = v ∧ v_{+} and f_{−}(v′) = v ∧ v_{−}. By Lemma 9.11.1 this implies that v′ = 0.

(2) The map q is the dual of p. □

Combining Lemmas 9.11.2 and 9.11.3 we have a monad

(9.11.3)  V′ ⊗ 𝒪(2) ⟶^{p} W ⊗ Λ^{2}𝒯 ⟶^{q} (V′)^{∗} ⊗ 𝒪(3).

Definition 9.11.4. The Horrocks–Mumford bundle ℱ_{HM} is defined to be the cohomology of the monad (9.11.3).

Proposition 9.11.5. The Chern polynomial of the Horrocks–Mumford bundle ℱ_{HM} is equal to 1 + 5h + 10h^{2}.

Proof. The Chern polynomial of Λ^{2}𝒯 is (1 + 2h)^{10}/(1 + h)^{5}. Applying Lemma 7.3.5 to the monad (9.11.3), we have

c(ℱ_{HM}) = ((1 + 2h)^{10}/(1 + h)^{5})^{2} / (1 + 2h)^{5}(1 + 3h)^{5},

which is equal to 1 + 5h + 10h^{2} as an element of ℤ[h]/(h^{5}). □

Theorem 9.11.6. The Horrocks–Mumford bundle ℱ_{HM} is indecomposable.

Proof. The Chern polynomial 1 + 5h + 10h^{2} does not factorise over the integers. See Lemma 7.3.9. □

Remark 9.11.7. The minimal resolution of the Horrocks–Mumford bundle by direct sums of line bundles (cf. Corollary 6.12.5) is as follows:

0 → 𝒪(−5)^{⊕2} → 𝒪(−3)^{⊕20} → 𝒪(−2)^{⊕35} → 𝒪^{⊕4} ⊕ 𝒪(−1)^{⊕15} → ℱ_{HM} → 0.

On the other hand, the construction above shows that there is a complex

0 → W ⊗ 𝒪 → W ⊗ V ⊗ 𝒪(1) → (W ⊗ Λ^{2}V)/V′ ⊗ 𝒪(2) → (V′)^{∗} ⊗ 𝒪(3) → 0

in which the maps are all linear, and which is exact except at the 𝒪(2) term, where the homology is ℱ_{HM}. This complex has the form

0 → 𝒪^{⊕2} → 𝒪(1)^{⊕10} → 𝒪(2)^{⊕15} → 𝒪(3)^{⊕5} → 0.

Writing a, b, c, d, e for the functions of degree one in V^{∗} dual to the basis v_{0}, …, v_{4} of V, the matrices for the maps in this complex may be taken to be

( \begin{matrix}a \\ b \\ c \\ d \\ e \\ & a \\ & b \\ & c \\ & d \\ & e \end{matrix} ) ,  ( \begin{matrix}−c & & a \\ & −d & & b \\ & & −e & & c \\ d & & & −a \\ & e & & & −b \\ −b & a & & & & & & e & & −c \\ & −c & b & & & −d & & & a \\ & & −d & c & & & −e & & & b \\ & & & −e & d & c & & −a \\ e & & & & −a & & d & & −b \\ & & & & & −b & a \\ & & & & & & −c & b \\ & & & & & & & −d & c \\ & & & & & & & & −e & d \\ & & & & & e & & & & −a \end{matrix} ) ,

( \begin{matrix}d & & & c & & & & a & & & e & & & & b \\ & e & & & d & & & & b & & c & a \\ e & & a & & & & & & & c & & d & b \\ & a & & b & & d & & & & & & & e & c \\ & & b & & c & & e & & & & & & & a & d \end{matrix} ) .

9.12. Automorphisms of the Horrocks–Mumford Bundle

In this section, we describe the automorphism group of ℱ_{HM}, under the assumption that k does not have characteristic 5.

Let ε be a primitive fifth root of unity in k. We define automorphisms of V using the following 5 × 5 matrices in SL(5, k):

α = (\begin{matrix}ε & 0 & 0 & 0 & 0 \\ 0 & ε & 0 & 0 & 0 \\ 0 & 0 & ε & 0 & 0 \\ 0 & 0 & 0 & ε & 0 \\ 0 & 0 & 0 & 0 & ε \end{matrix})  β = (\begin{matrix}1 & 0 & 0 & 0 & 0 \\ 0 & ε & 0 & 0 & 0 \\ 0 & 0 & ε^{2} & 0 & 0 \\ 0 & 0 & 0 & ε^{3} & 0 \\ 0 & 0 & 0 & 0 & ε^{4} \end{matrix})  γ = (\begin{matrix}0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 & 0 \end{matrix})

ζ = (\begin{matrix}1 & 0 & 0 & 0 & 0 \\ 0 & ε^{3} & 0 & 0 & 0 \\ 0 & 0 & ε^{2} & 0 & 0 \\ 0 & 0 & 0 & ε^{2} & 0 \\ 0 & 0 & 0 & 0 & ε^{3} \end{matrix}) η = −\frac{1}{\sqrt{5}} (\begin{matrix}1 & 1 & 1 & 1 & 1 \\ 1 & ε & ε^{2} & ε^{3} & ε^{4} \\ 1 & ε^{2} & ε^{4} & ε & ε^{3} \\ 1 & ε^{3} & ε & ε^{4} & ε^{2} \\ 1 & ε^{4} & ε^{3} & ε^{2} & ε \end{matrix}) κ = − (\begin{matrix}1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \end{matrix}) .

We write α′, …, κ′ for the automorphisms of V′ obtained by replacing each occurrence of ε^{i} by ε^{2i} in these matrices. Let α′′, …, κ′′ be the automorphisms of W given as follows:

α′′ = β′′ = γ′′ = (\begin{matrix}1 & 0 \\ 0 & 1 \end{matrix}) ζ′′ = (\begin{matrix}ε & 0 \\ 0 & ε^{4} \end{matrix}) η′′ = \frac{1}{\sqrt{5}} (\begin{matrix}ε^{2} − ε^{3} & ε − ε^{4} \\ ε − ε^{4} & ε^{3} − ε^{2} \end{matrix}) κ′′ = (\begin{matrix}0 & 1 \\ −1 & 0 \end{matrix}) .

The elements α, β, γ satisfy the relations

α^{5} = β^{5} = γ^{5} = 1,  αβ = βα,  αγ = γα,  γβ = αβγ.

So the group H = ⟨α, β, γ⟩ is an extraspecial group of order 5^{1+2} and exponent 5. Its centre is Z = ⟨α⟩ of order 5, and we write H̄ for H/Z.

The elements ζ and η satisfy

ζ^{5} = 1,  (ζη)^{3} = 1,  η^{2} = −1

and so the group L = ⟨ζ, η⟩ is isomorphic to SL(2, 5). Furthermore, κ is an element of L, and the subgroup Q = ⟨η, κ⟩ is a Sylow 2-subgroup of L, isomorphic to a quaternion group of order 8. The subgroup R = ⟨ζη⟩ is a Sylow 3-subgroup of L, of order 3.

The group L normalises H,

αζ = ζα,  αη = ηα,  βζ = ζβ,  βη = ηγ^{−1},  γζ = ζα^{3}βγ,  γη = ηβ,

and together they generate a semidirect product

N = ⟨α, β, γ, ζ, η⟩ = H ⋊ L,

a group of order 15, 000 and shape 5^{1+2}: SL(2, 5). The group N is the normaliser in SL(5, k) of H, and is the full automorphism group of ℱ_{HM}. Sylow 2-, 3- and 5-subgroups of N are given by Q, R and

S = ⟨α, β, γ, ζ⟩,

respectively.

The matrices α′, …, κ′ and the matrices α′′, …, κ′′ satisfy the same relations as those given for α, …, κ above, and so V′ and W are also representations of N. In fact, W is inflated from a representation of N̄ = N/H ≅ SL(2, 5). Corresponding to the isomorphism Λ^{2}(W) ≅ k, the representation W supports an invariant symplectic form given by the matrix

(9.12.1)  (\begin{matrix}0 & 1 \\ −1 & 0 \end{matrix}) .

It follows that W is self-dual.

Next we discuss the representation theory of H, and use it to explain why the normaliser of H in SL(5, k) is the group N of shape 5^{1+2}: SL(2, 5). In an Appendix to this section, we give a listing of code in Magma [63] for finding the ordinary character table of N.

We continue to assume that k does not have characteristic five. With this assumption, H has 25 irreducible representations of dimension one with Z in the kernel, and four faithful irreducible representations of dimension five. These four are distinguished by the scalar multiple of the identity matrix representing α. We write V_{1}, V_{2}, V_{3}, V_{4} for the irreducible representations of dimension five, where the scalar representing α on V_{i} is equal to ε^{i}.

Let ϕ be an automorphism of H that acts trivially on Z. If ρ: H → SL(5, k) is a homomorphism realising the representation V_{1} then ρ ∘ ϕ is another homomorphism realising an isomorphic representation. So there is a matrix A_{ϕ} such that for all h ∈ H we have ρ(ϕ(h)) = A_{ϕ}ρ(h)A_{ϕ}^{−1}. The matrix A_{ϕ} is well defined up to scalars, it can be chosen to be in SL(5, k), and A_{ϕ}A_{ϕ′} is a scalar multiple of A_{ϕϕ′}. Thus the matrices A_{ϕ} together with ρ(H) generate the normaliser N = N_{SL(5,k)}(H). The group N surjects onto the group N̄ of automorphisms of H acting trivially on Z, which is an extension of the outer automorphisms SL(2, 5) by the inner automorphisms H̄. The extension of SL(2, 5) by H̄ splits because the centraliser of an involution in the centre of a Sylow 2-subgroup is a complement for H̄. The kernel of N → Aut(H) is C_{SL(5,k)}(H), which consists just of the scalar matrices. There are five of these, and they are in ρ(H). The group N has order 15,000, and sits in an extension of the form

1 → H → N → SL(2, 5) → 1.

So we regard H as a subgroup of N and V_{1} as a representation of N. Since

H^{2}(SL(2, 5), ℤ/5) = 0

the extension splits, so that N is isomorphic to the semidirect product H ⋊ SL(2, 5).

[figure: 1; 1; 1; Z; H; H̄; 1; 1; Z; N; N̄; 1; SL(2, 5); SL(2, 5); 1; 1]

So V_{i} extends uniquely to a representation of N for 1 ≤ i ≤ 4 which we continue to denote V_{i}. Thus we have

V_{1} ≅ V, V_{2} ≅ V′, V_{3} ≅ (V′)^{∗} ≅ Λ^{4}(V′), V_{4} ≅ V^{∗} ≅ Λ^{4}(V).

Next, we apply this discussion to the construction of the Horrocks–Mumford bundle given in Section 9.11. We begin by explaining the roles of the maps ( \begin{matrix}f_{+} \\ f_{−} \end{matrix} ) : V′ → W ⊗ Λ^{2}V and (−f_{−}, f_{+}): W ⊗ Λ^{2}V^{∗} → (V′)^{∗} used in the constructions there.

Lemma 9.12.1. There is an isomorphism of kN-modules W ⊗ V_{2} ≅ Λ^{2}V_{1}.

Proof. The maps f_{+} and f_{−} of the previous section give rise to a map

(9.12.2)  W ⊗ V_{2} ⟶^{(f_{−},−f_{+})} Λ^{2}V_{1}
(9.12.3)  (\begin{matrix}v′_{i} \\ v′_{j} \end{matrix}) ↦ v_{i+1} ∧ v_{i−1} − v_{j+2} ∧ v_{j−2}.

We check that

(f_{−}, −f_{+}) ∘ (α′′ ⊗ α′) = Λ^{2}(α) ∘ (f_{−}, −f_{+}),

and similarly for β, …, κ. Therefore, (f_{1}, −f_{+}) is a kN-module homomorphism. It is nonzero, and Λ^{2}V_{1} is irreducible, so it is an isomorphism. □

Using the isomorphism W ≅ W^{∗} given by the symplectic form (9.12.1), the adjoint of the map given in Lemma 9.12.1 is the map

(\begin{matrix}f_{+} \\ f_{−} \end{matrix}) : V_{2} → W ⊗ Λ^{2}V_{1}

appearing in Section 9.11.

Dually, we have the following.

Lemma 9.12.2. There is an isomorphism

W ⊗ V_{3} ⟶^{(f_{+}^{∗},f_{−}^{∗})} Λ^{2}V_{4}.

The adjoint of this is the map

(−f_{−}^{∗}, f_{+}^{∗}): V_{3} → W ⊗ Λ^{2}V_{4}

used in Section 9.11.

We have thus seen that the maps used in the construction of ℱ_{HM} commute with the automorphisms given by the group N. Thus N acts as automorphisms of the Horrocks– Mumford bundle.

Appendix: Characters of N

Recall the shape of the group N of order 15,000:

N_{SL(2,5)} H_{(ℤ/5)^{2}} Z_{ℤ/5} 1 .

In this Appendix, we provide a listing of code in the Magma computer algebra language [63] to construct the group N as a subgroup of SL(5, p), provided p is congruent to 1 modulo 5, and produce its character table. The value of p may be changed from 11 to any other prime congruent to 1 modulo 5 by changing the first line. For primes not congruent to 1 modulo 5, it is necessary first to extend the ground field so that it contains fifth roots of unity.

p:=11;
G:=SL(5,p);
H:=SylowSubgroup(G,5);
Z:=Centre(H);
K:=H;
while Order(K) ne 5 do z:=Random(Z); K:=sub<H|z>; end while;
while IsAbelian(K) do
 while Order(K) ne 25 do x:=Random(H); K:=sub<H|z,x>; end while;
 while Order(K) ne 125 do y:=Random(H); K:=sub<H|z,x,y>; end while;
end while;
N:=Normaliser(G,K);
X:=CharacterTable(N);

The character degrees are as follows: 1 (trivial module k), 2 (twice, W and W′), 3 (twice, U and U′), 4 (twice, Y = W ⊗ W′ and Y′), 5 (five times, V_{1}, …, V_{4}, Z), 6 (T), 10 (eight times, W ⊗ V_{i} and W′ ⊗ V_{i} for 1 ≤ i ≤ 4), 15 (eight times, U ⊗ V_{i} and U′ ⊗ V_{i} for 1 ≤ i ≤ 4), 20 (eight times, Y ⊗ V_{i} and Y′ ⊗ V_{i} for 1 ≤ i ≤ 4), 24 (five times), 25 (four times, Z ⊗ V_{i} for 1 ≤ i ≤ 4) and 30 (four times, T ⊗ V_{i} for 1 ≤ i ≤ 4), for a total of 50 irreducible characters. Here are some tensor products of characters of low degree:

k W W′  U  U′  Y  Y′  Z
W k\U Y W′ ⊕ Y′  T  W′ ⊕ T  U ⊕ Z  Y′ ⊕ T
W′  k\U′  T  W′ ⊕ Y′  W ⊕ T  U′ ⊕ Z  Y′ ⊕ T
U  U\k ⊕ Z Y ⊕ Z U′ ⊕ Y ⊕ Z W ⊕ Y′ ⊕ T U ⊕ U′ ⊕ Y ⊕ Z
U′  U′\k ⊕ Z U ⊕ Y ⊕ Z W′ ⊕ Y′ ⊕ T U ⊕ U′ ⊕ Y ⊕ Z

Here, the backslash separates the exterior square from the symmetric square in the case of tensoring a character with itself. Further tensor products are given by Λ^{2}(V_{1}) = W ⊗ V_{2}, S^{2}(V_{1}) = U′ ⊗ V_{2}, Λ^{3}(V_{1}) = W ⊗ V_{3}, S^{3}(V_{1}) = (W ⊕ Z) ⊗ V_{3}, Λ^{4}(V_{1}) = V_{4}, S^{4}(V_{1}) = (U ⊕U′ ⊕Y ⊕Y′)⊗V_{4}, V_{1} ⊗V_{2} = (W′ ⊕U)⊗V_{3}, V_{1} ⊗V_{3} = (W ⊕U′)⊗V_{4}, and V_{1} ⊗V_{4} = k ⊕ irreducible 24.

Second proof of Lemma 9.11.2 in characteristic ≥ 7. Recall that we have

\xymatrix{V_{2} ⊗ 𝒪(2) \ar[r]^{^{( \begin{matrix}f_{+} \\ f_{−} \end{matrix} )}} \ar@{-->}[drr]_{p} & W ⊗ Λ^{2}V_{1} ⊗ 𝒪(2) \ar[rr]^{d} \ar@{-->}[dr] &  & W ⊗ Λ^{3}V_{1} ⊗ 𝒪(3) \ar[r]^{(−f_{−}^{∗},f_{+}^{∗})} & V_{3} ⊗ 𝒪(3). \\  &  & W ⊗ Λ^{2}𝒯 \ar@{-->}[ur] \ar@{-->}[urr]_{q}}

All of the maps commute with the action of N, and so the composite q ∘ p is represented by a kN-module homomomorphism V_{1} ⊗ V_{2} → V_{3}. But V_{1} ⊗ V_{2} ≅ (W′ ⊕ U) ⊗ V_{3} is a sum of two irreducible characters, and does not admit any non-zero kN-module homomorphisms to V_{3}. □

9.13. Realising the Horrocks–Mumford Bundle

We introduced the Horrocks–Mumford bundle ℱ_{HM} in Section 9.11. For the exterior algebra on five variables we can produce a module M of dimension 32 and constant Jordan type [2]^{15}[1]^{2} with ℱ_{1}(M) ≅ ℱ_{HM} in any characteristic, by using the matrices given in that section, see Figure 9.1. This figure shows the action of θ = aX_{1} + bX_{2} + cX_{3} + dX_{4} + eX_{5}.

In this section, we give a module theoretic construction for ℱ_{HM} which is similar to the construction of the functors ℱ_{i} but different.

Remark 9.13.1. In Chapter 8, we proved that given any vector bundle ℱ of rank s on ℙ^{r−1}, there is a finitely generated kE-module M of stable constant Jordan type [1]^{s} such that:

(i) if p = 2 then ℱ_{1}(M) ≅ ℱ, and (ii) if p is odd, then ℱ_{1}(M) ≅ F^{∗}(ℱ), where F: ℙ^{r−1} → ℙ^{r−1} is the Frobenius morphism.

However, we also proved in Section 10.2 that for a module M of stable constant Jordan type [1]^{s}, the Chern numbers c_{i}(ℱ_{1}(M)) are divisible by p for 1 ≤ i ≤ p − 2. So although F^{∗}(ℱ_{HM}) can be constructed by this general construction, no twist of ℱ_{HM} itself can be ℱ_{1}(M) for a module M of constant Jordan type [1]^{2}, since c_{1}(ℱ_{HM}(i)) = 2i + 5 and c_{2}(ℱ_{HM}(i)) = i^{2} + 5i + 10 cannot both be divisible by p for p ≥ 7. This demonstrates that we are forced to use a different construction.

( \begin{matrix}a \\ b \\ c \\ d \\ e \\ & a \\ & b \\ & c \\ & d \\ & e \\ & & −c & & a \\ & & & −d & & b \\ & & & & −e & & c \\ & & d & & & −a \\ & & & e & & & −b \\ & & −b & a & & & & & & e & & −c \\ & & & −c & b & & & −d & & & a \\ & & & & −d & c & & & −e & & & b \\ & & & & & −e & d & c & & −a \\ & & e & & & & −a & & d & & −b \\ & & & & & & & −b & a \\ & & & & & & & & −c & b \\ & & & & & & & & & −d & c \\ & & & & & & & & & & −e & d \\ & & & & & & & e & & & & −a \\ & & & & & & & & & & & & d & & & c & & & & a & & & e & & & & b \\ & & & & & & & & & & & & & e & & & d & & & & b & & c & a \\ & & & & & & & & & & & & e & & a & & & & & & & c & & d & b \\ & & & & & & & & & & & & & a & & b & & d & & & & & & & e & c \\ & & & & & & & & & & & & & & b & & c & & e & & & & & & & a & d \end{matrix}  )

Figure 9.1. Matrix for θ for the Horrocks–Mumford bundle example.

We begin with a description of extensions of simple modules over semidirect products of the form kP ⋊ G in preparation for the construction. Let P be a finite p-group, k be a field of characteristic p, G be a finite p′-group acting as algebra automorphisms on kP, and let A = kP ⋊ G, a finite dimensional algebra over k. Let V be the A-module given by J(kP)/J^{2}(kP). Note that P acts trivially on V, so that it is really just a kG-module. The simple A-modules all have P in the kernel, so they are the same as the simple kG-modules. The following proposition says that Ext^{1} between these simple modules can be computed just using character theory.

Proposition 9.13.2. If S_{1} and S_{2} are A-modules with P in the kernel then

Ext_{A}^{1}(S_{1}, S_{2}) ≅ Hom_{kG}(V ⊗_{k} S_{1}, S_{2}).

Proof. As kP -modules, S_{1} and S_{2} are trivial, so

Ext_{kP}^{1} (S_{1}, S_{2}) ≅ Ext_{kP}^{1} (k, k) ⊗_{k} Hom_{k}(S_{1}, S_{2})
≅ V^{∗} ⊗_{k} Hom_{k}(S_{1}, S_{2})
≅ Hom_{k}(V ⊗_{k} S_{1}, S_{2}).

Since G is a p′-group, we have

Ext_{A}^{1}(S_{1}, S_{2}) ≅ Ext_{kP}^{1} (S_{1}, S_{2})^{H} ≅ Hom_{kG}(V ⊗_{k} S_{1}, S_{2}).  □

More explicitly, given a kG-module homomorphism ρ: V ⊗_{k} S_{1} → S_{2}, there is an adjoint kG-module homomorphism ρ′: V → Hom_{k}(S_{1}, S_{2}). We can describe the corresponding extension of S_{1} by S_{2} as follows. As a module for kG, the extension splits as S_{1} ⊕ S_{2}. The action of P has J^{2}(kP) in the kernel, and v ∈ J(kP)/J^{2}(kP) acts as the matrix ( \begin{matrix}0 & 0 \\ ρ′(v) & 0 \end{matrix} ).

Note that if G is not a p′-group then Hom_{kG}(V ⊗_{k} S_{1}, S_{2}) is canonically isomorphic to a subspace of Ext_{A}^{1}(S_{1}, S_{2}), but there may also be extensions coming from extensions of kG-modules.

Let us apply this in the case of kE⋊N-modules, where E = (ℤ/p)^{5} and N = 5^{1+2}: SL(2, 5) acting on X_{1}, …, X_{5} via the matrices given in Section 9.12. Recall from Sections 9.11 and 9.12 that we have a diagram of sheaves on ℙ^{4}, (9.13.1)

\xymatrix{V_{2} ⊗ 𝒪(2) \ar[r]^{^{( \begin{matrix}f_{+} \\ f_{−} \end{matrix} )}} \ar@{-->}[drr]_{p} & W ⊗ Λ^{2}V_{1} ⊗ 𝒪(2) \ar[rr]^{d} \ar@{-->}[dr] &  & W ⊗ Λ^{3}V_{1} ⊗ 𝒪(3) \ar[r]^{(−f_{−}^{∗},f_{+}^{∗})} & V_{3} ⊗ 𝒪(3), \\  &  & W ⊗ Λ^{2}𝒯 \ar@{-->}[ur] \ar@{-->}[urr]_{q}}

and that the Horrocks–Mumford bundle ℱ_{HM} is defined to be Ker (q)/Im (p) as a subquotient of W ⊗ Λ^{2}𝒯.

Using Proposition 9.13.2, we realise the Koszul differential

d: W ⊗ Λ^{2}V_{1} ⊗ 𝒪(2) → W ⊗ Λ^{3}V_{1} ⊗ 𝒪(3)

by a homomorphism V_{1} ⊗ (W ⊗ Λ^{2}V_{1}) → W ⊗ Λ^{3}V_{1} and hence by an extension of kE ⋊ Nmodules

0 → W ⊗ Λ^{3}V_{1} → M → W ⊗ Λ^{2}V_{1} → 0.

Thus M is a kE-module of Loewy length two. Write N_{1} for the kernel of (−f_{−}, f_{+}): W ⊗ Λ^{3}V_{1} ↠ V_{3} and N_{2} for the cokernel of ( \begin{matrix}f_{+} \\ f_{−} \end{matrix} ) : V_{2} ↣ W ⊗ Λ^{2}V_{1}. Then we have a complex of kE ⋊ N-modules

N_{1} ↣^{ϕ} M ↠^{ψ} N_{2}.

This gives rise to a diagram of sheaves on ℙ^{4}:

(9.13.2)  \xymatrix{ & M̃ \ar[r]^{ψ̃} \ar[d]^{θ} & Ñ_{2}. \\ Ñ_{1}(1) \ar[r]^{ϕ̃(1)} & M̃(1)}

Theorem 9.13.3. In Diagram (9.13.2), we have

θ(Ker ψ̃) ⊆ Im (ϕ̃(1))

and

Im (θ) ∩ Im (ϕ̃(1)) / θ(Ker ψ̃) ≅ ℱ_{HM}(−2),

the Horrocks–Mumford bundle untwisted twice.

Proof. In Diagram (9.13.2), { \atop Soc(M)} is in the kernel of ψ̃ and θ, so we may quotient it out. Similarly, the images of ϕ̃(1) and θ are contained in { \atop Soc(M)}(1) so we obtain a diagram

\xymatrix{0 \ar[r] & Ṽ_{2} \ar[r] & { \atop W ⊗ Λ^{2}V_{1}} \ar[r]^{ψ̃} \ar[d]^{θ} & Ñ_{2} \ar[r] & 0 \\ 0 \ar[r] & Ñ_{1}(1) \ar[r]^{ϕ̃(1)} & { \atop W ⊗ Λ^{3}V_{1}}(1) \ar[r] & Ṽ_{3}(1) \ar[r] & 0}

where the rows are exact. Twisting this twice and comparing with Diagram (9.13.1) proves the theorem. □

9.14. The Horrocks Parent Bundle and the Tango Bundle

In this section, we describe the Horrocks parent bundle of rank three on ℙ^{5} in any characteristic. Further information on this bundle can be found in Horrocks [160] as well as the papers [11, 101, 236]. Let V be a vector space of dimension six over k, and choose a decomposition V = U ⊕ U^{∗} as a sum of a vector space of dimension three and its dual. Choose a basis v_{1}, v_{2}, v_{3} of U, and let v_{4}, v_{5}, v_{6} be the dual basis of U^{∗}. We have an action of SL(3, k) on U as matrices with determinant one, and on U^{∗} as the transpose inverses of these matrices. This gives an action of G = SL(3, k) ⋊ ℤ/2 on V. Here, the ℤ/2 acts on SL(3, k) through transpose inverse on matrices. The group G will be the full automorphism group of the Horrocks parent bundle.

Using the determinant form given by these bases, we have isomorphisms U ≅ Λ^{2}(U^{∗}) and U^{∗} ≅ Λ^{2}(U) given by

(9.14.1)  v_{1} ⟷ v_{5} ∧ v_{6},  v_{2} ⟷ v_{6} ∧ v_{4},  v_{3} ⟷ v_{4} ∧ v_{5}

and

(9.14.2)  v_{4} ⟷ v_{2} ∧ v_{3},  v_{5} ⟷ v_{3} ∧ v_{1},  v_{6} ⟷ v_{1} ∧ v_{2}.

So we have

Λ^{2}(V) = Λ^{2}(U) ⊕ Λ^{2}(U^{∗}) ⊕ (U ⊗ U^{∗}) ≅ U^{∗} ⊕ U ⊕ (U ⊗ U^{∗}) ≅ V ⊕ (U ⊗ U^{∗}).

The isomorphisms (9.14.1) and (9.14.2) combine to give maps

V ⟶^{ϕ} Λ^{2}(V) ⟶^{ψ} V

with ψϕ = Id_{V}.

There is a non-degenerate G-invariant symplectic form ω on V given by

ω((u, ρ), (u′, ρ′)) = ρ′(u) − ρ(u′)

for u, u′ ∈ U and ρ, ρ′ ∈ U^{∗}. With respect to the basis v_{1}, …, v_{6} of V this has matrix ( \begin{matrix}0 & I \\ −I & 0 \end{matrix} ). This symplectic form induces a G-invariant isomorphism V ≅ V^{∗}, under which the image of ω is ω^{∗} = v_{1} ∧ v_{4} + v_{2} ∧ v_{5} + v_{3} ∧ v_{6}.

The form ω induces a non-degenerate G-invariant symmetric bilinear form on Λ^{2}V via

((v ∧ v′), (v′′ ∧ v′′′)) ↦ ω(v, v′)ω(v′′, v′′′) + ω(v, v′′)ω(v′′′, v′) + ω(v, v′′′)ω(v′, v′′)

and hence an isomorphism Λ^{2}V ≅ (Λ^{2}V)^{∗} = Λ^{2}V^{∗}. The following lemma is easy to check.

Lemma 9.14.1. (i) ϕ sends every element v ∈ V to a sum of wedges of elements v′ ∧ v′′ satisfying ω(v, v′) = ω(v, v′′) = 0.

(ii) For all v, v′, v′′ in V we have ω(v, ψ(v′ ∧ v′′)) = ω(v′, ψ(v′′ ∧ v)) = ω(v′′, ψ(v ∧ v′)). In particular, we have ω(v, ψ(v, v′)) = 0. □

These maps allow us to define a complex of sheaves on ℙ^{5} = ℙ(V):

(9.14.3) 0 → 𝒪(−2) → (k ⊕ V) ⊗ 𝒪(−1) → Λ^{2}(V) ⊗ 𝒪 → (V ⊕ k) ⊗ 𝒪(1) → 𝒪(2) → 0.

The first two maps in the complex are given by the Koszul differential combined with the map ϕ. The third and fourth maps are related to these by taking the duals using the isomorphisms above, and then conjugating by the matrix ( \begin{matrix}0 & I \\ −I & 0 \end{matrix} ). More explicitly, these maps are as follows:

(i) 𝒪(−2) → (k ⊕ V) ⊗ 𝒪(−1) (ii) (k ⊕ V) ⊗ 𝒪(−1) → Λ^{2}V ⊗ 𝒪 v: λ ↦ (0, λv) v: (λ, v′) ↦ λϕ(v) + v ∧ v′

(iii) Λ^{2}V ⊗ 𝒪 → (V ⊕ k) ⊗ 𝒪(1)

v: v′ ∧ v′′ ↦ (ω(v, v′)v′′ + ω(v′, v′′)v + ω(v′′, v)v′, ω(v, ψ(v′ ∧ v′′)))

(iv) (V ⊕ k) ⊗ 𝒪(1) → 𝒪(2)

v: (v′, λ) ↦ ω(v, v′).

Theorem 9.14.2. The complex (9.14.3) only has homology in the middle term, and is exact elsewhere.

Proof. It is easy to check using the explicit formulae above together with Lemma 9.14.1 that the composites are zero. Alternatively, it can be verified by multiplying the matrices given below for the maps. More abstractly, it comes from the fact that the construction is based on the exterior square of a null correlation bundle for the symplectic form, see [160].

Exactness at the second term amounts to showing that if 0 ≠ v ∈ V, and (λ, w) ∈ k ⊕ V satisfies

(9.14.4)  λϕ(v) + v ∧ w = 0

then λ = 0 and w is a multiple of v. To see this, let v = (v′, v′′), w = (w′, w′′) with v′, w′ ∈ U and v′′, w′′ ∈ U^{∗}. Projecting the equation (9.14.4) onto the summand U ⊗ U^{∗} of Λ^{2}(V), the term λϕ(v) vanishes and we are left with v′ ∧ w′′ + v′′ ∧ w′ = 0 in Λ^{2}(V). As elements of U ⊗ U^{∗} this gives v′ ⊗ w′′ = w′ ⊗ v′′. Since v ≠ 0 this implies that for some scalar μ ∈ k we have w′ = μv′ and w′′ = μv′′. Thus w = μv. Plugging this back into (9.14.4), we see that λϕ(v) = 0. Since ϕ is injective, we conclude that λ = 0.

For exactness at the fourth term, we first check that (0, 1) is in the image. Given 0 ≠ v = λ_{1}v_{1} + ⋯ + λ_{6}v_{6}, some λ_{i} is non-zero. Since there are elements of G sending any v_{i} to any other, we may assume that i = 1 without loss of generality. Consider the element

−λ_{1}v_{2} ∧ v_{3} − λ_{6}v_{2} ∧ v_{4} + λ_{5}v_{3} ∧ v_{4}.

A short computation using the definitions shows that v sends this element to (0, λ_{1}), and hence dividing by λ_{1}, (0, 1) is also in the image.

Having established this, we can ignore the second term in the image, and prove that given v′ ∈ V with ω(v, v′) = 0, there exists an element in Λ^{2}V taken by v to something with v′ in its first component. If v′ is not a multiple of v, then by nondegeneracy of ω we may choose v′′ ∈ V with ω(v′′, v) = 1 and ω(v′, v”) = 0. Then

v: v′ ∧ v′′ ↦ (v′, ω(v, ψ(v′ ∧ v′′)))

does the trick. If v′ = v, we choose v′′ and v′′′ with ω(v, v′′) = 0, ω(v, v′′′) = 0, ω(v′′, v′′′) = 1. Then

v: v′′ ∧ v′′′ ↦ (v, ω(v, ψ(v′′ ∧ v′′′)))

does the trick. So we are done. □

Definition 9.14.3. The Horrocks parent bundle is the homology at the middle term of the complex (9.14.3).

Remark 9.14.4. From the construction, we see that the dual of the Horrocks parent bundle is the pullback of this bundle through the automorphism ( \begin{matrix}0 & I \\ −I & 0 \end{matrix} ) of ℙ^{5}.

Writing a, b, c, d, e, f for the functions of degree one in V^{∗} dual to the basis v_{1}, …, v_{6} of V, the matrices for the maps in this complex may be taken to be

( \begin{matrix}0 \\ a \\ b \\ c \\ d \\ e \\ f \end{matrix} ) , ( \begin{matrix}a & & & & & f & −e \\ b & & & & −f & & d \\ c & & & & e & −d \\ d & & c & −b \\ e & −c & & a \\ f & b & −a \\ & d & & & −a \\ & e & & & & −a \\ & f & & & & & −a \\ & & d & & −b \\ & & e & & & −b \\ & & f & & & & −b \\ & & & d & −c \\ & & & e & & −c \\ & & & f & & & −c \end{matrix} ) ,

( \begin{matrix}& & & & f & −e & & b & c & & −a & & & & −a \\ & & & −f & & d & −b & & & a & & c & & & −b \\ & & & e & −d & & −c & & & & −c & & a & b \\ & c & −b & & & & & & & −e & d & & −f & & d \\ −c & & a & & & & e & −d & & & & & & −f & e \\ b & −a & & & & & f & & −d & & f & −e \\ d & e & f & −a & −b & −c \end{matrix} ) , ( d e f a b c 0 ) .

Proposition 9.14.5. The Horrocks parent bundle has rank three and Chern polynomial 1 + 3h^{2}.

Proof. Using the defining sequence (9.14.3), we see that the rank is 1−7+15−7+1 = 3 and the Chern polynomial is

(1 − 2h)(1 − h)^{−7}(1 + h)^{−7}(1 + 2h)

which is equal to 1 + 3h^{2} as an element of ℤ[h]/(h^{6}). □

Theorem 9.14.6. If k does not have characteristic two, then the Horrocks parent bundle is irreducible.

( \begin{matrix}a \\ b \\ c \\ d \\ e \\ f \\ & a & & & & & f & −e \\ & b & & & & −f & & d \\ & c & & & & e & −d \\ & d & & c & −b \\ & e & −c & & a \\ & f & b & −a \\ & & d & & & −a \\ & & e & & & & −a \\ & & f & & & & & −a \\ & & & d & & −b \\ & & & e & & & −b \\ & & & f & & & & −b \\ & & & & d & −c \\ & & & & e & & −c \\ & & & & f & & & −c \\ & & & & & & & & & & & & f & −e & & b & c & & −a & & & & −a \\ & & & & & & & & & & & −f & & d & −b & & & a & & c & & & −b \\ & & & & & & & & & & & e & −d & & −c & & & & −c & & a & b \\ & & & & & & & & & c & −b & & & & & & & −e & d & & −f & & d \\ & & & & & & & & −c & & a & & & & e & −d & & & & & & −f & e \\ & & & & & & & & b & −a & & & & & f & & −d & & f & −e \\ & & & & & & & & d & e & f & −a & −b & −c \\ & & & & & & & & & & & & & & & & & & & & & & & d & e & f & a & b & c \end{matrix}  )

Figure 9.2. Matrix for θ for the Parent bundle example.

Proof. Since it has rank three, if it is not irreducible, then either it or its dual contains a line bundle 𝒪(j). The quotient is then a vector bundle of rank two with Chern polynomial (1 + 3h^{2})/(1 + jh), and hence j = 0.

Now restrict to the projective subspace ℙ(U). The restriction of (9.14.3) is a direct sum of two twists of the (exact) Koszul complex and a monad:

\xymatrix{0 \ar[r] & 𝒪(−2) \ar[r] & U ⊗ 𝒪(−1) \ar[r] & U^{∗} ⊗ 𝒪 \ar[r] & 𝒪(1) \ar[r] & 0 \\  & 0 \ar[r] & 𝒪(−1) \ar[r] & U ⊗ 𝒪 \ar[r] & U^{∗} ⊗ 𝒪(1) \ar[r] & 𝒪(2) \ar[r] & 0 \\  & 0 \ar[r] & U^{∗} ⊗ 𝒪(−1) \ar[r] & U ⊗ U^{∗} ⊗ 𝒪 \ar[r] & U ⊗ 𝒪(1) \ar[r] & 0.}

The maps in the monad are u: ρ → u ⊗ ρ and u: u′ ⊗ ρ → ρ(u′)u − ρ(u)u′ for u, u′ ∈ U and ρ ∈ U^{∗}.

It follows that the restriction of the Horrocks parent bundle to ℙ(U) is the homology of this monad, which is the quotient of 𝒯 ⊗Ω = ℋom(𝒯, 𝒯) by the copy of 𝒪 given by multiples of the identity. If k does not have characteristic two, then ℋom(𝒯, 𝒯) decomposes as a direct sum of 𝒪 and a self-dual bundle of rank three which does not have a subbundle isomorphic to 𝒪. □

( \begin{matrix}a \\ b \\ c \\ d \\ e \\ f \\ & a & & & & & f & e \\ & b & & & & f & & d \\ & c & & & & e & d \\ & d & & c & b \\ & e & c & & a \\ & f & b & a \\ & & d & & f & a & & c \\ & & e & & & & a \\ & & f & & & & & a \\ & & & d & & b \\ & & & e & f & & b & c \\ & & & f & & & & b \\ & & & & d & c \\ & & & & e & & c \\ & & & & & & & & & & & & f & e & & b & c & & a \\ & & & & & & & & & & & f & & d & b & & & a & & c \\ & & & & & & & & & & & e & d & & c & & & & c & & a & b \\ & & & & & & & & & c & b & & & & & & & e & d & & f \\ & & & & & & & & c & & a & & & & e & d & & & & & & f \\ & & & & & & & & b & a & & & & & f & & d & & f & e \\ & & & & & & & & d & e & f & a & b & c \\ & & & & & & & & & & & & & & & & & & & & & & d & e & f & a & b & c \end{matrix}  )

Figure 9.3. Matrix for θ for the Tango bundle example.

Theorem 9.14.7. If k has characteristic two, then the Horrocks parent bundle contains a copy of 𝒪, and the quotient is an irreducible vector bundle of rank two on ℙ^{5} with Chern polynomial 1 + 3h^{2}.

Proof. Under the map (iii) above, v sends the symplectic form ω^{∗} ∈ Λ^{2}V to 2v. This is zero in characteristic two. Since ω^{∗} is not in the image of the map (ii) for any v ∈ V, it follows that this defines a trivial subbundle of the Horrocks parent bundle. The quotient is irreducible because its Chern polynomial does not factorise as a product of two linear factors. □

Definition 9.14.8. The Tango bundle is the vector bundle of rank two on ℙ^{5} in characteristic two described in the theorem. It is thus the homology in the middle of a complex of the form

0 → 𝒪(−2) → (k ⊕ V) ⊗ 𝒪(−1) → Λ^{2}(V)/⟨ω^{∗}⟩ ⊗ 𝒪 → (V ⊕ k) ⊗ 𝒪(1) → 𝒪(2) → 0

which is exact elsewhere.

For the exterior algebra on six variables, in odd characteristic, we can produce a module M of dimension 31 and constant Jordan type [2]^{14}[1]^{3} with ℱ_{1}(M) isomorphic to the Horrocks parent bundle on ℙ^{5}, by using the matrices given in Section 9.14, see Figure 9.2 on page 200. This matrix shows the action of θ = aX_{1} + bX_{2} + cX_{3} + dX_{4} + eX_{5} + fX_{6}.

In characteristic two, the exterior algebra is isomorphic to the group algebra of an elementary abelian 2-group. So again using the matrices from Section 9.14, we obtain a module for E = (ℤ/2)^{6} of dimension 30 and stable constant Jordan type [2]^{14}[1]^{2} in characteristic two such that ℱ_{1}(M) is isomorphic to the Tango bundle on ℙ^{5}. See Figure 9.3 on page 201.

10 Restrictions Coming from Chern Numbers

The purpose of this chapter is to use the theory of Chern numbers and the Hirzebruch– Riemann–Roch theorem to obtain information about modules of constant Jordan type and the vector bundles associated with them.

The method works more generally for studying linear varieties of matrices of constant rank, and we begin with an easy example of how to use Chern numbers to study this situation.

10.1. Matrices of Constant Rank

In this section, we use the theory of Chern numbers to prove theorems of Sylvester and Westwick on matrices of constant rank. We then use this to obtain information about modules of constant Jordan type.

Recall from Section 7.1 that the Chow ring of ℙ^{r−1} is

A^{∗}(ℙ^{r−1}) ≅ ℤ[h]/(h^{r}).

If ℱ is a vector bundle on ℙ^{r−1}, we write

c(ℱ) = ∑_{j≥0} c_{j}(ℱ)h^{j} ∈ A^{∗}(ℙ^{r−1})

for the Chern polynomial, where c_{0}(ℱ) = 1 and the c_{i}(ℱ) ∈ ℤ (1 ≤ i ≤ r − 1) are the Chern numbers of ℱ. If 0 → ℱ → ℱ′ → ℱ′′ → 0 is a short exact sequence of vector bundles then we have c(ℱ′) = c(ℱ)c(ℱ′′).

Theorem 10.1.1 (Sylvester [231]). Let A_{1}, …, A_{r} span a linear variety of n×n matrices of constant rank n − 1 with n > 1. Then r ≤ 3, and if r = 3 then n is odd.

Proof. We use A_{1}, …, A_{r} to construct an operator on vector bundles on ℙ^{r−1},

θ: 𝒪^{⊕n} → 𝒪(1)^{⊕n}

as described in Section 8.2. Since A_{1}, …, A_{r} span a linear variety of constant rank n − 1, it follows using Theorem 6.2.1 that θ is a map of vector bundles whose kernel and cokernel are vector bundles of rank one. By Theorem 6.3.1 the kernel and cokernel are twists of the structure sheaf. So we obtain an exact sequence of vector bundles

0 → 𝒪(a) → 𝒪^{⊕n} ⟶^{θ} 𝒪(1)^{⊕n} → 𝒪(b) → 0

for some a, b ∈ ℤ. Applying Chern polynomials, we obtain

c(𝒪(a))c(𝒪(1)^{⊕n}) = c(𝒪^{⊕n})c(𝒪(b))

as elements of ℤ[h]/(h^{r}). Thus

(1 + ah)(1 + h)^{n} ≡ 1 + bh (mod h^{r}).

If r ≥ 2 then equating coefficients of h gives b = n + a. If r ≥ 3 then equating coefficients of h^{2} gives an + \binom{n}{2} = 0, so that a = −(n − 1)/2 and it follows that n is odd. If r ≥ 4 then equating coefficients of h^{3} gives a\binom{n}{2} + \binom{n}{3} = 0. It follows that n = 1, contradicting the hypotheses. □

To see that the theorem is sharp, we exhibit the following examples. For r = 2 we have the matrices

Y_{1}A_{1} + Y_{2}A_{2} = (\begin{matrix}Y_{2} & & & & & 0 \\ Y_{1} & Y_{2} & & & & 0 \\ & Y_{1} & ... & & & ⋮ \\ & & ... & Y_{2} & & ⋮ \\ & & & Y_{1} & Y_{2} & 0 \\ & & & & Y_{1} & 0 \end{matrix})

while for r = 3 and n odd we have the matrices

Y_{1}A_{1} + Y_{2}A_{2} + Y_{3}A_{3} = ( \begin{matrix}& −Y_{3} & & & & & −Y_{1} \\ Y_{3} & & & & & & −Y_{2} & −Y_{1} \\ & & & & & & −Y_{3} & −Y_{2} & −Y_{1} \\ & & & & & & & ... & ... & ... \\ & & & & & & & & −Y_{3} & −Y_{2} & −Y_{1} \\ & & & & & & & & & −Y_{3} & −Y_{2} \\ Y_{1} & Y_{2} & Y_{3} \\ & Y_{1} & Y_{2} & Y_{3} \\ & & ... & ... & ... \\ & & & Y_{1} & Y_{2} & Y_{3} \\ & & & & Y_{1} & Y_{2} \end{matrix} )

(Westwick [245, Theorem 4.1]).

Using the same method together with a determinant argument, we have the following generalisation of Theorem 10.1.1.

Theorem 10.1.2 (Westwick [246]). Let A_{1}, …, A_{r} span a linear variety of n×m matrices of constant rank s > 0. Then r ≤ m + n − 2s + 1.

Proof. By transposing if necessary, we may assume that m ≤ n. Again we use A_{1}, …, A_{r} to construct an operator on vector bundles on ℙ^{r−1}

θ: 𝒪^{⊕m} → 𝒪(1)^{⊕n}.

This is a map of vector bundles whose kernel and cokernel are vector bundles of rank m − s and n − s respectively. So we obtain an exact sequence of vector bundles

0 → ℱ → 𝒪^{⊕m} → 𝒪(1)^{⊕n} → ℱ′ → 0.

We obtain

c(ℱ)(1 + h)^{n} = c(ℱ′)

as elements of ℤ[h]/(h^{r}), with deg c(ℱ) ≤ m − s and deg c(ℱ′) ≤ n − s. If r ≥ m + n − 2s + 2 then comparing coefficients of h^{n−s+1}, …, h^{(m−s)+(n−s)+1} gives us a system of m − s + 1 equations of the form

\binom{n}{n − s + 1} +  \binom{n}{n − s} c_{1}(ℱ) + ⋯ + \binom{n}{n − m + 1}c_{m−s}(ℱ) = 0
\binom{n}{n − s + 2} + \binom{n}{n − s + 1}c_{1}(ℱ) + ⋯ + \binom{n}{n − m + 2}c_{m−s}(ℱ) = 0
…  …  …  …
\binom{n}{n + m − 2s + 1} + \binom{n}{n + m − 2s}c_{1}(ℱ) + ⋯ + \binom{n}{n − s + 1}c_{m−s}(ℱ) = 0.

Now the determinant of the coefficients in this set of equations is equal to

±\begin{matrix}\binom{n}{n − s + 1} & \binom{n + 1}{n − s + 1} & … & \binom{n + m − s}{n − s + 1} \\ \binom{n − s + 1}{n − s + 1} & \binom{n − s + 2}{n − s + 1} & … & \binom{n + m − 2s + 1}{n − s + 1} \end{matrix}

(see Muir and Metzler [195], Chapter XX, Section 733, page 682), and this is clearly non-zero. Therefore the system of equations is inconsistent. Since we assumed that r ≥ m + n − 2s + 2 in order to obtain a square matrix of coefficients, this proves that r ≤ m + n − 2s + 1. □

Example 10.1.3. Example 5.1.4 shows that there are 10 × 10 matrices of constant rank 8 with r = 4, but it is not known whether there are examples of spaces of dimension five consisting of n × n matrices of constant rank n − 2 (Westwick [248]).

Theorem 10.1.4. Let p > 2 and let M be a k(ℤ/p)^{r}-module of constant Jordan type [2]^{s}[1]^{t}. If s > 0 then t ≥ r − 1.

Proof. By Proposition 5.1.17, we have Rad^{2}(M) = 0. Let A_{1}, …, A_{r} be the matrices for the actions of

X_{1}, …, X_{r}: M/Rad(M) → Rad(M).

This is a linear variety of n×m matrices of constant rank s, where m+n = dim_{k}(M) = 2s+t. Applying Theorem 10.1.2, we obtain r ≤ m + n − 2s + 1 = t + 1. □

10.2. Congruences on Chern Numbers

In this section we obtain congruences for Chern numbers that will be applied to give information about modules of constant Jordan type.

Lemma 10.2.1. For a vector bundle ℱ of rank s on ℙ^{r−1} we have

c(ℱ)c(ℱ(1)) ⋯ c(ℱ(p − 1)) ≡ 1 − sh^{p−1} (mod (p, h^{p})).

as an element of ℤ[h]/(h^{r}).

Proof. We give two proofs, the first in terms of the Chern roots, and the second directly from the definition of Chern classes (see Definitions 7.1.6 and 7.2.6).

We write

c(ℱ) = ∏_{j=1}^{s}(1 + α_{j}h),

where the α_{j} are the Chern roots. Then by formula (7.6.1) we have

c(ℱ(i)) = ∏_{j=1}^{s}(1 + (α_{j} + i)h).

Thus we have

c(ℱ)c(ℱ(1)) ⋯ c(ℱ(p − 1)) = ∏_{j=1}^{s}(1 + α_{j}h)(1 + (α_{j} + 1)h) ⋯ (1 + (α_{j} + p − 1)h).

Now by Fermat’s little theorem, we have the identity

(10.2.1)  x(x + y) ⋯ (x + (p − 1)y) ≡ x^{p} − xy^{p−1} (mod p)

and so putting x = 1 + α_{j}h, y = h we obtain

c(ℱ)c(ℱ(1)) ⋯ c(ℱ(p − 1)) ≡ ∏_{j=1}^{s}((1 + α_{j}h)^{p} − (1 + α_{j}h)h^{p−1}) (mod p)

≡ ∏_{j=1}^{s}(1 − h^{p−1} + (α_{j}^{p} − α_{j})h^{p}) (mod p)

≡ 1 − sh^{p−1} (mod (p, h^{p})).

A priori, this is a congruence between polynomials with algebraic integer coefficients. But if two rational integers are congruent modulo p as algebraic integers then they are also congruent modulo p as rational integers. This is because their difference, divided by p, is both an algebraic integer and a rational number, therefore an integer. □

The second proof gives slightly more information, so formulate it as a separate statement. The first proof can probably be made to give the same information, but it is less clear.

Lemma 10.2.2. For a vector bundle ℱ of rank s on ℙ^{r−1} we have

c(ℱ)c(ℱ(1)) ⋯ c(ℱ(p − 1)) ≡ 1 − sh^{p−1} (mod (p, h^{2p−2})).

as an element of ℤ[h]/(h^{r}).

Proof. The argument is similar to the proof of Lemma 10.2.1, but avoids algebraic integers. Recall that if ℱ = M^{∼} where M is a finitely generated R-module with Poincaré series

p_{M}(t) = \frac{∑_{j} a_{j}t^{j}}{(1 − t)^{r}}

then

c(ℱ) = ∏_{j}(1 + jh)^{a_{j}} ∈ ℤ[h]/(h^{r}).

So again using (10.2.1) we have

c(ℱ)c(ℱ(1)) … c(ℱ(p − 1)) = ∏_{j}((1 + jh)(1 + (j + 1)h) … (1 + (j + p − 1)h))^{a_{j}}

≡ ∏_{j}((1 + jh)^{p} − (1 + jh)h^{p−1})^{a_{j}} (mod p)

≡ ∏_{j}(1 − h^{p−1} + (j^{p} − j)h^{p})^{a_{j}} (mod p)

≡ ∏_{j}(1 − h^{p−1})^{a_{j}} (mod p)

≡ 1 − ∑_{j} a_{j}h^{p−1} (mod (p, h^{2p−2})).

Recalling from Definition 7.1.2 that

s = rank (ℱ) = ∑_{j} a_{j},

this proves the lemma. □

10.3. Restrictions on Stable Jordan Type, p Odd

The main theorem of this section, Theorem 10.3.1 complements the Realisation Theorem 8.1.1.

Theorem 10.3.1 (Benson and Pevtsova [53]). Suppose that M has stable constant Jordan type [1]^{s}. Then p divides the Chern numbers c_{m}(ℱ_{1}(M)) for 1 ≤ m ≤ p − 2.

Proof. The trivial vector bundle M̃ has a filtration with filtered quotients (not in order) ℱ_{1}(M), ℱ_{p}(M), ℱ_{p}(M)(1), …, ℱ_{p}(M)(p − 1). So we have

1 = c(M̃)
= c(ℱ_{1}(M))c(ℱ_{p}(M))c(ℱ_{p}(M)(1)) ⋯ c(ℱ_{p}(M)(p − 1))
≡ c(ℱ_{1}(M)) (mod (p, h^{p−1}))

by Lemma 10.2.1. It follows that the coefficients c_{m}(ℱ_{1}(M)) are divisible by p for 1 ≤ m ≤ p − 2. □

Remark 10.3.2. For p = 2 this theorem says nothing. But for p odd, it at least forces c_{1}(ℱ_{1}(M)) to be divisible by p. As an explicit example, the twists of the Horrocks–Mumford bundle ℱ_{HM}(i) have c_{1} = 2i + 5 and c_{2} = i^{2} + 5i + 10. For p ≥ 7 these cannot both be divisible by p, and so there is no module M of stable constant Jordan type [1]^{2} and integer i such that ℱ_{1}(M) ≅ ℱ_{HM}(i).

Remark 10.3.3. In Section 9.7, we constructed for p odd and r even a kE-module Y of stable constant Jordan type [1]^{r−2} with ℱ_{1}(Y) ≅ F^{∗}(ℱ_{N}), the inverse image of the null correlation bundle along the Frobenius map on ℙ^{r−1}.

The Chern polynomial of the null correlation bundle was computed in Proposition 7.3.6 to be

(1 − h)^{−1}(1 + h)^{−1} = 1 + h^{2} + h^{4} + ⋯ + h^{r−2} ∈ ℤ[h]/(h^{r}).

Since the coefficient of h^{2} is not divisibe by p, it follows from Theorem 10.3.1 that the null correlation bundle is not of the form ℱ_{1}(M) for any kE-module M of stable constant Jordan type [1]^{r−2} as long as p ≥ 5.

The same is true for p = 3 but needs slightly more argument, along the lines of the method for Theorem 5.12.1. If M is such a module then Λ^{r−1}M is projective by Dade’s lemma, so the dimension of M is congruent to r − 2 modulo p^{r}. Since r > 1, it follows that ℱ_{p}(M) is a bundle of rank divisible by p. So in the application of Lemma 10.2.1 in the proof of Theorem 10.3.1, the constant s is divisible by p and so the proof shows that c_{m}(ℱ_{1}(M)) is divisible by p for 1 ≤ m ≤ p − 1. If p = 3 then p − 1 = 2 and looking at the coefficient of h^{2} again shows that the null correlation bundle has to be pulled back through the Frobenius map before it can be realised in this way.

10.4. Eliminating More Stable Jordan Types

In this section, we come back to the question of what stable types can occur for modules of constant Jordan type. The new tool is Lemma 10.2.1, and the main theorem of this section is as follows.

Theorem 10.4.1 (Benson [43]). If a module for (ℤ/p)^{r} has stable constant Jordan type [a_{1}] … [a_{t}] with

∑_{i=1}^{t} a_{i} ≤ min(r − 1, p − 2)

then a_{1} = ⋯ = a_{t} = 1.

Proof. Let M be a module with stable constant Jordan type [a_{1}] … [a_{t}]. Since M̃ is a direct sum of copies of 𝒪, we have c(M̃) = 1. Furthermore, by Lemmas 8.4.6 and 8.4.8, M̃ has a filtration in which the filtered quotients are isomorphic to ℱ_{i}(M)(j) for 0 ≤ j < i ≤ p. Let a = ∑_{i=1}^{t} a_{i}. By hypothesis, a ≤ p − 2, so using Lemma 10.2.1 we have

∏_{j=1}^{p−2} c(ℱ_{j}(M))c(ℱ_{j}(M)(1)) … c(ℱ_{j}(M)(j − 1)) ≡ 1 (mod (p, h^{p−1})).

The left-hand side is a polynomial of degree a ≤ p − 2. Furthermore, we have a ≤ r − 1 and h^{r−1} ≠ 0. So this congruence may be read as an equation in the polynomial ring 𝔽_{p}[h]. The units in this ring are the constant polynomials, so all factors in the product are equal to one. If j ≥ 2, both c(ℱ_{j}(M)) and c(ℱ_{j}(M)(1)) then have to be congruent to 1 modulo p. Since

c_{1}(ℱ_{j}(M)(1)) = c_{1}(ℱ_{j}(M)) + rank (ℱ_{j}(M))

(see Lemma 7.5.1), this implies that the rank of ℱ_{j}(M) is divisible by p. Since a < p, this implies that ℱ_{j}(M) = 0 for j ≥ 2, so each a_{i} = 1. □

This proves a weak version of Conjecture 5.15.2 provided p is large:

Corollary 10.4.2. If M is a k(ℤ/p)^{r}-module of stable constant Jordan type [2] [1]^{j} with p > r then j ≥ r − 2. □

We can get slightly more information in the case where j = r − 2 by examining the coefficients more carefully. We illustrate this in the case r = 3, j = 1.

Theorem 10.4.3. If M is a module of stable constant Jordan type [2] [1] for (ℤ/p)^{3} with p ≥ 5 then p ≡ 1 (mod 3).

Proof. Using Lemma 7.5.1 as before, we have

(1 + αh)(1 + (α + 1)h)(1 + βh) ≡ 1 (mod (p, h^{3}))

where α = c_{1}(ℱ_{2}(M)) and β = c_{1}(ℱ_{1}(M)). Comparing coefficients of h and h^{2}, we obtain

2α + β + 1 ≡ 0
α(α + 1) + (2α + 1)β ≡ 0 (mod p).

Substituting β = −(2α + 1) into the second equation gives

3α^{2} + 3α + 1 ≡ 0 (mod p).

This has a solution if and only if −3 is a square modulo p. Since p ≥ 5, this is equivalent to the condition that p ≡ 1 (mod 3) by quadratic reciprocity. □

If p ≡ 1 (mod 3) then the Chern class information for a module of stable constant Jordan type [2] [1] for (ℤ/p)^{3} is consistent. For example, if p = 7 then the two solutions of the equations in the proof of the theorem are α ≡ 1, β ≡ 4 and α ≡ 5, β ≡ 3 (mod 7).

θ ↓ 12  4  θ ↓ 56  3

These two possibilities correspond to the equations

(1 + h)(1 + 2h)(1 + 4h) ≡ 1 + h^{3} (mod 7)
(1 + 3h)(1 + 5h)(1 + 6h) ≡ 1 − h^{3} (mod 7).

It is not clear whether such a module exists if p = 7, or more generally if p ≡ 1 (mod 3), but if it does then its restriction to (ℤ/p)^{2} is not isomorphic to any of the modules of type [2] [1] that we know how to construct (see Section 9.5 for some examples), because these examples all have either α ≡ −1, β ≡ 1 or α ≡ 0, β ≡ −1 (mod p), and in particular 3α^{2} + 3α + 1 ≢ 0 (mod p).

The general theorem obtained by the method of Theorem 10.4.3 is as follows.

Theorem 10.4.4. If M is a kE-module of stable constant Jordan type [2] [1]^{r−2} for (ℤ/p)^{r} with p > r and r odd then p ≡ 1 modulo some prime factor of r.

Proof. Using Lemma 7.5.1 as before, we have

(1 + αh)(1 + (α + 1)h)c(ℱ_{1}(M)) ≡ 1 (mod p, h^{r})

where α = c_{1}(ℱ_{2}(M)). So the left-hand side is congruent to 1−ah^{r} (mod p) for some integer a. By the factor theorem, both α and α + 1 are rth roots of a modulo p, and so p does not divide α, α + 1 or a. Furthermore, writing α^{−1} for an inverse of α modulo p, it follows that 1 + α^{−1} is an rth root of unity modulo p. Since it is not equal to one, it follows that for some prime factor q of r, there is a primitive qth root of unity modulo p and so p ≡ 1 (mod q). □

10.5. Restrictions on Jordan Type for p = 2

For p = 2 it does not make sense to try to restrict the stable Jordan type, but there are interesting restrictions on Jordan type. Some of these come from Dade’s lemma, but we shall deduce them instead from the theory of Chern numbers. As soon as the rank is at least four, we shall have to make use of the Hirzebruch–Riemann–Roch theorem to get some of the information. We obtain complete information up to rank four, but it already becomes apparent that the pattern for higher rank will be complicated.

Throughout this section, M will be a module of constant Jordan type [2]^{n}[1]^{m} for E ≅ (ℤ/2)^{r}. We shall write c_{i} for the Chern numbers of ℱ_{2}(M) and d_{i} for the Chern numbers of ℱ_{1}(M).

We begin with some lemmas giving information for m large.

Lemma 10.5.1. If m ≥ r−1 then for each n ≥ 0 there exists a module of constant Jordan type [2]^{n}[1]^{m} for (ℤ/2)^{r}.

Proof. Example 5.6.16 gives modules of constant Jordan type [2]^{n}[1]^{r−1} for every value of n. Direct sums of these with copies of the trivial module of constant Jordan type [1] show that as long as m ≥ r − 1, there are no restrictions on the Jordan type. □

The following should be compared with Theorem 10.1.4.

Lemma 10.5.2. If r is even and m = r − 2 then for each even value of n ≥ 0 there exists a module of constant Jordan type [2]^{n}[1]^{r−2} for (ℤ/2)^{r}.

Proof. This follows from the construction in Section 9.6 of modules M_{k} such that ℱ_{1}(M_{k}) is an instanton bundle of charge k on ℙ^{r−1}. The modules M_{k} constructed there have type [2]^{2k}[1]^{r−2}. □

For r odd we get much weaker information.

Lemma 10.5.3. If r is odd and m = r − 2 then there are modules of constant Jordan types [2]^{r}[1]^{r−2} and [2]^{2^{r−1}−2r+2}[1]^{r−2} for (ℤ/2)^{r}.

Proof. This follows from the construction in Section 9.10 of modules M of constant Jordan type [2]^{r}[1]^{r−2} associated with Tango’s rank r − 2 examples of simple vector bundles on ℙ^{r} for any r. These modules cannot compete with the instanton examples used to prove Lemma 10.5.2 when r is even, but they’re all we’ve got for r odd. The modules Ω(M) have constant Jordan type [2]^{2^{r−1}−2r+2}[1]^{r−2}. □

The method we shall adopt in order to get restrictions on Jordan type is as follows. Recall from Theorem 8.4.9 that the trivial sheaf M̃ = M ⊗_{k} 𝒪 has a filtration in which the quotients are ℱ_{1}(M), ℱ_{2}(M) and ℱ_{2}(M)(1). Using Lemma 7.5.1 for the Chern numbers of the twisted bundle ℱ_{2}(M)(1), we obtain the following equation:

(1 + d_{1}h + d_{2}h^{2} + d_{3}h^{3} + ⋯)(1 + c_{1}h + c_{2}h^{2} + c_{3}h^{3} + ⋯)

(1 + (c_{1} + n)h + (c_{2} + (n − 1)c_{1} + \binom{n}{2})h^{2} + (c_{3} + (n − 2)c_{2} + \binom{n−1}{2}c_{1} + \binom{n}{3})h^{3} + ⋯) = 1.

This may be used inductively to express the d_{i} in terms of the c_{i}. The result is the following:

d_{1} = −2c_{1} − n
d_{2} = −2c_{2} + 3c_{1}^{2} + (2n + 1)c_{1} + \binom{n+1}{2}
d_{3} = −2c_{3} + 6c_{1}c_{2} − 4c_{1}^{3} + (n + 1)(2c_{2} − 3c_{1}^{2}) − (n + 1)^{2}c_{1} − \binom{n+2}{3}
d_{4} = −2c_{4} + 6c_{1}c_{3} + 3c_{2}^{2} − 12c_{1}^{2}c_{2} + 5c_{1}^{4} + (2n + 3)(c_{3} − 3c_{1}c_{2} + 2c_{1}^{3})
+ \frac{3n^{2}+9n+8}{2}c_{1}^{2} − (n^{2} + 3n + 3)c_{2} + \frac{(n+1)(n+2)(2n+3)}{6}c_{1} + \binom{n+3}{4}
⋯

From these formulas, information then comes from Remark 7.3.8. Namely, for m < j < r we get an equation between the c_{i} from the fact that d_{j} = 0.

Rank Two. For r = 2, there is only one restriction on Jordan type. Namely, if m = 0 then n is even. This restriction comes from Dade’s lemma, but to deduce it using Chern numbers, we argue as follows. We have d_{1} = 0, so

c_{1}(ℱ_{2}(M)) + c_{1}(ℱ_{2}(M)(1)) = 0.

Since c_{1}(ℱ_{2}(M)(1)) = c_{1}(ℱ_{2}(M)) + n, this gives

n = −2c_{1}.

It follows that n is even. Since kE has constant Jordan type [2]^{2}, Lemma 10.5.1 shows that this is the only restriction.

Rank Three. For r = 3 and m = 0, again Dade’s lemma shows that n is divisible by 4. To show this using Chern classes, we note that again n = −2c_{1}(ℱ_{2}(M)). The equation

(1 − \frac{n}{2}h + c_{2}h^{2})(1 + \frac{n}{2}h + (c_{2} + (n − 1)(−\frac{n}{2}) + \binom{n}{2})h^{2}) = 1

shows that c_{2} = \frac{n^{2}}{8} , and hence n is divisible by 4.

For r = 3 and m = 1, Theorem 5.4.2 shows that n is congruent to −1 or 0 modulo four. To show this using Chern classes, we have

(1 + c_{1}h + c_{2}h^{2})(1 + (c_{1} + n)h + (c_{2} + (n − 1)c_{1} + \binom{n}{2})h^{2})(1 + d_{1}h) = 1

in ℤ/(h^{3}). Comparing coefficients of h gives

d_{1} = −2c_{1} − n,

and then comparing coefficients of h^{2} gives

c_{1}(c_{1} + n) + 2c_{2} + (n − 1)c_{1} + \binom{n}{2} + (−2c_{1} − n)(2c_{1} + n) = 0.

Reading this equation modulo two, we get

\binom{n}{2} + n^{2} ≡ 0 (mod 2).

Thus n is congruent to −1 or 0 modulo four.

To see that these are the only restrictions for r = 3, we note that the modules kE, k and Ω(k) have constant Jordan type [2]^{4}, [1] and [2]^{3}[1] respectively, and then use Lemma 10.5.1.

Here is a diagram of these restrictions for rank three. Dots indicate possible Jordan types, and the pattern repeats to the right with period 4.

n m0  4  8 • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •

Rank Four. For r = 4 and m = 0, we shall see that the Dade’s lemma restriction that n is divisible by 8 can be proved using Chern classes, with the use of the Hirzebruch–Riemann– Roch theorem.

The method of the previous two cases shows that the Chern polynomial of ℱ_{2}(M) has the form

1 − \frac{n}{2} h + \frac{n^{2}}{8} h^{2} − \frac{n^{3} − 4n}{48} h^{3}.

Without resorting to the Hirzebruch–Riemann–Roch theorem, all we get from this is the restriction that n is divisible by 4. But it follows from Hirzebruch–Riemann–Roch that for any vector bundle on ℙ^{3}, we have

c_{1}c_{2} + c_{3} ≡ 0 (mod 2),

see Example 7.7.11. Thus

\frac{n^{3} − 4n}{48} + (−\frac{n}{2})(\frac{n^{2}}{8} ) ≡ 0 (mod 2),

which forces n to be divisible by 8.

For r = 4 and m = 1, Theorem 5.4.2 shows that n is congruent to 0 or −1 modulo 8. This information may also be obtained from Hirzebruch–Riemann–Roch, but we defer the proof of this until Section 10.8.

For r = 4 and m = 2, we find that d_{1} ≡ n (mod 2) and then d_{2} ≡ \binom{n}{2}+n^{2} (mod 2). The Hirzebruch–Riemann–Roch Theorem implies that d_{1}d_{2} is even, which in turn implies that n is not congruent to one modulo four.

Continuing with r = 4 and m = 2, if n ≡ 3 (mod 8), then working mod 8 and h^{4} we have

(1+c_{1}h+c_{2}h^{2} +c_{3}h^{3})(1+(c_{1} +3)h+(c_{2} +2c_{1} +3)h^{2} +(c_{3} +c_{2} +c_{1} +1)h^{3})(1+d_{1}h+d_{2}h^{2}) ≡ 1.

Equating coefficients of h and h^{2} gives

d_{1} ≡ −2c_{1} − 3,
d_{2} ≡ −2c_{2} + 3c_{1}^{2} − c_{1} − 2.

Equating coefficients of h^{3} then gives

2c_{3} + 2c_{1}c_{2} + 4c_{1}^{3} + 4c_{1}^{2} + 2 ≡ 0.

By Hirzebruch–Riemann–Roch, c_{3} + c_{1}c_{2} is even and so we obtain a contradiction. Thus n cannot be 3 modulo 8.

So the limitations we have found for rank four are:

(i) If m = 0 then n is divisible by 8.

(ii) If m = 1 then n is congruent to 0 or −1 modulo 8.

(iii) If m = 2 then n is not congruent to 1, 3 or 5 modulo 8.

To see that these form a complete set of conditions, kE has type [2]^{8}, k has type [1], and Rad(kE) has type [2]^{7}[1]. Sums of these together with Lemmas 10.5.1 and 10.5.2 show that all possibilities for n allowed by the above conditions do actually occur.

Here is a diagram of the restrictions for rank four. Dots indicate possible Jordan types, and the pattern repeats to the right with period 8.

n m0  8 • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •

We shall go some way towards explaining this pattern in Sections 10.6–10.8.

10.6. Applying Hirzebruch–Riemann–Roch for p = 2: The Case m = 0

We continue with the notation of the previous section. At this stage, it becomes apparent that there should be a better way to organise the information, especially the congruences coming from the Hirzebruch–Riemann–Roch Theorem 7.8.7. Here is how to recover the information coming from Dade’s lemma this way, for p = 2.

Lemma 10.6.1. If m = 0 then the Chern character of ℱ_{2}(M) is given by

Ch(ℱ_{2}(M)) = 2n/(1 + e^{h}).

Proof. We have a short exact sequence

0 → ℱ_{2}(M) → M̃ → ℱ_{2}(M)(1) → 0

and M has dimension 2n, so we have

Ch(ℱ_{2}(M)) + Ch(ℱ_{2}(M)(1)) = 2n.

By Lemma 7.5.1 we have

Ch(ℱ_{2}(M)(1)) = e^{h}Ch(ℱ_{2}(M)).

Hence

(1 + e^{h})Ch(ℱ_{2}(M)) = 2n.  □

Lemma 10.6.2. The coefficient of h^{r−1} in the power series expansion of

(\frac{h}{1 − e^{−h}} )^{r} \frac{1}{1 + e^{h}}

is equal to \frac{1}{2^{r}}.

Proof. By Cauchy’s integral formula, the coefficient is

\frac{1}{2πi} ∮ \frac{dh}{(1 − e^{−h})^{r}(1 + e^{h})}

where the contour of integration is a small anticlockwise circle around the origin. Substituting z = 1 − e^{−h} (see Remark 7.8.8), this is equal to

\frac{1}{2πi} ∮ \frac{dz}{z^{r}(2 − z)} = \frac{1}{2πi} ∮ (\frac{2^{r−1} + 2^{r−2}z + ⋯ + z^{r−1}}{z^{r}} + \frac{1}{2 − z} ) \frac{dz}{2^{r}} = \frac{1}{2^{r}} .  □

Theorem 10.6.3. If m = 0 then n is divisible by 2^{r−1}.

Proof. Applying the Hirzebruch–Riemann–Roch Theorem 7.8.7 to ℱ_{2}(M) and using Lemma 10.6.1 we find that χ(ℱ_{2}(M)) is the coefficient of h^{r−1} in the power series

(\frac{h}{1 − e^{−h}} )^{r} \frac{2n}{1 + e^{h}} .

By Lemma 10.6.2 this is equal to

\frac{2n}{2^{r}}.

Since χ(ℱ_{2}(M)) is an integer this shows that n is divisible by 2^{r−1}. □

Remark 10.6.4. In contrast with Dade’s lemma, this method works for nilvarieties of constant Jordan type just as well as for modules of constant Jordan type. So we obtain the following theorem, which holds in any characteristic.

Theorem 10.6.5. Let k be an algebraically closed field, and let M be a nilvariety of rank r over k of constant Jordan type [2]^{n}. Then n is divisible by 2^{r−1}.

Remark 10.6.6. This theorem should not be so surprising, because if A_{1}, …, A_{r} span a nilvariety of rank r consisting of matrices that square to zero then for 1 ≤ i < j ≤ r we have (A_{i} + A_{j})^{2} = 0, so that

A_{j}A_{i} = −A_{i}A_{j}.

So the matrices generate a representation of an exterior algebra. Dade’s lemma holds in any characteristic for exterior algebras, see for example Theorem 3.1 of Aramova, Avramov and Herzog [12].

Remark 10.6.7. In contrast with this theorem, Example 4.1.6 shows that for p odd, if a nilvariety of rank r has constant Jordan type [p]^{n}, it does not follow that n is divisible by p^{r−1}. We investigate this further in Section 10.9.

10.7. Bypassing Hirzebruch–Riemann–Roch

We have seen that the integrality conditions on Chern numbers coming from the Hirzebruch–Riemann–Roch theorem are consequences of the integrality of the coefficients in the Poincaré seires of a corresponding graded module. We ought therefore to be able to obtain the results of the last section directly using Poincaré series.

Second proof of Theorem 10.6.5. Let M be a kE-module of constant Jordan type [2]^{n}, let R = k[Y_{1}, …, Y_{r}], and let N be a finitely generated graded R-module such that N^{∼} ≅ ℱ_{2}(M). Let

p_{N}(t) = \frac{∑_{i} a_{i}t^{i}}{(1 − t)^{r}}

with ∑_{i} a_{i}t^{i} ∈ ℤ[t, t^{−1}]. Then since M̃ has a filtration with filtered quotients ℱ_{2}(M) and ℱ_{2}(M)(1), we have

(1 + t)p_{N}(t) = \frac{2n}{(1 − t)^{r}} + q(t)

with q(t) ∈ ℤ[t, t^{−1}]. Substitute t = −1 to get

0 = \frac{2n}{2^{r}} + q(−1).

Since q(−1) ∈ ℤ, it follows that n is divisible by 2^{r−1}. □

10.8. Applying and Bypassing Hirzebruch–Riemann–Roch for p = 2: The Case 1 ≤ m ≤ r − 3

Recall from Section 6.3 that if ℱ is a vector bundle of rank m on ℙ^{r−1} with 1 ≤ m ≤ r −3 then there are three possibilities:

(i) ℱ is a direct sum of line bundles,

(ii) r ≤ 6 and ℱ has a summand isomorphic to one of the examples mentioned in Section 6.3, or

(iii) ℱ has a summand which is a new indecomposable vector bundle of low rank on projective space.

In this section we investigate case (i). Let k be an algebraically closed field of characteristic two and E ≅ (ℤ/2)^{r}. Let M be a kE-module of constant Jordan type [2]^{n}[1]^{m}.

Lemma 10.8.1. If ℱ_{1}(M) is a direct sum of line bundles 𝒪(a_{1}) ⊕ ⋯ ⊕ 𝒪(a_{m}) then the Chern character of ℱ_{2}(M) is given by

Ch(ℱ_{2}(M)) = \frac{2n + m − e^{a_{1}h} − ⋯ − e^{a_{m}h}}{1 + e^{h}} .

Proof. The vector bundle M̃ has a filtration with filtered quotients ℱ_{2}(M), ℱ_{2}(M)(1) and ℱ_{1}(M). So

(1 + e^{h})Ch(ℱ_{2}(M)) + e^{a_{1}h} + ⋯ + e^{a_{m}h} = 2n + m.  □

Lemma 10.8.2. The coefficient of h^{r−1} in the power series expansion of

(\frac{h}{1 − e^{−h}} )^{r} \frac{2n + m − e^{a_{1}h} − ⋯ − e^{a_{m}h}}{1 + e^{h}}

differs from

\frac{2n + m − (−1)^{a_{1}} − ⋯ − (−1)^{a_{m}}}{2^{r}}

by an integer.

Proof. As usual, by Cauchy’s integral formula, the coefficient is

\frac{1}{2πi} ∮ \frac{(2n + m − e^{a_{1}h} − ⋯ − e^{a_{m}h})dh}{(1 − e^{−h})^{r}(1 + e^{h})}.

Substituting z = 1 − e^{−h} (see Remark 7.8.8), this is equal to

\frac{1}{2πi} ∮ \frac{2n + m − ∑_{i=1}^{m}(1 − z)^{−a_{i}}}{z^{r}(2 − z)} dz.

This is the coefficient of z^{r−1} in the power series

\frac{2n + m − ∑_{i=1}^{m}(1 − z)^{−a_{i}}}{2 − z}

= (\frac{1}{2} + \frac{1}{2^{2}} z + \frac{1}{2^{3}} z^{2} + ⋯)(2n + m − ∑_{i=1}^{m}(1 + a_{i}z + \binom{a_{i}+1}{2}z^{2} + \binom{a_{i}+2}{3}z^{3} + ⋯)).

So this coefficient is equal to

(10.8.1)  \frac{2n + m − ∑_{i=1}^{m}(1 + 2a_{i} + 2^{2}\binom{a_{i}+1}{2} + 2^{3}\binom{a_{i}+2}{3} + ⋯ + 2^{r−1}\binom{a_{i}+r−2}{r−1})}{2^{r}}

Now the power series expansion

(10.8.2)  (1 − 2)^{−a_{i}} = 1 + 2a_{i} + 2^{2}\binom{a_{i}+1}{2} + ⋯

converges 2-adically, and so

1 + 2a_{i} + 2^{2}\binom{a_{i} + 1}{2} + 2^{3}\binom{a_{i} + 2}{3} + ⋯ + 2^{r−1}\binom{a_{i} + r − 2}{r − 1} ≡ (−1)^{a_{i}} (mod 2^{r}).

It follows that the expression (10.8.1) differs by an integer from

\frac{2n + m − ∑_{i=1}^{m}(−1)^{a_{i}}}{2^{r}}

as required. □

Theorem 10.8.3. If ℱ_{1}(M) ≅ 𝒪(a_{1}) ⊕ ⋯ ⊕ 𝒪(a_{m}) then n is congruent to one of the integers 0, −1, …, −m modulo 2^{r−1}. More precisely, it is congruent to minus the number of a_{i} that are odd.

As in the case of Theorem 10.6.5, we provide two proofs, the first using the Hirzebruch– Riemann–Roch theorem and the second avoiding it. The relationship between these two proofs is quite instructive, but the second one is clearly easier.

First proof. Applying the Hirzebruch–Riemann–Roch Theorem 7.8.7 and Lemma 10.8.1, we find that χ(ℱ_{2}(M)) is the coefficient of h^{r−1} in the power series

(\frac{h}{1 − e^{−h}} )^{r} (\frac{2n + m − e^{a_{1}h} − ⋯ − e^{a_{m}h}}{1 + e^{h}} ) .

By Lemma 10.8.2 this is differs from

\frac{2n + m − ∑_{i=1}^{m}(−1)^{a_{i}}}{2^{r}}

by an integer. Thus 2n + m − ∑_{i=1}^{m}(−1)^{a_{i}} is divisible by 2^{r}. It follows that n is congruent modulo 2^{r−1} to minus the number of a_{i} that are odd. □

Second proof. Let R = k[Y_{1}, …, Y_{r}], and let N_{1}, N_{2} be finitely generated graded Rmodules with N_{1}^{∼} ≅ ℱ_{1}(M) and N_{2}^{∼} ≅ ℱ_{2}(M). Since ℱ_{1}(M) ≅ 𝒪(a_{1}) ⊕ ⋯ ⊕ 𝒪(a_{m}), we may take for N_{1} a sum of shifts of R by degrees a_{1}, …, a_{m}. The Poincaré series of N_{1} and N_{2} are then given by

p_{N_{1}}(t) = \frac{t^{a_{1}} + ⋯ + t^{a_{m}}}{(1 − t)^{r}} , p_{N_{2}}(t) = \frac{f(t)}{(1 − t)^{r}}

with f(t) ∈ ℤ[t, t^{−1}]. Since M̃ is a trivial bundle of rank 2n + m having a filtration with filtered quotients ℱ_{2}(M), ℱ_{2}(M)(1) and ℱ_{1}(M), we have

(1 + t)p_{N_{2}}(t) + \frac{t^{a_{1}} + ⋯ + t^{a_{m}}}{(1 − t)^{r}} = \frac{2n + m}{(1 − t)^{r}} + q(t)

where q(t) ∈ ℤ[t, t^{−1}]. Substitute t = −1 to get

\frac{(−1)^{a_{1}} + ⋯ + (−1)^{a_{m}}}{2^{r}} = \frac{2n + m}{2^{r}} + q(−1).

Since q(−1) ∈ ℤ, it follows that 2n+m−∑_{i=1}^{m}(−1)^{a_{i}} is divisible by 2^{r}, and so n is congruent modulo 2^{r−1} to minus the number of a_{i} that are odd. □

Note that all the possibilities allowed by Theorem 10.8.3 actually occur. For example we can take

M = Ω^{−a_{1}}(k) ⊕ ⋯ ⊕ Ω^{−a_{m}}(k).

If m = 1 then we are in the case of Theorem 5.4.2, and that theorem shows that n is congruent to 0 or −1 modulo 2^{r−1}. In this case ℱ_{1}(M) has to be a line bundle, and so Theorem 10.8.3 gives the same congruence for n.

If 2 ≤ m ≤ r − 3, then vector bundles that are not sums of line bundles are hard to come by. In particular, no examples are known with r > 6 (cf. Section 6.3). So we have the following theorem.

Theorem 10.8.4. Let k be an algebraically closed field of characteristic two and E ≅ (ℤ/2)^{r}. If M is a module of constant Jordan type [2]^{n}[1]^{m} and n is not congruent to one of the integers 0, −1, …, −m modulo 2^{r−1} then ℱ_{1}(M) is a vector bundle on ℙ^{r−1} which is not a sum of line bundles. In particular, if r > 6 and m ≤ r − 3 then ℱ_{1}(M) is a new vector bundle of low rank on projective space. □

This theorem explains why, in the case r = 4 in Section 10.5, we had to resort to examples where ℱ_{1}(M) is not a direct sum of line bundles in order to obtain existence for Jordan type [2]^{n}[1]^{2} with n congruent to 2 or 4 modulo 8.

For r = 6, in Section 9.14 we constructed a module of constant Jordan type [2]^{14}[1]^{2} associated with the Tango bundle. This does not satisfy the congruences of the theorem.

Combining the information given in this section with Lemma 10.5.1, the case that remains for discussion is the case m = r − 2. In this case, we have Lemmas 10.5.2 and 10.5.3, which give somewhat incomplete information.

10.9. Nilvarieties of Constant Jordan Type [p]^{n} for p ≥ 3

In this section we investigate nilvarieties of rank r and constant Jordan type [p]^{n} for p ≥ 3. In the case of kE-modules of constant Jordan type, Dade’s lemma implies that p^{r−1}|n. However, we already know from Example 4.1.6 that there are nilvarieties of rank two and constant Jordan type [p], so the divisibility statement coming from Dade’s lemma does not apply to nilvarieties. In this section we obtain a lower bound on the power of p dividing n. This lower bound is exact for p = 3, but the situation is unclear for p ≥ 5.

Again we have a choice of using Hirzebruch–Riemann–Roch or working directly with Poincaré series. We choose the latter approach, as it turns out to be more straightforward.

Theorem 10.9.1. Let p ≥ 3 and let M be a rank r nilvariety of constant Jordan type [p]^{n}. Then

p^{⌊^{\frac{r−1}{p−1}}⌋}|n

where ⌊\frac{r−1}{p−1}⌋ denotes the largest integer less than or equal to \frac{r−1}{p−1}.

Proof. Let R = k[Y_{1}, …, Y_{r}], and let N be a finitely generated graded R-module with ℱ_{p}(M) ≅ N^{∼}. Write the Poincaré series of N as

p_{N}(t) = \frac{f(t)}{(1 − t)^{r}} .

The vector bundle M̃ has a filtration with filtered quotients

ℱ_{p}(M), ℱ_{p}(M)(1), …, ℱ_{p}(M)(p − 1),

so we have

(10.9.1)  (1 + t + ⋯ + t^{p−1})p_{N}(t) = \frac{np}{(1 − t)^{r}} + q(t)

where q(t) ∈ ℤ[t, t^{−1}]. Let ε be a primitive pth root of unity in ℂ, so that

1 + ε + ⋯ + ε^{p−1} = 0.

Then ℤ[ε] is the ring of cyclotomic integers in the cyclotomic number field ℚ[ε]. This ring is a Dedekind domain in which the prime ideal lying above (p) is (1 − ε), and we have (1 − ε)^{p−1} = (p).

Substituting t = ε in equation (10.9.1), we obtain the equation

\frac{np}{(1 − ε)^{r}} + q(ε) = 0

in ℚ[ε]. Since q(ε) ∈ ℤ[ε], it follows that np is in the rth power of the ideal (1 − ε). Since it is a rational integer, it follows that it is divisible by p^{a}, where a is the smallest integer greater than or equal to \frac{r}{p−1}, namely ⌊\frac{r−1}{p−1}⌋ + 1. So n is divisible by p^{⌊^{\frac{r−1}{p−1}}⌋}. □

The following theorem shows that Theorem 10.9.1 is sharp in the case p = 3.

Theorem 10.9.2. Let k be an algebraically closed field of characteristic three. Then there exists a nilvariety M of rank r of constant Jordan type [3]^{n} with n = 3^{[^{\frac{r−1}{2}}]}.

Proof. It suffices to prove this with r even. In this case, we may take a tensor product of r/2 copies of Example 4.1.6. In characteristic three, a tensor product of two Jordan blocks of length three is a sum of three Jordan blocks of length three. □

Example 10.9.3. If r = 4, the tensor product of two copies of Example 4.1.6 gives a nilvariety of constant Jordan type where the operator θ acts as the matrix

(\begin{matrix}0 & Y_{2} & 0 & Y_{4} & 0 & 0 & 0 & 0 & 0 \\ Y_{1} & 0 & Y_{2} & 0 & Y_{4} & 0 & 0 & 0 & 0 \\ 0 & −Y_{1} & 0 & 0 & 0 & Y_{4} & 0 & 0 & 0 \\ Y_{3} & 0 & 0 & 0 & Y_{2} & 0 & Y_{4} & 0 & 0 \\ 0 & Y_{3} & 0 & Y_{1} & 0 & Y_{2} & 0 & Y_{4} & 0 \\ 0 & 0 & Y_{3} & 0 & −Y_{1} & 0 & 0 & 0 & Y_{4} \\ 0 & 0 & 0 & −Y_{3} & 0 & 0 & 0 & Y_{2} & 0 \\ 0 & 0 & 0 & 0 & −Y_{3} & 0 & Y_{1} & 0 & Y_{2} \\ 0 & 0 & 0 & 0 & 0 & −Y_{3} & 0 & −Y_{1} & 0 \end{matrix}) .

If k has characteristic 3 then the Jordan type is [3]^{3} while if k has characteristic at least 5 or characteristic zero, the Jordan type is [5] [3] [1].

10.10. Nilvarieties with a Single Jordan Block

In contrast with Theorem 10.9.2, for p ≥ 5 the situation is less clear. Theorem 10.9.1 allows for the existence of nilvarieties of rank p − 1 and constant Jordan type [p], whereas we have the following theorem saying that this cannot occur.

Theorem 10.10.1 (Causa, Re and Teodorescu [85]). Let M be a nilvariety of rank r and constant Jordan type [m]. Then r ≤ 2, and if r = 2 then m is odd.

Proof. Suppose that M is a nilvariety of constant Jordan type [m]. Then ℱ_{m}(M) is a line bundle, so by Theorem 6.3.1 we have ℱ_{m}(M) ≅ 𝒪(a) for some integer a. The bundle M̃ ≅ 𝒪^{⊕m} has a filtration with filtered quotients

𝒪(a), 𝒪(a + 1), …, 𝒪(a + m − 1)

and so

0 = c_{1}(M̃) = a + (a + 1) + ⋯ + (a + m − 1) = ma + m(m − 1)/2.

Thus a = −(m − 1)/2 and so m is odd.

If r ≥ 3 then for all i and j we have

Ext_{𝒪_{ℙ^{r−1}}}^{1}(𝒪(i), 𝒪(j)) ≅ H^{1}(ℙ^{r−1}, 𝒪(j − i)) = 0

and so the filtration splits, giving

𝒪^{⊕m} ≅ 𝒪(a) ⊕ 𝒪(a + 1) ⊕ ⋯ ⊕ 𝒪(a + m − 1),

which contradicts Theorem 6.1.6. □

Theorem 10.10.2 (Causa, Re and Teodorescu [85]). For each odd value of m ≥ 3 there exists a nilvariety of rank two and constant Jordan type [m] of the form

Y_{1}A_{1} + Y_{2}A_{2} = (\begin{matrix}0 & Y_{1} \\ α_{1}Y_{2} & 0 & Y_{1} & & 0 \\ & α_{2}Y_{2} & 0 & Y_{1} \\ & & α_{3}Y_{2} & ... & ... \\ & 0 & & ... & 0 & Y_{1} \\ & & & & α_{m−1}Y_{2} & 0 \end{matrix}) .

The set of values of (α_{1}, …, α_{m−1}) ∈ k^{m−1} that give such nilvarieties form a non-empty dense open subset of a complete intersection of dimension (m − 1)/2 (i.e., projectively of dimension (m − 3)/2).

Proof. The given matrices form a nilvariety of constant Jordan type [m] if and only if the α_{i} are all non-zero, and

(10.10.1)  (Y_{1}A_{1} + Y_{2}A_{2})^{m} = 0.

We claim that condition (10.10.1) is equivalent to the condition that (Y_{1}A_{1} + Y_{2}A_{2})^{m} annihilates the first basis vector v_{1} of k^{m}. One direction is clear; for the other direction, the observation that (Y_{1}A_{1}+Y_{2}A_{2})^{i}v_{1} is Y_{2}^{i} times the ith basis vector v_{i} plus a linear combination of previous basis vectors v_{1}, …, v_{i−1} shows inductively that it annihilates all basis vectors.

The condition that (Y_{1}A_{1} + Y_{2}A_{2})^{m}v_{1} = 0 gives (m − 1)/2 equations. The ith equation says that the (2i)th coordinate is zero, and is of the form

(polynomial in α_{1}, …, α_{(m+2i−1)/2}) + α_{2i+2} … α_{(m+2i+1)/2} = 0.

These equations describe a complete intersection in k^{m−1} of dimension (m − 1)/2. The open subset defined by the non-vanishing of α_{1}, … α_{m−1} is non-empty and dense. □

Example 10.10.3. If m = 5 the general solution to the equations of Theorem 10.10.2 is of the form

A_{1} = (\begin{matrix}0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 \end{matrix}) A_{2} = λ (\begin{matrix}0 & 0 & 0 & 0 & 0 \\ a & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & −(a + 1)^{2} & 0 & 0 \\ 0 & 0 & 0 & a(a + 1) & 0 \end{matrix})

(λ ≠ 0, a ≠ 0, −1).

Example 10.10.4. We can do slightly better than rank two in Theorem 10.10.1 if we look for a nilvariety of constant Jordan type [m]^{2}. The following is an example of constant Jordan type [5]^{2} and rank three, with k an algebraically closed field of characteristic not equal to two.

(\begin{matrix}0 & Y_{2} & Y_{3} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 2Y_{1} & 0 & Y_{2} & −4Y_{3} & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & −Y_{1} & 0 & Y_{2} & Y_{3} & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & Y_{1} & 0 & Y_{2} & Y_{3} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & −2Y_{1} & 0 & 0 & Y_{3} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & Y_{2} & Y_{3} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 2Y_{1} & 0 & Y_{2} & −4Y_{3} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & −Y_{1} & 0 & Y_{2} & Y_{3} \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & Y_{1} & 0 & Y_{2} \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & −2Y_{1} & 0 \end{matrix})

So now the question is, how sharp is Theorem 10.9.1 for p ≥ 5? For example, does there exist a nilvariety of constant Jordan type [p]^{n} and rank p − 1 with n not divisible by p? The following extended version of Theorem 10.10.1 shows that for p large we cannot expect n to be small.

Theorem 10.10.5. Let M be a nilvariety of rank r ≥ 3 and constant Jordan type [m]^{n}. Then ℱ_{m}(M) is a vector bundle of rank n on ℙ^{r−1} which is not a direct sum of line bundles. In particular, if n ≤ r − 2 and r ≥ 7 this gives a new vector bundle of low rank on projective space (cf. Section 6.3).

Proof. Suppose to the contrary that ℱ_{m}(M) is a direct sum of line bundles. Arguing as in the proof of Theorem 10.10.1, the filtration of M̃ splits and we obtain

𝒪^{⊕nm} ≅ ℱ_{m}(M) ⊕ ℱ_{m}(M)(1) ⋯ ⊕ ℱ_{m}(M)(m − 1)

contradicting Theorem 6.1.6. □

10.11. Babylonian Towers

Recall from Remark 6.3.4 the Babylonian tower theorem for vector bundles on projective spaces. It states that a vector bundle on ℙ^{r−1} which extends to a vector bundle on ℙ^{r+n−1} for all n ≥ 0 has to be a direct sum of line bundles. In this section we examine the corresponding situation for modules of constant Jordan type.

Definition 10.11.1. We say that a k(ℤ/p)^{r}-module M of constant Jordan type stably extends to (ℤ/p)^{r+n} if there exists a k(ℤ/p)^{r+n}-module M′ of constant Jordan type such that M′↓_{k(ℤ/p)^{r}} ≅ M ⊕ (projective).

Theorem 10.11.2. Let p be an odd prime, and suppose that M is a k(ℤ/p)^{r}-module of stable constant Jordan type [a_{1}] … [a_{t}] which stably extends to k(ℤ/p)^{r+n} for all n ≥ 0, and suppose that ∑_{i=1}^{t} a_{i} ≤ p − 2. Then a_{i} = 1 for 1 ≤ i ≤ t, so that the stable constant Jordan type of M is [1]^{t}. Furthermore, ℱ_{1}(M) is a direct sum of line bundles ⨁_{i=1}^{t} 𝒪(pn_{i}).

Proof. The fact that we have a_{i} = 1 for 1 ≤ i ≤ t is proved in Theorem 10.4.1. By the Babylonian tower theorem for vector bundles (Remark 6.3.4), ℱ_{1}(M) is a direct sum of line bundles ⨁_{i=1}^{t} 𝒪(m_{i}). It remains to prove that each m_{i} is divisibe by p. If n is large enough so that r + n > t, then the Chern polynomial c(ℱ_{1}(M)) is a polynomial of degree at most t in ℤ[h]/(h^{r+n}). So the factorisation

c(ℱ_{1}(M)) = ∏_{i=1}^{t}(1 + m_{i}h)

may be regarded as an equation in ℤ[[h]] = lim_{\overleftarrow{n}} A^{∗}(ℙ^{r+n−1}). By Theorem 10.3.1, we have

p|c_{i}(ℱ_{1}(M)) for 1 ≤ i ≤ p − 2. Since t ≤ p − 2, this implies that

p|c_{t}(ℱ_{1}(M)) = ∏_{i=1}^{t} m_{i},

and so for some value of i, m_{i} is divisible by p. Thus

(1 + m_{i}h)^{−1}c(ℱ_{1}(M)) ∈ ℤ[[h]]

is a polynomial of degree at most t − 1, all of whose coefficients of positive powers of h are divisible by p, and so ∏_{j≠i} m_{j} is divisible by p. Continuing this way, stripping off one factor at a time, we see that all the m_{j} are divisible by p, as required. □

Conjecture 10.11.3 (Babylonian tower conjecture for modules of constant Jordan type). Let p be an odd prime, and suppose that M is a k(ℤ/p)^{r}-module of stable constant Jordan type [a_{1}] … [a_{t}] which stably extends to k(ℤ/p)^{r+n} for all n ≥ 0, and suppose that ∑_{i=1}^{t} a_{i} ≤ p − 2. Then M decomposes as a direct sum of modules of the form Ω^{−n_{i}}(k) and projective modules (cf. Corollary 8.5.3).

11 Orlov’s Correspondence

11.1. Introduction

This chapter is a lightly modified version of the paper [45]. The goal is to describe an equivalence between the derived category of an elementary abelian p-group of rank r, and the category of (graded) reduced matrix factorisations of the polynomial y_{1}X_{1}^{p} + ⋯ + y_{r}X_{r}^{p}. This gives another interpretation of the Realisation Theorem 8.1.1 for vector bundles on projective space from modular representations of constant Jordan type.

Orlov [204] established an equivalence of singularity categories which he called “reduction of dimension”. This has been further investigated by many authors, including for example Baranovsky and Pecharich [29], Burke and Stevenson [69], Burke and Walker [70], Chen [87]. Our purpose here is to investigate the consequences for representation theory of an elementary abelian p-group E over a field k of characteristic p as explicitly as possible. In this context, we explain the Orlov correspondence and how it lifts to the derived category. We use the Orlov correspondence to give a new interpretation of the construction given in Section 8.8 for proving Theorem 8.1.1, realising vector bundles from modules of constant Jordan type, see Remark 11.12.4.

I should point out that Theorem 11.3.3, the main theorem of this chapter, is a special case of Theorem 7.5 of Burke and Stevenson [69]; even the functors realising the equivalences in the theorem are the same. The proof presented here uses a minimum of heavy machinery, taking advantage of the special situation in hand to reduce to an explicit computation involving the “bidirectional Koszul complex,” introduced in Section 11.6.

Let R_{0} be the polynomial ring k[X_{1}, …, X_{r}], and let

R = R_{0}[y_{1}, …, y_{r}].

We make this a graded ring with the X_{i} in degree zero and the y_{i} in degree one. Let f be the degree one polynomial

f = y_{1}X_{1}^{p} + ⋯ + y_{r}X_{r}^{p} ∈ R,

and let

A = R/(f), B = R/(X_{1}^{p}, …, X_{r}^{p}).

Let E = ⟨g_{1}, …, g_{r}⟩ ≅ (ℤ/p)^{r} be an elementary abelian p-group of rank r, and let k be a field of characteristic p. The map sending X_{i} to g_{i} − 1 ∈ kE gives the identification

kE = k[X_{1}, …, X_{r}]/(X_{1}^{p}, …, X_{r}^{p})

so that

B = k[y_{1}, …, y_{r}] ⊗_{k} kE.

The elements y_{1}, …, y_{r} will eventually be identified with the generators for the polynomial part of the cohomology ring H^{∗}(E, k). So if p is odd, the y_{i} here correspond to the x_{i} in Section 1.8 while if p = 2 they correspond to the elements y_{i}^{2} given there. We thus have the following diagram of graded rings and homomorphisms:

\xymatrix{B & A \ar@{->>}[l] & R \ar@{->>}[l] \\ kE \ar[u] &  & k[X_{1}, …, X_{r}] \ar[u] \ar@{-->}[ul] \ar@{->>}[ll]}

The ring kE is a complete intersection and A is a hypersurface. Our goal is to investigate the Orlov correspondence as an equivalence of triangulated categories

\xymatrix{D^{b}(kE) \ar[r]^{Ψ} & D_{sg}(A) \ar[l]^{Φ} & ≃ \underline{RMF}(f) ≃ \underline{MCM}(A)}

Here, D_{sg}(A) is the singularity category of A, namely the Verdier quotient of D^{b}(A) by the thick subcategory of perfect complexes. This is equivalent to \underline{RMF}(f), the category of reduced matrix factorisations of f, which in turn is equivalent to the stable category \underline{MCM}(A) of maximal Cohen–Macaulay A-modules. See Theorem 11.9.3. This descends to an equivalence

\xymatrix{stmod(kE) \ar[r]^{Ψ̄} & \underline{\underline{RMF}}(f) \ar[l]^{Φ̄} & ≃ \underline{\underline{MCM}}(A)}

where \underline{\underline{MCM}}(A) is the Verdier quotient of \underline{MCM}(A) by the thick subcategory of maximal Cohen–Macaulay approximations of modules sitting in a finite set of degrees, and \underline{\underline{RMF}}(f) is the corresponding quotient of \underline{RMF}(f).

The functor Ψ is easy to describe. Given a bounded complex of kE-modules M_{∗}, we tensor over k with k[y_{1}, …, y_{r}] to get a bounded complex of B-modules, pull back to get a bounded complex of A-modules, and then apply the quotient functor D^{b}(A) → D_{sg}(A). For the functor Φ, a reduced matrix factorisation of f is given by a pair of free R-modules F and F′ together with maps F′[−1] ⟶^{V} F ⟶^{U} F′ with UV and V U equal to multiplication by f. Given such data, we apply B ⊗_{R} − to obtain free B-modules B ⊗_{R} F, B ⊗_{R} F′ and then take graded pieces to get free kE-modules:

⋯ ⟶^{V} (B ⊗_{R} F)_{0} ⟶^{U} (B ⊗_{R} F′)_{0} ⟶^{V} (B ⊗_{R} F)_{1} ⟶^{U} (B ⊗_{R} F′)_{1} ⟶^{V} ⋯

This complex only goes finitely far to the left, since F and F′ are finitely generated, but it goes infinitely far to the right. For technical reasons, we then need to index this complex so that the term (B ⊗_{R} F)_{0} is in homological degree r. The resulting complex is not exact, but it is exact sufficiently far to the right. So it is a semi-injective resolution of a well-defined object in D^{b}(kE). We shall show that this defines a functor Φ: \underline{RMF}(f) → D^{b}(kE) such that ΦΨ and ΨΦ are naturally isomorphic to the identity functors.

Our main tools are a bidirectional Koszul complex, which replaces Orlov’s use of local duality, and a theorem of Schoutens [223] relating the singularity category to the singular set.

Grading Conventions. We grade everything homologically, so that the differential decreases degree. When we talk of complexes of graded modules, there are two subscripts. The first subscript gives the homological degree and the second gives the internal degree. If C is a complex of graded modules with components C_{i,j} then we write C[n] for the homological shift: C[n]_{i,j} = C_{i+n,j}, and C(n) for the internal shift: C(n)_{i,j} = C_{i,j+n}.

11.2. Maximal Cohen–Macaulay Modules

Recall from Eisenbud [110] that there is an equivalence of categories between the stable category of maximal Cohen–Macaulay A-modules and matrix factorisations of the polynomial f. This correspondence is also explained in detail in Chapter 7 of Yoshino [251]. Our exposition is broadly taken from there.

We write mod(A) for the abelian category of finitely generated A-modules and module homomorphisms. A maximal Cohen–Macaulay A-module is a module whose depth is equal to 2r − 1, which is the Krull dimension of A. We write MCM(A) for the full subcategory of mod(A) consisting of maximal Cohen–Macaulay A-modules. This is an exact category with enough projectives and enough injectives, and the projective and injective objects coincide. Such a category is called a Frobenius category, see Section I.2 of Happel [144].

The stable category \underline{MCM}(A) of maximal Cohen–Macaulay modules has the same objects as MCM(A), but the homomorphisms are given by

\underline{Hom}_{A}(M, N) = Hom_{A}(M, N)/P Hom_{A}(M, N)

where P Hom_{A}(M, N) is the subspace of Hom_{A}(M, N) consisting of homomorphisms that factor through a free A-module. Note that \underline{MCM}(A) is a triangulated category.

So let M be a maximal Cohen–Macaulay A-module. By the Auslander–Buchsbaum formula, as an R-module M has projective dimension one. So it has a resolution of the form

0 → F ⟶^{U} F′ → M → 0

where F and F′ are free R-modules, which necessarily have the same number of generators. Here, the generators of a free module are allowed to be in different degrees. We can therefore think of U as a square matrix with entries in R. Since fM = 0, we have fF′ ⊆ UF. So for any x ∈ F′ there exists a unique y ∈ F such that fx = Uy. This defines for us an R-module homomorphism V : F′ → F [1] sending x to y. It satisfies UV = f.I, and therefore also V U = f.I since R is an integral domain.

Definition 11.2.1. A matrix factorisation (U, V) of f consists of a pair of square matrices U and V of the same size, with entries in R, satisfying

UV = V U = f.I

where I denotes the identity matrix. This can be regarded as a pair of maps between free R-modules

U: F → F′, V : F′[−1] → F.

A map of matrix factorisations is given by a pair of maps of free modules making the two obvious squares commute. This gives matrix factorisations the structure of an exact category which we denote MF(f).

Given a matrix factorisation of f, let F̄ = A ⊗_{R} F, F̄′ = A ⊗_{R} F′ to give a complex

(11.2.1)  ⋯ → F̄[−1] ⟶^{U} F̄′[−1] ⟶^{V} F̄ ⟶^{U} F̄′.

In fact, this is an exact sequence, since if x ∈ F with Ux̄ = 0 then there exists y ∈ F′ such that Ux = fy, so fx = V Ux = V fy = fV y and hence x = V y (and similarly if x ∈ F′ with V x̄ = 0). It follows that (11.2.1) is a free resolution of the cokernel of U.

The matrix factorisation (U, V) = (1, f) (with F = F′ = R) has Coker (U) = 0, so to get a correspondence between maximal Cohen–Macaulay modules and matrix factorisations we need to work modulo copies of this one. Furthermore, the matrix factorisation (U, V) = (f, 1) (with F = R[−1], F′ = R) has Coker (U) ≅ A. So for the stable category of maximal Cohen– Macaulay modules we need to work modulo this one as well.

Definition 11.2.2. The category \underline{MF}(f) = MF(f)/\{(1, f)\} has the same objects as MF(f), but the homomorphisms are obtained by quotienting out those that factor through a direct sum of copies of (1, f).

The category \underline{RMF}(f) = MF(f)/\{(1, f), (f, 1)\} of reduced matrix factorisations has the same objects as MF(f) but the homomorphisms are obtained by quotienting out those that factor through a direct sum of copies of (1, f) and (f, 1).

Theorem 11.2.3 (Eisenbud [110]). The processes described above induce equivalences of categories between \underline{MF}(f) and MCM(A) and between \underline{RMF}(f) and \underline{MCM}(A).

Given a matrix factorisation F′[−1] ⟶^{V} F ⟶^{U} F′, the shift τ in the triangulated category \underline{RMF}(f) takes it to F ⟶^{−U} F′ ⟶^{−V} F [1]. Doing the same again gives F′ ⟶^{V} F [1] ⟶^{U} F′[1]. So the square of the shift is equal to the internal degree shift for graded A-modules:

τ^{2} = [1].

Now given any A-module N, its (2r −1)st syzygy Ω^{2r−1}N is a maximal Cohen–Macaulay module, and hence has a periodic resolution of the form (11.2.1). So the minimal resolution of N eventually agrees with the minimal resolution of a maximal Cohen–Macaulay module N′. The pitchfork lemma of Auslander and Buchweitz [21] allows us to extend to a map of resolutions, so that we obtain a map N′ → N, well defined in the stable category. The module N′ is called a maximal Cohen–Macaulay approximation to N. Adding a free module to N′ to make this map surjective, we obtain a short exact sequence

0 → N′′ → N′ → N → 0

where N′′ has finite injective dimension.

This allows us to identify the maximal Cohen–Macaulay modules with the singularity category.

Theorem 11.2.4 (Buchweitz [66]). The inclusion of the maximal Cohen–Macaulay modules in the derived category induces an equivalence between \underline{MCM}(A) and D_{sg}(A).

11.3. The Orlov Correspondence

In this section we give a version of the Orlov correspondence for a complete intersection of dimension zero. Let C = k[X_{1}, …, X_{r}]/(f_{1}, …, f_{r}) where f_{1}, …, f_{r} is a regular sequence contained in the square of the maximal ideal (X_{1}, …, X_{r}).

Example 11.3.1. Let

E = ⟨g_{1}, …, g_{r}⟩ ≅ (ℤ/p)^{r}

and let k be a field of characteristic p. Let kE be the group algebra of E over k, and let

X_{i} = g_{i} − 1 ∈ kE.

Then kE = k[X_{1}, …, X_{r}]/(X_{1}^{p}, …, X_{r}^{p}) is a complete intersection of codimension r and dimension zero.

Let R_{0} = k[X_{1}, …, X_{r}] and let R = k[y_{1}, …, y_{r}] ⊗_{k} R_{0}. We regard R as a graded polynomial ring with the y_{i} in degree one and the X_{i} in degree zero. Let

f = y_{1}f_{1} + ⋯ + y_{r}f_{r} ∈ R,

an element of degree one. Let

A = R/(f)

and

B = R/(f_{1}, …, f_{r}) = k[y_{1}, …, y_{r}] ⊗_{k} C.

We have a diagram

\xymatrix{B & A \ar@{->>}[l] & R \ar@{->>}[l] \\ C \ar[u] &  & k[X_{1}, …, X_{r}] \ar@{->>}[ll] \ar@{-->}[ul] \ar[u]}

Taking Proj of these graded rings, we get the diagram in Section 2 of Orlov [204]:

\xymatrix{Z \ar[d]_{p} \ar@{^{(}->}[r]^{i} & Y \ar@{^{(}->}[r]^{u} \ar[dr]^{π} & S′ \ar[d]^{q} \\ X \ar@{^{(}->}[rr]^{j} &  & S}

Theorem 11.3.2 (Orlov). The functor ℝi_{∗}p^{∗}: D^{b}(X) → D^{b}(Y) descends to an equivalence of categories stmod(C) = D_{sg}(X) → D_{sg}(Y). The right adjoint

(11.3.1)  ℝp_{∗}i^{♭} = ℝp_{∗}𝕃i^{∗}(− ⊗ ω_{Z/Y})[−r + 1]

gives the inverse equivalence.

Our goal is to prove the following lift of the Orlov correspondence to the derived category.

Theorem 11.3.3. There is an equivalence of categories D^{b}(C) ≃ D_{sg}(A) lifting the equivalence D_{sg}(X) ≃ D_{sg}(Y) of Orlov.

Remark 11.3.4. In Orlov’s version, he makes use of local duality as described in Chapter III, Corollary 7.3 of Hartshorne [146] to identify the right adjoint (11.3.1). In our version, this is replaced by the self-duality (11.7.1) of a complex _{A}Δ̂_{A} giving a Tate resolution of A as an A-A-bimodule. The discrepancy between the shift of −r + 1 in (11.3.1) and the shift of −r in (11.7.1) is explained by the fact that in our situation the sheaf ω_{Z/Y} is just 𝒪(−1).

In the following corollary, we spell out the consequences for the modular representation theory of elementary abelian p-groups.

Corollary 11.3.5. Let E be an elementary abelian p-group of rank r. Then the following triangulated categories are equivalent:

(1) The derived category D^{b}(kE).

(2) The singularity category of graded A-modules D_{sg}(A) where

A = R/(f), R = k[y_{1}, …, y_{r}, X_{1}, …, X_{r}], f = y_{1}X_{1}^{p} + ⋯ + y_{r}X_{r}^{p},

the X_{i} have degree zero and the y_{i} have degree one.

(3) The stable category of maximal Cohen–Macaulay graded A-modules.

(4) The category of reduced graded matrix factorisations of f over R.

We shall see that under the correspondence given by this Corollary, the image of the trivial kE-module k is a 2^{r−1} × 2^{r−1} matrix factorisation given by taking the even and odd terms in a bidirectional Koszul complex. The perfect complexes correspond to the maximal Cohen–Macaulay approximations to the A-modules which are non-zero only in finitely many degrees, so that the equivalence descends to Orlov’s equivalence

stmod(kE) ≃ D_{sg}(Proj (A)).

The elements y_{i} correspond to the basis for the primitive elements in H^{2}(E, k) obtained by applying the Bockstein map to the basis of H^{1}(E, k) dual to X_{1}, …, X_{r}.

11.4. The Functors

First we describe the functor Φ: D^{b}(C) → D_{sg}(A). If M_{∗} is a bounded complex of finitely generated C-modules then the tensor product

k[y_{1}, …, y_{r}] ⊗_{k} M_{∗}

is a bounded complex of finitely generated B-modules, which may then be regarded as a bounded complex of finitely generated A-modules. Passing down to the singularity category D_{sg}(A), we obtain Φ(M_{∗}). Thinking of D_{sg}(A) as equivalent to \underline{MCM}(A), we can view Φ(M_{∗}) as a maximal Cohen–Macaulay approximation to k[y_{1}, …, y_{r}] ⊗_{k} M_{∗}.

Next we describe the functor Ψ: D_{sg}(A) → D^{b}(C). An object N in D_{sg}(A) can be thought of as a maximal Cohen–Macaulay A-module. It is therefore represented by a reduced graded matrix factorisation of the polynomial f over R. Namely, we have a pair of finitely generated free R-modules F and F′ and maps

F ⟶^{U} F′ ⟶^{V} F (1)

such that UV and V U are both equal to f times the identity map. Then we have the following sequence of free C-modules:

\xymatrix{… & … \ar[dl]_{V} \\ C ⊗_{R_{0}} F_{0} \ar[r]^{U} & C ⊗_{R_{0}} F′_{0} \ar[dl]_{V} \\ C ⊗_{R_{0}} F_{1} \ar[r]^{U} & C ⊗_{R_{0}} F′_{1} \ar[dl]_{V} \\ … & …}

Since the free modules are finitely generated, this is zero far enough up the page. We shall see in Lemma 11.4.1 below, that the resulting complex only has homology in a finite number of degrees. It is therefore a complex of free C-modules, bounded to the left, and whose homology is totally bounded. It is therefore a semi-injective resolution of a well-defined object in D^{b}(C). We shift in degree so that the term F_{0} ⊗_{R_{0}} C appears in degree −r, and this is the object Ψ(N) in D^{b}(C).

Another way of viewing the object Ψ(N) is to take a complete resolution of N as an A-module:

\xymatrix{⋯ \ar[r] & P_{1} \ar[r] & P_{0} \ar[rr] \ar@{-->}[dr] &  & P_{−1} \ar[r] & ⋯ \\  &  &  & N \ar@{-->}[ur] \ar[dr] \\  &  & 0 \ar[ur] &  & 0}

and then Ψ(N) is the complex (B ⊗_{A} P_{∗})_{0}[−r], whose degree n term is (B ⊗_{A} P_{∗})_{(n−r,0)}.

Lemma 11.4.1. If M is a maximal Cohen–Macaulay A-module then for all j ≥ 0, Tor_{j}^{A}(B, M) is non-zero only in finitely many degrees.

Proof. This follows from the fact that B is locally (but not globally) a complete intersection as an A-module. More explicitly, for 1 ≤ i ≤ r, in the ring R[y_{i}^{−1}] we have the following equation:

X_{i}^{p} = y_{i}^{−1}(f − y_{1}f_{1} − ⋯ ↑^{i} ⋯ − y_{r}f_{r}).

It follows that

B[y_{i}^{−1}] = A[y_{i}^{−1}]/(f_{1}, ⋯ ↑^{i} ⋯, f_{r})

is a complete intersection of codimension r − 1 over A[y_{i}^{−1}]. Using a Koszul complex, it follows that for all A-modules M and all j ≥ r we have

Tor_{j}^{A}(B, M)[y_{i}^{−1}] = Tor_{j}^{A[y_{i}^{−1}]}(B[y_{i}^{−1}], M[y_{i}^{−1}]) = 0.

If M is a maximal Cohen–Macaulay module then the minimal resolution of M is periodic and so Tor_{j}^{A}(B, M)[y_{i}^{−1}] = 0 for all j ≥ 0. Since Tor_{j}^{A}(B, M) is finitely generated, it is annihilated by a high enough power of each y_{i} and is hence it is non-zero only in finitely many degrees. □

11.5. An Example

Before delving into proofs, let us examine an example in detail. Let E = (ℤ/p)^{2} = ⟨g_{1}, g_{2}⟩, an elementary abelian group of rank two, and let k be a field of characteristic p. Then setting X_{1} = g_{1} − 1, X_{2} = g_{2} − 1, we have

C = kE = k[X_{1}, X_{2}]/(X_{1}^{p}, X_{2}^{p})
A = k[y_{1}, y_{2}, X_{1}, X_{2}]/(y_{1}X_{1}^{p} + y_{2}X_{2}^{p}).

Let us compute Φ(k), where k is the trivial kE-module. This means we should resolve k[y_{1}, y_{2}] as an A-module, and look at the corresponding matrix factorisation. This minimal resolution has the form

A(−1) ⊕ A(−1) \underrightarrow{( \begin{matrix}y_{2}X_{2}^{p−1} & −y_{1}X_{1}^{p−1} \\ X_{1} & X_{2} \end{matrix} )} A ⊕ A(−1) \underrightarrow{( \begin{matrix}X_{2} & y_{1}X_{1}^{p−1} \\ −X_{1} & y_{2}X_{2}^{p−1} \end{matrix} )} A ⊕ A ⟶^{( X_{1},X_{2} )} A → k[y_{1}, y_{2}]

This pair of 2 × 2 matrices gives a matrix factorisation of the polynomial y_{1}X_{1}^{p} + y_{2}X_{2}^{p}, and it is the matrix factorisation corresponding to the trivial kE-module.

Applying the functor Ψ to this matrix factorisation, we obtain the minimal injective resolution of the trivial kE-module, shifted in degree by two. The elements y_{1} and y_{2} give the action of the polynomial generators of H^{∗}(E, k) of degree two on the minimal resolution.

Similarly, we compute Φ(kE) using the following resolution:

A(−1) ⊕ A(−1) \underrightarrow{( \begin{matrix}y_{2} & −y_{1} \\ X_{1}^{p} & X_{2}^{p} \end{matrix} )} A ⊕ A(−1) \underrightarrow{( \begin{matrix}X_{2}^{p} & y_{1} \\ −X_{1}^{p} & y_{2} \end{matrix} )} A ⊕ A \underrightarrow{(^{X_{1}^{p},X_{2}^{p}})} A → kE ⊗_{k} k[y_{1}, y_{2}].

This should be compared with the resolution of k[X_{1}, X_{2}]:

A(−2)⊕A(−2)\underrightarrow{( \begin{matrix}y_{2} & −y_{1} \\ X_{1}^{p} & X_{2}^{p} \end{matrix} )} A(−1)⊕A(−2) \underrightarrow{( \begin{matrix}X_{2}^{p} & y_{1} \\ −X_{1}^{p} & y_{2} \end{matrix} )} A(−1)⊕A(−1) ⟶^{( y_{2},−y_{1} )} A → k[X_{1}, X_{2}].

This is eventually the same resolution as above, but shifted two places to the left. Thus Φ(kE) is a maximal Cohen–Macaulay approximation to a module concentrated in a single degree.

11.6. The Bidirectional Koszul Complex

In this section, we construct the bidirectional Koszul complex. This first appears in a paper of Tate [237], and reappears in many places. This allows us to describe the minimal resolution of B as an A-module. This computes the value of the functor Φ on the free C-module of rank one.

Let Λ_{n} (0 ≤ n ≤ r) be the free R-module of rank \binom{r}{n} on generators

e_{j_{1}} ∧ ⋯ ∧ e_{j_{n}}

with 1 ≤ j_{1} < ⋯ < j_{n} ≤ r. We use the convention that the wedge is alternating, in the sense that e_{i} ∧ e_{j} = −e_{j} ∧ e_{i} and e_{i} ∧ e_{i} = 0, to give meaning to wedge products with indices out of order or repeated indices.

We give Λ_{∗} a differential d: Λ_{n} → Λ_{n−1} described by

(11.6.1)  d(e_{j_{1}} ∧ ⋯ ∧ e_{j_{n}}) = ∑_{i}(−1)^{i−1}f_{j_{i}}(e_{j_{1}} ∧ ⋯ ↑^{j_{i}} ⋯ ∧ e_{j_{n}})

where the vertical arrow indicates a missing term. We also give Λ_{∗} a differential in the other direction δ: Λ_{n} → Λ_{n+1}(1) described by

(11.6.2)  δ(e_{j_{1}} ∧ ⋯ ∧ e_{j_{n}}) = ∑_{j} y_{j}e_{j} ∧ (e_{j_{1}} ∧ ⋯ ∧ e_{j_{n}}).

We call the graded R-module Λ_{∗} with these two differentials the bidirectional Koszul complex with respect to the pair of sequences f_{1}, …, f_{r} and y_{1}, …, y_{r}:

[figure: Λ_{r}; d; Λ_{r−1}; δ; (1); …; Λ_{1}; d; Λ_{0}; δ; (1)] .

Lemma 11.6.1. The map dδ + δd: Λ_{n} → Λ_{n}(1) is equal to multiplication by f = ∑_{i} y_{i}f_{i}.

Proof. We have

δd(e_{j_{1}} ∧ ⋯ ∧ e_{j_{n}}) = ∑_{i,j}(−1)^{i−1}y_{j}f_{j}e_{j} ∧ (e_{j_{1}} ∧ ⋯ ↑^{j_{i}} ⋯ ∧ e_{j_{n}}),

whereas

dδ(e_{j_{1}} ∧ ⋯ ∧ e_{j_{n}}) = ∑_{i,j}(−1)^{i−1}y_{j}f_{j}e_{j} ∧ (e_{j_{1}} ∧ ⋯ ↑^{j_{i}} ⋯ ∧ e_{j_{n}})

+ ∑_{i} y_{i}f_{i}(e_{j_{1}} ∧ ⋯ ∧ e_{j_{n}}).  □

Thus, taking even and odd parts of Λ_{∗}, we see that

(11.6.3)  ⨁_{n=0}^{⌊ \frac{r}{2} ⌋} Λ_{2n}(n − 1) ⟶^{d+δ} ⨁_{n=1}^{⌊ \frac{r+1}{2} ⌋} Λ_{2n−1}(n − 1) ⟶^{d+δ} ⨁_{n=0}^{⌊ \frac{r}{2} ⌋} Λ_{2n}(n)

is a matrix factorisation of f, called the Koszul factorisation. Note that the free R-modules in this matrix factorisation all have rank 2^{r−1}, because this is the sum of the even binomial coefficients as well as the sum of the odd binomial coefficients. For notation, we write

(11.6.4)  K_{0}(−1) ⟶^{d+δ} K_{1} ⟶^{d+δ} K_{0}

for the Koszul factorisation, and we write K for the cokernel of d + δ: K_{1} → K_{0}. For example, if r = 4 we get the Koszul factorisation

Λ_{4}(1) ⊕ Λ_{2} ⊕ Λ_{0}(−1) \underrightarrow{(\begin{matrix}d & δ & 0 \\ 0 & d & δ \end{matrix})} Λ_{3}(1) ⊕ Λ_{1} \underrightarrow{(\begin{matrix}δ & 0 \\ d & δ \\ 0 & d \end{matrix})} Λ_{4}(2) ⊕ Λ_{2}(1) ⊕ Λ_{0}.

The minimal resolution of the A-module B is obtained by applying A ⊗_{R} − to

(11.6.5) ⋯ → Λ_{4} ⊕ Λ_{2}(−1) ⊕ Λ_{0}(−2) → Λ_{3} ⊕ Λ_{1}(−1) ⟶^{d+δ} Λ_{2} ⊕ Λ_{0}(−1) ⟶^{d+δ} Λ_{1} ⟶^{d} Λ_{0}.

This takes r steps to settle down to the Koszul factorisation, but in large degrees (i.e., far enough to the left) it agrees with

(11.6.6)  ⋯ → K_{0}(−2) ⟶^{d+δ} K_{1}(−1) ⟶^{d+δ} K_{0}(−1) ⟶^{d+δ} K_{1} ⟶^{d+δ} K_{0}.

For notation, we write Λ̄_{i} = A ⊗_{R} Λ_{i}, so that the minimal resolution of B as an A-module takes the form

(11.6.7) ⋯ → Λ̄_{4} ⊕ Λ̄_{2}(−1) ⊕ Λ̄_{0}(−2) → Λ̄_{3} ⊕ Λ̄_{1}(−1) ⟶^{d+δ} Λ̄_{2} ⊕ Λ̄_{0}(−1) ⟶^{d+δ} Λ̄_{1} ⟶^{d} Λ̄_{0}.

Let us write Δ_{i,j} for this complex. As usual, i denotes the homological degree and j the internal degree. Thus Δ → B is a free resolution.

A complete resolution of B as an A-module is also easy to write down at this stage. Namely, we just continue (11.6.6) to the right in the obvious way. Let us write Δ̂_{i,j} for this complete resolution. Then we notice a self-duality up to shift:

(11.6.8)  Hom_{A}(Δ̂, A) ≅ Δ̂[−r]

and a periodicity

Δ̂[2] ≅ Δ̂(1).

Next we observe that the minimal resolution of A/(y_{1}, …, y_{r}) = k[X_{1}, …, X_{r}] as an A-module takes the form

(11.6.9)  ⋯ → Λ̄_{r−3}(−3) ⊕ Λ̄_{r−1}(−2) ⟶^{d+δ} Λ̄_{r−2}(−2) ⊕ Λ̄_{r}(−1) ⟶^{d+δ} Λ̄_{r−1}(−1) ⟶^{δ} Λ̄_{r}.

This again takes r steps to settle down to the Koszul factorisation, but in large degrees (i.e., far enough to the left) it agrees with the result of applying A ⊗_{R} − to

(11.6.10)  ⋯ → K_{1}(⌊\frac{−r−2}{2}⌋) ⟶^{d+δ} K_{0}(⌊\frac{−r−1}{2}⌋) ⟶^{d+δ} K_{1}(⌊\frac{−r}{2}⌋) ⟶^{d+δ} K_{0}(⌊\frac{−r+1}{2}⌋).

Theorem 11.6.2. The minimal resolutions over A of A/(y_{1}, …, y_{r}) = k[X_{1}, …, X_{r}] after r steps and of B after 2r steps are equal.

Proof. Compare (11.6.10) with (11.6.6). □

Now let M_{∗} be a bounded complex of C-modules, regarded as an object in D^{b}(C), and let X_{i,j} be a free resolution of k[y_{1}, …, y_{r}] ⊗_{k} M_{∗} as an A-module. Thus for large positive homological degree i, this is a periodic complex corresponding to a matrix factorisation of f, namely Φ(M_{∗}). The maps Δ_{∗,∗} → B and X_{∗,∗} → k[y_{1}, …, y_{r}] ⊗_{k} M_{∗} induce homotopy equivalences

B ⊗_{A} X_{∗,∗} ← Δ_{∗,∗} ⊗_{A} X_{∗,∗} → Δ_{∗,∗} ⊗_{A} (k[y_{1}, …, y_{r}] ⊗_{k} M_{∗}).

Now f_{1}, …, f_{r} annihilate M_{∗}, and so they act as zero in Δ_{∗,∗} ⊗_{A} (k[y_{1}, …, y_{r}] ⊗_{k} M_{∗}). So the operator d in the complex Δ_{∗,∗} acts as zero in the tensor product, which therefore decomposes as a direct sum of pieces, each living in a finite set of degrees. To be more explicit, it decomposes as a sum of the following pieces tensored over A with (k[y_{1}, …, y_{r}] ⊗_{k} M_{∗}):

\xymatrix{ &  &  &  &  &  &  & Λ̄_{0} \\  &  &  &  &  & Λ̄_{0}(−1) \ar[r]^{δ} & Λ̄_{1} \\  &  &  & Λ̄_{0}(−2) \ar[r]^{δ} & Λ̄_{1}(−1) \ar[r]^{δ} & Λ̄_{2} \\  & Λ̄_{0}(−3) \ar[r]^{δ} & Λ̄_{1}(−2) \ar[r]^{δ} & Λ̄_{2}(−1) \ar[r]^{δ} & Λ̄_{3} \\ … &  & …}

Eventually, this just consists of copies of the Koszul complex for parameters y_{1}, …, y_{r} on k[y_{1}, …, y_{r}]⊗_{k}M_{∗}, shifted in degree by (2n, −n+r). This Koszul complex is quasi-isomorphic to M_{∗} shifted (2n, −n + r).

It follows that if we take a complete resolution over A of k[y_{1}, …, y_{r}] ⊗_{k} M_{∗}, apply B ⊗_{A} − to it, and take the part with internal degree zero, we obtain a complex which is quasi-isomorphic to M_{∗} shifted in degree by r. This process is exactly the functor Ψ applied to Φ(M_{∗}). To summarise, we have proved the following:

Theorem 11.6.3. The composite functor Ψ ∘ Φ: D^{b}(C) → D^{b}(C) is naturally isomorphic to the identity functor. □

If we restrict just to C-modules rather than complexes, we have the following formulation:

Theorem 11.6.4. Let M be a C-module. Then for i ≥ 0 we have

Tor_{i+r,j}^{A}(B, k[y_{1}, …, y_{r}] ⊗_{k} M) ≅ Tor_{i,j}^{A}(k[X_{1}, …, X_{r}], k[y_{1}, …, y_{r}] ⊗_{k} M)
≅ \begin{cases}M & i = 2j \\ 0 & otherwise. \end{cases}

Proof. This follows from Theorems 11.6.2 and 11.6.3. □

11.7. A Bimodule Resolution

A similar bidirectional Koszul complex can be used to describe the minimal resolution of A as an A-A-bimodule. This works more generally for any hypersurface (or indeed with suitable modifications for any complete intersection, see Section 3 of Wolffhardt [249]), so we introduce it in that context. Let S = k[u_{1}, …, u_{n}] where each u_{i} is a homogeneous variable of non-negative degree. Let ϕ(u_{1}, …, u_{n}) ∈ S be a homogeneous polynomial of positive degree and let H = S/(ϕ) be the corresponding hypersurface. We write

S ⊗_{k} S = k[u′_{1}, …, u′_{n}, u′′_{1}, …, u′′_{n}]
H ⊗_{k} H = S ⊗_{k} S/(ϕ(u′_{1}, …, u′_{n}), ϕ(u′′_{1}, …, u′′_{n})).

Then we can form the bidirectional Koszul complex on the two sequences

u′_{1} − u′′_{1}, …, u′_{n} − u′′_{n}

and

(ϕ(u′_{1}, u′_{2} …, u′_{n}) − ϕ(u′′_{1}, u′_{2}, …, u′_{n}))/(u′_{1} − u′′_{1}),
(ϕ(u′′_{1}, u′_{2}, u′_{3}, …, u′_{n}) − ϕ(u′′_{1}, u′′_{2}, u′_{3}, …, u′_{n}))/(u′_{2} − u′′_{2}),
…
(ϕ(u′′_{1}, …, u′′_{n−1}, u′_{n}) − ϕ(u′′_{1}, …, u′′_{n−1}, u′′_{n}))/(u′_{n} − u′′_{n}).

Note that the latter is indeed a sequence of polynomials, and that the sum of the products of corresponding terms in these two sequences gives

ϕ(u′_{1}, …, u′_{n}) − ϕ(u′′_{1}, …, u′′_{n}).

We therefore obtain a matrix factorisation of this difference over S ⊗_{k} S looking much like (11.6.3). The construction corresponding to (11.6.7) in this situation gives a resolution of the module S = (S ⊗_{k} S)/(u′_{1} −u′′_{1}, …, u′_{r} −u′′_{r}) over the hypersurface (S ⊗_{k} S)/(ϕ′ −ϕ′′). Since ϕ′ is a not a zerodivisor on either the module or the hypersurface, we can mod it out, retaining exactness, to obtain a resolution of S/ϕ = H as a module over (S⊗_{k}S)/(ϕ′−ϕ′′, ϕ′) = H⊗_{k}H (i.e., as an H-H-bimodule). We write _{H}Δ_{H} for this resolution. It is eventually periodic with period two. The corresponding complete resolution is periodic with period two, and we denote it by _{H} Δ̂_{H}.

Applying this in the particular case of A as an A-A-bimodule, we have

A ⊗_{k} A = k[y′_{1}, .. , y′_{r}, y′′_{1}, .. , y′′_{r}, X′_{1}, .. , X′_{r}, X′′_{1}, .. , X′′_{r}]/(y′_{1}f′_{1} + · · +y′_{r}f′_{r}, y′′_{1}f′′_{1} + · · +y′′_{r}f′′_{r}).

Let _{A}Δ̂_{A} be the corresponding complete resolution. Exactly as in (11.6.8) we have a selfduality up to shift:

(11.7.1)  Hom_{A⊗_{k}A}(_{A}Δ̂_{A}, A ⊗_{k} A) ≅ _{A}Δ̂_{A}[−r]

and a periodicity

(11.7.2)  _{A}Δ̂_{A}[2] ≅ _{A}Δ̂_{A}(1).

Now regarding B as an A-B-bimodule via the map A → B, we have a free resolution given by _{A}Δ_{A} ⊗_{A} _{A}B_{B}. We write _{A}Δ_{B} for this resolution, and _{A}Δ̂_{B} for the corresponding complete resolution _{A}Δ̂_{A} ⊗_{A} _{A}B_{B}. Similarly, if we regard B as a B-A-bimodule, we have a free resolution _{B}Δ_{A} = _{B}B_{A} ⊗_{A} _{A}Δ_{A} and a complete resolution _{B}Δ̂_{A} = _{B}B_{A} ⊗_{A} _{A}Δ̂_{A}. The duality (11.7.1) gives

(11.7.3)  Hom_{A⊗_{k}B}(_{A}Δ̂_{B}, A ⊗_{k} B) ≅ _{B}Δ̂_{A}[−r]

and the periodicity (11.7.2) gives

(11.7.4)  _{A}Δ̂_{B}[2] ≅ _{A}Δ̂_{B}(1).

Finally, for any left A-module N we have a free resolution _{A}Δ_{A} ⊗_{A} N and a complete resolution _{A}Δ̂_{A} ⊗_{A} N.

11.8. The Adjunction

Proposition 11.8.1. The functor Ψ is right adjoint to Φ.

Proof. Let M_{∗} be a bounded complex of C-modules. If N is a maximal Cohen– Macaulay A-module, then _{A}Δ̂_{A} ⊗_{A} N is a complete resolution of N as an A-module. Then

Ψ(N) = (_{B}B_{A} ⊗_{A} (_{A}Δ̂_{A} ⊗_{A} N))_{0}[−r] ≅ (_{B}Δ̂_{A} ⊗_{A} N)_{0}[−r].

Write \underline{Hom} for homomorphisms of complexes modulo homotopy. Since (_{B}Δ̂_{A} ⊗_{A} N)_{0}[−r] is semi-injective, homomorphisms in D^{b}(C) from an object to it are just homotopy classes of maps of complexes. So using the duality (11.7.3), we have

Hom_{D^{b}(C)}(M_{∗}, Ψ(N)) = \underline{Hom}_{C}^{n}(M_{∗}, (_{B}Δ̂_{A} ⊗_{A} N)_{0}[−r])
≅ \underline{Hom}_{B}(k[y_{1}, …, y_{r}] ⊗_{k} M_{∗},_{B}Δ̂_{A} ⊗_{A} N[−r])
≅ \underline{Hom}_{B}(k[y_{1}, …, y_{r}] ⊗_{k} M_{∗}, Hom_{A}(_{A}Δ̂_{B}, N))
≅ \underline{Hom}_{A}(_{A}Δ̂_{B} ⊗_{B} (k[y_{1}, …, y_{r}] ⊗_{k} M_{∗}), N)
≅ \underline{Hom}_{A}(Φ(M), N).

In the last line, we are using the fact that

AΔ̂_{B} ⊗_{B} (k[y_{1}, …, y_{r}] ⊗_{k} M_{∗}) = _{A}Δ̂_{A} ⊗_{A} (k[y_{1}, …, y_{r}] ⊗_{k} M_{∗})

is a complete resolution over A of a maximal Cohen–Macaulay approximation to the complex k[y_{1}, …, y_{r}] ⊗_{k} M_{∗}. □

11.9. The Equivalence

Proposition 11.9.1. The category D_{sg}(A) is generated by the A-module k[y_{1}, …, y_{r}].

Proof. The singular locus of A is defined by the equations \frac{∂f}{∂y_{i}} = 0 and \frac{∂f}{∂X_{i}} = 0. The former give the equations f_{1} = 0, …, f_{r} = 0; since C is a zero dimensional complete intersection, these equations define the same variety as X_{1} = 0, …, X_{r} = 0. The latter give the equations ∑_{j} y_{j} \frac{∂f_{j}}{∂X_{i}} = 0. Since f_{1}, …, f_{r} are in the square of the maximal ideal, \frac{∂f_{j}}{∂X_{i}} has zero constant term and so no new conditions are imposed by these equations. So we have shown that the prime ideals in the singular locus are those containing (X_{1}, …, X_{r}).

By Theorem 1.7.7, the singularity category D_{sg}(A) is generated by the modules A/𝔭 with 𝔭 ⊇ (X_{1}, …, X_{r}), namely the quotients of k[y_{1}, …, y_{r}] by prime ideals. These in turn are generated by the single object k[y_{1}, …, y_{r}], by the Hilbert syzygy theorem. □

Theorem 11.9.2. The composite functor Φ∘Ψ: D_{sg}(A) → D_{sg}(A) is naturally isomorphic to the identity functor.

Proof. Consider the adjunction of Proposition 11.8.1. By Theorem 11.6.3, the unit of this adjunction gives an isomorphism k → ΨΦ(k), where k is the residue field of C, regarded as an object in D^{b}(C) by putting it in degree zero. It follows that ΦΨ(Φ(k)) = Φ(ΨΦ(k)) ≅ Φ(k). An easy diagram chase shows that this isomorphism is given by the counit of the adjunction. It follows that the counit of the adjunction is an isomorphism for every object in the thick subcategory of D_{sg}(A) generated by Φ(k). Since Φ(k) ≅ k[y_{1}, …, y_{r}], by Proposition 11.9.1 this is the whole of D_{sg}(A). □

Theorem 11.9.3. The functors Φ: D^{b}(C) → D_{sg}(A) and Ψ: D_{sg}(A) → D^{b}(C) are inverse equivalences of categories.

Proof. This follows from Theorems 11.6.3 and 11.9.2. □

Theorem 11.9.4. The equivalence \xymatrix{D^{b}(C) \ar[r]^{Φ} & D_{sg}(A) \ar[l]^{Ψ}} descends to an equivalence

\xymatrix{stmod(C) \ar[r]^{Φ̄} & \underline{\underline{MCM}}(A) \ar[l]^{Ψ̄}} .

Proof. It follows from Theorem 11.6.2 that Φ(C) is a shift of k[X_{1}, …, X_{r}]. The theorem now follows, because stmod(C) is the quotient of D^{b}(C) by the thick subcategory generated by C and \underline{\underline{MCM}}(A) is the quotient of \underline{MCM}(A) by the thick subcategory generated by k[X_{1}, …, X_{r}]. □

11.10. The Trivial Module

So, in the case C = kE, where does the trivial kE-module k go to under the correspondence of Theorem 11.9.3? To answer this, we must find the minimal resolution of k[y_{1}, …, y_{r}] as an A-module. This is again given in terms of a bidirectional Koszul complex, this time for the pair of sequences X_{1}, …, X_{r} and y_{1}X_{1}^{p−1}, …, y_{r}X_{r}^{p−1}. The sum of the products of corresponding terms in these sequences again gives f = ∑_{i} y_{i}X_{i}^{p}, and so we obtain a matrix factorisation of f by taking the even and odd parts of this bidirectional Koszul complex.

Adding up the even and odd binomial coefficients, we see that this is a 2^{r−1} × 2^{r−1} matrix factorisation of f. The minimal resolution of k[y_{1}, …, y_{r}] as an A-module is given by (11.6.7) with respect to this version of the bidirectional Koszul complex.

In a similar way, we can find the image of any module of the form kE/(X_{1}^{a_{1}}, …, X_{r}^{a_{r}}) under the correspondence of Theorem 11.9.3 by doing the same process with a bidirectional Koszul complex for the pair of sequences X_{1}^{a_{1}}, …, X_{r}^{a_{r}} and y_{1}X_{1}^{p−a_{1}}, …, y_{r}X_{r}^{p−a_{r}}.

Let us look at some examples. First we look at the case r = 1. In this case E is cyclic of order p, kE = k[X]/(X^{p}), A = k[y, X]/(yX^{p}) and B = k[y, X]/(X^{p}). The indecomposable kE-modules are the Jordan blocks J_{n} = k[X]/(X^{n}) (1 ≤ n ≤ p). Resolving J_{n} ⊗_{k} k[y] we get

⋯ → A[−1] ⟶^{X^{n}} A[−1] ⟶^{yX^{p−n}} A ⟶^{X^{n}} A.

So the matrix factorisation corresponding to J_{n} is given by the 1 × 1 matrices

A[−1] ⟶^{(yX^{p−n})} A ⟶^{(X^{n})} A.

Next let r = 2, so that A = k[y_{1}, y_{2}, X_{1}, X_{2}]/(y_{1}X_{1}^{p} + y_{2}X_{2}^{p}). To find the matrix factorisation corresponding to the trivial module, we resolve the A-module k[y_{1}, y_{2}] = A/(X_{1}, X_{2}). Using the construction in Section 11.6, we obtain the following minimal resolution:

A[−1] ⊕ A[−1] \underrightarrow{(\begin{matrix}y_{2}X_{2}^{p−1} & −y_{1}X_{1}^{p−1} \\ X_{1} & X_{2} \end{matrix})} A ⊕ A[−1] \underrightarrow{(\begin{matrix}X_{2} & y_{1}X_{1}^{p−1} \\ −X_{1} & y_{2}X_{2}^{p−1} \end{matrix})} A ⊕ A \underrightarrow{(X_{1} X_{2})} A.

The two square matrices in this resolution alternate, and so the matrix factorisation corresponding to the trivial module is as follows:

A ⊕ A[−1] \underrightarrow{(\begin{matrix}X_{2} & y_{1}X_{1}^{p−1} \\ −X_{1} & y_{2}X_{2}^{p−1} \end{matrix})} A ⊕ A \underrightarrow{(\begin{matrix}y_{2}X_{2}^{p−1} & −y_{1}X_{1}^{p−1} \\ X_{1} & X_{2} \end{matrix})} A[1] ⊕ A.

In rank three, the minimal resolution of k[y_{1}, y_{2}, y_{3}] as an A-module takes the following form.

A[−1]^{⊕3} ⊕ A[−2] \underrightarrow{(\begin{matrix}0 & X_{3} & −X_{2} & y_{1}X_{1}^{p−1} \\ −X_{3} & 0 & X_{1} & y_{2}X_{2}^{p−1} \\ X_{2} & −X_{1} & 0 & y_{3}X_{3}^{p−1} \\ y_{1}X_{1}^{p−1} & y_{2}X_{2}^{p−2} & y_{3}X_{3}^{p−1} & 0 \end{matrix})} A[−1]^{⊕3} ⊕ A \underrightarrow{(\begin{matrix}0 & −y_{3}X_{3}^{p−1} & y_{2}X_{2}^{p−1} & X_{1} \\ y_{3}X_{3}^{p−1} & 0 & −y_{1}X_{1}^{p−1} & X_{2} \\ −y_{2}X_{2}^{p−1} & y_{1}X_{1}^{p−1} & 0 & X_{3} \\ X_{1} & X_{2} & X_{3} & 0 \end{matrix})}

A^{⊕3} ⊕ A[−1] \underrightarrow{(\begin{matrix}0 & X_{3} & −X_{2} & y_{1}X_{1}^{p−1} \\ −X_{3} & 0 & X_{1} & y_{2}X_{2}^{p−1} \\ X_{2} & −X_{1} & 0 & y_{3}X_{3}^{p−1} \end{matrix})} A^{⊕3} \underrightarrow{(X_{1} X_{2} X_{3})} A

The left-hand pair of matrices therefore gives the matrix factorisation of f corresponding to the trivial kE-module.

11.11. Computer Algebra

Here is some code in the computer algebra language Macaulay2 for computing the functor mod(kE) ↪ D^{b}(kE) ⟶^{Φ} \underline{RMF}(f). It is given here for the trivial module for a group of rank two with p = 7, but the code is easy to modify. The last two commands print out the sixth and seventh matrices in the minimal resolution, which in this case is easily far enough to give a matrix factorisation.

p=7
R = ZZ/p[X1,X2,y1,y2]
f = X1^p * y1 + X2^p * y2
A = R/(f)
U = cokernel matrix {{X1,X2}}
F = resolution (U,LengthLimit=>8)
F.dd_6
F.dd_7

To modify the code to work for other kE-modules M, the fifth line should be changed to give a presentation of

U = k[y_{1}, …, y_{r}] ⊗_{k} M

as an A-module. Don’t forget the relations saying that X_{i}^{p} annihilates M. For example, if M = Ω(k) for the same group of rank two above, then U has two generators and three relations; it is the cokernel of the matrix

(\begin{matrix}X_{1}^{p} & X_{2} & 0 \\ 0 & X_{1} & X_{2}^{p} \end{matrix}) : A^{⊕3} → A^{⊕2}.

So the fifth line should be changed to

U = cokernel matrix {{X1^p,X2,0},{0,X1,X2^p}}.

11.12. Cohomology

For this section, we stick with the case C = kE. The elements y_{1}, …, y_{r} ∈ A act on maximal Cohen–Macaulay modules N as maps N → N(1). Now N(1) is isomorphic to Ω^{−2}(N), and Ω^{−1} is the shift functor in the triangulated category \underline{MCM}(A). It follows that under equivalence of categories of Theorem 11.9.3, these elements correspond to maps in D^{b}(kE) from M_{∗} to M_{∗}[2]. We claim that these elements act as the polynomial part of the cohomology ring, namely the subring generated by the Bocksteins of the elements of degree one.

Recall that kE is a Hopf algebra, either via the group theoretic diagonal map defined by Δ(g_{i}) = g_{i} ⊗ g_{i}, Δ(X_{i}) = X_{i} ⊗ 1 + 1 ⊗ X_{i} + X_{i} ⊗ X_{i} or via the restricted Lie algebra diagonal map defined by Δ̃(X_{i}) = X_{i} ⊗1+1⊗X_{i}. In both cases, we make A into a right kE-comodule via Δ, Δ̃: A → A⊗kE defined by the same formula on X_{i} and via Δ(y_{i}) = Δ̃(y_{i}) = y_{i} ⊗1. It is easy to check that this is a coaction: (1⊗Δ)∘Δ = (Δ⊗1)∘Δ and (1⊗Δ̃)∘Δ̃ = (Δ̃⊗1)∘Δ̃. We denote the corresponding tensor products N ⊗_{k} M and N ˜⊗_{k} M, where M is a kE-module and N and the tensor product are A-modules.

Lemma 11.12.1. If N is a maximal Cohen–Macaulay A-module and M is a kE-module then N ⊗_{k} M and N ˜⊗_{k} M are maximal Cohen–Macaulay A-modules.

Proof. The kE-module M has a finite filtration in which the filtered quotients are copies of the trivial kE-module k. So the tensor products N ⊗_{k} M and N ˜⊗_{k} M both have finite filtrations in which the quotients are isomorphic to N. The lemma now follows from the fact that every extension of maximal Cohen–Macaulay modules is maximal Cohen–Macaulay. □

Recall that the maximal Cohen–Macaulay module K corresponding to the Koszul factorisation (11.6.4) is the image of the trivial kE-module k under Φ.

Proposition 11.12.2. If M is a kE-module then Φ(M) may be taken to be the maximal Cohen–Macaulay module K ⊗_{k} M (or K ˜⊗_{k} M).

Proof. It follows from Lemma 11.12.1 that K ⊗_{k} M, resp. K ˜⊗_{k} M is a maximal Cohen– Macaulay approximation to k[y_{1}, …, y_{r}] ⊗_{k} M. □

Let V be the linear space spanned by X_{1}, …, X_{r}. Then we may use the polynomial f to identify the space spanned by y_{1}, …, y_{r} with the Frobenius twist of the dual F (V^{∗}). There is an action of GL(V) on kE induced by linear substitutions of the X_{i}, and this induces an action on the linear space F (V^{∗}) spanned by the y_{i}.

Theorem 11.12.3. (i) The maps

Ψ(y_{1}), …, Ψ(y_{r}): k → k[2]

form a vector space basis for the image of the Bockstein map H^{1}(E, k) → H^{2}(E, k). (ii) For any M in stmod(kE), the induced map

Ψ̄(y_{i}): M → Ω^{−2}(M)

is equal to the map

Ψ̄(y_{i}) ⊗ 1: k ⊗_{k} M → Ω^{−2}(k) ⊗ M

and also to the map

Ψ̄(y_{i}) ˜⊗ 1: k ˜⊗_{k} M → Ω^{−2}(k) ˜⊗ M.

Proof. (i) Consider the action of GL(V) on H^{2}(E, k). For p odd, we have

H^{2}(E, k) ≅ F (V^{∗}) ⊕ Λ^{2}(V^{∗}).

For p = 2, we have H^{2}(E, k) ≅ S^{2}(V); in this case S^{2}(V^{∗}) has two composition factors as a GL(V)-module, given by the non-split short exact sequence

0 → F (V^{∗}) → S^{2}(V^{∗}) → Λ^{2}(V^{∗}) → 0.

In both cases, there is a unique GL(V)-invariant subspace of H^{2}(E, k) isomorphic to F (V^{∗}), and this is the image of the Bockstein map. It therefore suffices to prove that the Ψ̄(y_{i}) are not all equal to zero. To see this, take the Koszul complex for K with respect to the parameters y_{1}, …, y_{r}. Since K is a maximal Cohen–Macaulay approximation to k[y_{1}, …, y_{r}], the homology of this complex disappears after applying Ψ̄. If Ψ̄(y_{i}) were zero, this could not be the case. It follows that Ψ̄(y_{i}) ≠ 0 and the theorem is proved.

(ii) This follows from Proposition 11.12.2. □

Remark 11.12.4. Theorem 11.12.3 gives an interpretation of the realisation theorem of Benson and Pevtsova, Theorem 8.1.1. Namely, given any k[y_{1}, …, y_{r}]-module N, inflating it and applying Ψ̄ to a maximal Cohen–Macaulay approximation to the resulting A-module gives the same kE-module M as the process described in Section 8.8. In particular, it follows from the analysis in Section 8.9 that ℱ_{1}(M) is the Frobenius twist of the sheaf on ℙ^{r−1} corresponding to N and ℱ_{i}(M) = 0 for 1 < i < p. So if N is locally free over k[y_{1}, …, y_{r}] then M has constant Jordan type.

11.13. Twisted Versions of kE

In this section we indicate how to modify the constructions of this chapter to deal with the twisted versions of kE discussed in Section 1.18. This is similar to the modifications of the Koszul complex used by Avramov, Gasharov and Peeva in §2 of [23]. Recall that q ∈ Mat_{r}(k) is a matrix satisfying q_{ij}q_{ji} = 1 for 1 ≤ i, j ≤ r and q_{ii} = 1 for 1 ≤ i ≤ r, n = (n_{1}, …, n_{r}) is an r-tuple of strictly positive integers, and C_{q,n} is the algebra with generators X_{1}, …, X_{r} satisfying X_{i}X_{j} = q_{ij}X_{j}X_{i} for 1 ≤ i, j ≤ r and X_{i}^{p^{n_{i}}} = 0 for 1 ≤ i ≤ r; these algebras arise in discussions of blocks with normal abelian defect group, abelian inertial quotient and one isomorphism class of simple module.

In this case, we let R be the twisted polynomial ring with generators y_{1}, …, y_{r}, X_{1}, …, X_{r} satisfying the following relations:

(11.13.1)  y_{i}y_{j} = q_{ji}^{p^{n_{i}+n_{j}}} y_{j}y_{i}; X_{i}X_{j} = q_{ij}X_{j}X_{i}; X_{j}y_{i} = y_{i}X_{j}.

with the variables X_{i} in degree zero and the variables y_{i} in degree one. Thus y_{1}X_{1}^{p^{n_{1}}}, …, y_{r}X_{r}^{p^{nr}} are commuting elements of R, and we set f = y_{1}X_{1}^{p^{n_{1}}} + ⋯ + y_{r}X_{r}^{p^{nr}}.

In the bidirectional Koszul complex, the sense in which the wedges are alternating is modified to

e_{i} ∧ e_{j} = −q_{ij}^{p^{n_{i}+n_{j}}} e_{j} ∧ e_{i}.

The definition of the differential d in (11.6.1) then becomes

d(e_{j_{1}} ∧ ⋯ ∧ e_{j_{n}}) = ∑_{i}(−1)^{i−1} (∏_{t=1}^{i−1} q_{j_{t}j_{i}}^{p^{n_{j_{t}} +n_{j_{i}}}} ) (e_{j_{1}} ∧ … ↑^{j_{i}} ⋯ ∧ e_{j_{n}})

while the definition of δ in (11.6.2) needs no alteration.

With these definitions, we obtain the following theorem.

Theorem 11.13.1. The following triangulated categories are equivalent:

(1) The derived category D^{b}(C_{q,n}).

(2) The singularity category of graded A-modules D_{sg}(A) where A = R/f, R is the twisted polynomial ring with generators y_{1}, …, y_{r} in degree one, X_{1}, …, X_{r} in degree zero, and commutation relations (11.13.1), and f = y_{1}X_{1}^{p^{n_{1}}} + ⋯ + y_{r}X_{r}^{p^{nr}}.

(3) The stable category of maximal Cohen–Macaulay graded A-modules.

(4) The category of reduced graded matrix factorisations of f over R.

Proof. The proofs are the same as in the untwisted case, with the modifications outlined above. □

12 Phenomenology of Modules over Elementary Abelian p-Groups

12.1. Introduction

This chapter is about the following question. How small can a module be, and still have an interesting variety? Methods for producing such modules are similar to those used in discussions of constant Jordan type in previous chapters. We consider various measures of size, including dimension and Loewy length.

Example 12.1.1. My interest in this question was kindled by the discovery that the 42 dimensional Specht module S^{(3^{3})} for the symmetric group 𝔖_{9} in characteristic 3, when restricted to the elementary abelian subgroup (ℤ/3)^{3}, has variety given by the equation

Y_{1}^{2}Y_{2}^{2} + Y_{1}^{2}Y_{3}^{2} + Y_{2}^{2}Y_{3}^{2} = 0

in affine space 𝔸^{3}. This was computed by Jon Carlson using the computer algebra package Magma [63] around 2005, and an abstract argument appears in the paper of Lim [187].

The question is, in odd characteristic, how can a module of such small dimension have such an interesting variety? All the methods we knew, at the time when this was computed, produced much larger modules. We shall see later in this chapter, that there is in fact a module of dimension 36 for (ℤ/3)^{3} with the same variety.

Let k be an algebraically closed field of characteristic p, and let E = (ℤ/p)^{r} be an elementary abelian p-group of rank r. We use the notation from Sections 1.3 and 1.9. In particular, E = ⟨g_{1}, …, g_{r}⟩, X_{i} = g_{i} − 1, so that

kE = k[X_{1}, …, X_{r}]/(X_{1}^{p}, …, X_{r}^{p}).

The affine space V_{E}^{rk} ≅ 𝔸^{r}(k) has X_{1}, …, X_{r} as a basis, and its coordinate ring is k[V_{E}^{rk}] = k[Y_{1}, …, Y_{r}], regarded as a graded ring with each Y_{i} in degree one. Thus Y_{i} is the function on V_{E}^{rk} defined by Y_{i}(X_{j}) = δ_{i,j} (Kronecker delta). If α = (λ_{1}, …, λ_{r}) ∈ V_{E}^{rk} then we write X_{α} for λ_{1}X_{1} + ⋯ + λ_{r}X_{r}.

We can think of Y_{1}, …, Y_{r} as corresponding to the elements y_{1}, …, y_{r} in H^{1}(E, k) = Ext_{kE}^{1}(k, k), see Section 1.8. If p = 2, these generated the polynomial ring H^{∗}(E, k), and so we may identify H^{∗}(E, k) with k[V_{E}^{rk}]. If p is odd, the y_{i} anticommute, and their Bocksteins x_{i} generate the polynomial part of the cohomology ring. Since the Bockstein map is semilinear with respect to the Frobenius map, this means that the x_{i} correspond with Y_{i}^{p}, as we saw in Chapter 8.

Example 12.1.2. Let W be a linear subspace of V_{E}^{rk} of dimension c. Choose a basis X_{β_{1}}, …, X_{β_{c}} for W, and let

kE′ = k[X_{β_{1}}, …, X_{β_{c}}]/(X_{β_{1}}^{p}, …, X_{β_{c}}^{p}).

This is a subalgebra flatly embedded in kE. Let M(W) be the kE-module kE ⊗_{kE′} k. Then dim_{k}(M(W)) = p^{r−c}, the Loewy length of M(W) is (p−1)(r −c)+1, and V_{E}^{rk}(M(W)) = W.

Let U be a linear subspace of V_{E}^{rk} of dimension r − c such that U ∩ W = {0}. Choose a basis α_{1}, …, α_{r−c} for U, and set

(12.1.1)  kE′′ = k[X_{α_{1}}, …, X_{α_{r−c}}]/(X_{α_{1}}^{p}, …, X_{α_{r−c}}^{p}).

Then M(W)↓_{kE′′} is a free module of rank one.

Similarly in the exterior algebra case, if W is a linear subspace of V_{Λ}^{rk} of dimension c with basis X_{β_{1}}, …, X_{β_{c}}, let

Λ′ = Λ(X_{β_{1}}, …, X_{β_{c}})

and let M(W) be the Λ-module Λ ⊗_{Λ′} k. Then dim_{k}(M(W)) = 2^{r−c}, the Loewy length of M(W) is r − c + 1, and V_{Λ}^{rk}(M(W)) = W.

These modules have as small a dimension as possible among modules with rank variety W, as the following theorem shows.

Theorem 12.1.3 (Carlson [78, §4]). If M is a kE-module such that V_{E}^{rk}(M) has dimension m and degree d then dim_{k} M/p^{r−m} is an integer at least as big as d.

Proof. By Bézout’s theorem, a generic subspace V′ ⊆ V_{E}^{rk} of dimension r − m + 1 intersects V_{E}^{rk}(M) in d distinct lines through the origin. So by parts (4) and (6) of Proposition 1.9.11, M(V) ⊗_{k} M decomposes as a direct sum of d modules, each of whose rank varieties is a single line through the origin. For such a summand, the restriction to a subgroup of rank r − 1 not containing this line is projective, so the dimension is divisible by p^{r−1}. So dim_{k}(M(V) ⊗_{k} M) = p^{m−1} dim_{k}(M) is p^{r−1} times an integer at least d. □

Essentially the same proof gives the following.

Theorem 12.1.4. If M is a kE-module such that V_{E}^{rk}(M) has an irreducible component of dimension m and degree d then dim_{k}(M) ≥ d.p^{r−m}.

Remarks 12.1.5. The conclusion no longer states that dim_{k}(M) is divisible by p^{r−m}, because of the possible existence of other irreducible components of V_{E}^{rk}(M) of larger dimension than m.

The adjustment of the proof is to notice that V′ intersects the given irreducible component of V_{E}^{rk}(M) in d distinct lines through the origin, none of which is in any other irreducible component.

The corresponding theorem for exterior algebras is as follows.

Theorem 12.1.6. Let Λ = Λ(X_{1}, …, X_{r}).

(i) If M is a Λ-module such that V_{Λ}^{rk}(M) has dimension m and degree d, then dim_{k} M/2^{r−m} is an integer at least as big as d.

(ii) If M is a Λ-module such that V_{Λ}^{rk}(M) has an irreducible component of dimension m and degree d then dim_{k}(M) ≥ d.2^{r−m}.

Given this situation, it is natural to regard the rank variety of a kE-module or of a Λ-module as “interesting” if it is not a finite union of linear subspaces of V_{E}^{rk}, respectively V_{Λ}^{rk}.

One surprising aspect of the discussion is that it seems to be much easier to produce small modules for elementary abelian groups with interesting varieties in characteristic two than in odd characteristic. In a lot of ways, in characteristics other than two, exterior algebras are much closer in behaviour to elementary abelian 2-groups than elementary abelian p-groups are. For this reason, we include these in our discussions whenever possible.

12.2. Module Constructions

We begin by looking at Carlson’s “favourite example” [73, pp. 418–9], for E = (ℤ/2)^{3} over a field k of characteristic two. The representing matrices in this example are given by

g_{1} ↦ (\begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \end{matrix}) g_{2} ↦ (\begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ a & 0 & 1 & 0 \\ 0 & b & 0 & 1 \end{matrix}) g_{3} ↦ (\begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & c & 1 & 0 \\ 1 & 0 & 0 & 1 \end{matrix}) .

This module is an extension of k ⊕ k by k ⊕ k, so it is determined by a 2 × 2 matrix of elements of Ext_{kE}^{1}(k, k). This matrix is

(\begin{matrix}y_{1} + ay_{2} & y_{3} \\ cy_{3} & y_{1} + by_{2} \end{matrix}) .

In this way he obtained a three-parameter family of modules of dimension four whose varieties are the quadrics

(12.2.1)  (Y_{1} + aY_{2})(Y_{1} + bY_{2}) = cY_{3}^{2}.

To verify this, we note that

X_{1} ↦ (\begin{matrix}0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \end{matrix}) X_{2} ↦ (\begin{matrix}0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ a & 0 & 0 & 0 \\ 0 & b & 0 & 0 \end{matrix}) X_{3} ↦ (\begin{matrix}0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & c & 0 & 0 \\ 1 & 0 & 0 & 0 \end{matrix})

and so

Y_{1}X_{1} + Y_{2}X_{2} + Y_{3}X_{3} ↦ (\begin{matrix}0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ Y_{1} + aY_{2} & Y_{3} & 0 & 0 \\ cY_{3} & Y_{1} + bY_{2} & 0 & 0 \end{matrix}) .

The rank of this is less than half the dimension of the module precisely when the determinant of the bottom left 2 × 2 block vanishes.

We can modify Carlson’s example by making the group a bit bigger, E = (ℤ/2)^{4}, and using the matrix

(\begin{matrix}y_{1} & y_{3} \\ y_{2} & y_{4} \end{matrix})

to get a module of dimension four whose variety is the ruled surface Y_{2}Y_{3} = Y_{1}Y_{4}. More explicitly, the matrices for this module are given by

g_{1} ↦ (\begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}) g_{2} ↦ (\begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 1 \end{matrix}) g_{3} ↦ (\begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 1 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}) g_{4} ↦ (\begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \end{matrix}) .

The module M defined by these matrices sits in a triangle in stmod(kE) of the form

Ω(k) ⊕ Ω(k) \underrightarrow{(\begin{matrix}y_{1} & y_{3} \\ y_{2} & y_{4} \end{matrix})} k ⊕ k → M → k ⊕ k.

Restricting to shifted subgroups of kE of rank three gives the modules in Carlson’s family.

This construction generalises in an obvious way to give a 2n dimensional module for an elementary abelian group of rank n^{2} whose variety is given by the vanishing of the n × n generic determinant in n^{2} variables. This variety has dimension n^{2} − 1 = r − 1 and degree n, so this module hits the Bézout bound from Theorem 12.1.3.

12.3. Odd Primes Are More Difficult

The mysterious question at this point is this. If we try to make the same sort of example for p odd, it is impossible to produce a module of dimension 2p over an elementary abelian group of order p^{3} whose variety is given by equation (12.2.1). So the bound from Theorem 12.1.3 is not sharp in this case. More generally, we have the following theorem:

Theorem 12.3.1. If p is an odd prime, E ≅ (ℤ/p)^{r} and M is a kE-module of dimension 2p then V_{E}^{rk}(M) is the whole of affine space 𝔸^{r}(k), a linear subspace of dimension r − 1, or the union of two such subspaces intersecting in a linear subspace of dimension r − 2.

Proof. First we note that V_{E}^{rk}(M) has to have dimension r − 1 or r by Theorem 12.1.3, because dim_{k}(M) = 2p < p^{2}.

Now suppose that V_{E}^{rk}(M) is not the whole of 𝔸^{r}(k). Then some linear combination of X_{1}, …, X_{r} acts as the matrix

(\begin{matrix}0 & 0 & & & & & & 0 & 0 \\ 0 & 0 & & & & & & 0 & 0 \\ 1 & 0 \\ 0 & 1 \\ & & . & . & . \\ 0 & 0 & & & & 1 & 0 & 0 & 0 \\ 0 & 0 & & & & 0 & 1 & 0 & 0 \end{matrix}) .

The action of each X_{i} commutes with this matrix, and is hence of the form

(\begin{matrix}A_{1,i} & 0 & … & 0 \\ A_{2,i} & A_{1,i} & & 0 \\ ⋮ & & ... & ⋮ \\ A_{p,i} & A_{p−1,i} & ⋯ & A_{1,i} \end{matrix})

where each A_{n,i} is a 2 × 2 matrix.

The matrices A_{1,i} all commute and are nilpotent, and so after a change of basis within the 2 × 2 blocks if necessary, we have

A_{1,i} = (\begin{matrix}0 & 0 \\ λ_{i} & 0 \end{matrix}) .

Now we have two cases. The first case is where some λ_{i} is non-zero. Then the fact that A_{1,i}A_{n,j} = A_{n,j}A_{1,i} forces all the A_{n,j} to be lower triangular. So M has a submodule M_{1}, spanned by the even-numbered basis vectors, of dimension p. Then V_{E}^{rk}(M_{1}) and V_{E}^{rk}(M/M_{1}) are both hyperplanes. So V_{E}^{rk}(M) is a subvariety of codimension one contained in the union of these two hyperplanes (which may be equal), and is therefore equal to this union.

The second case is where all the λ_{i} are equal to zero. In this case we have A_{2,i}A_{2,j} = A_{2,j}A_{2,i}, and so after a change of basis within the 2 × 2 blocks if necessary, the matrices A_{2,i} are simultaneously lower triangular:

A_{2,i} = (\begin{matrix}μ_{i} & 0 \\ κ_{i} & ν_{i} \end{matrix}) .

Then V_{E}^{rk}(M) is given by the vanishing of the (2p − 2) × (2p − 2) minor in the lower left of the matrix, and therefore has the form

(μ_{1}Y_{1} + ⋯ + μ_{r}Y_{r})^{p−1}(ν_{1}Y_{1} + ⋯ + ν_{p}Y_{p})^{p−1} = 0.

It is therefore either a single hyperplane or a union of two hyperplanes. □

12.4. Relative Cohomology

In this section, we present a variation on Carlson’s L_{ζ} modules from Section 1.10, making use of relative cohomology.

Let W be a linear subspace of V_{E}^{rk} of dimension c, and let X_{β_{1}}, …, X_{β_{c}} be a basis for W. Then we have a shifted subgroup E′ generated by g_{β_{1}} = 1 + X_{β_{1}}, …, g_{β_{c}} = 1 + X_{β_{c}}, and

kE′ = k[X_{β_{1}}, …, X_{β_{c}}]/(X_{β_{1}}^{p}, …, X_{β_{c}}^{p})

is flatly embedded in kE. If U is a linear subspace of dimension r − c complementary to W, then choosing a basis X_{α_{1}}, …, X_{α_{r−c}} for U, we have another shifted subgroup E′′ generated by g_{α_{1}} = 1 + X_{α_{1}}, …, g_{α_{r−c}} = 1 + X_{α_{r−c}} and a flat embedding

kE′′ = k[X_{α_{1}}, …, X_{α_{r−c}}]/(X_{α_{1}}^{p}, …, X_{α_{r−c}}^{p}) → kE.

Then

kE = k[X_{β_{1}}, …, X_{β_{c}}, X_{α_{1}}, …, X_{α_{r−c}})/(X_{β_{1}}^{p}, …, X_{β_{c}}^{p}, X_{α_{1}}^{p}, …, X_{α_{r−c}}^{p}).

Letting E_{0} be the shifted subgroup of kE generated by E′ and E′′, we have

kE = kE_{0} = kE′ ⊗_{k} kE′′.

The cohomology of E relative to E′ is given by

H^{∗}(kE, kE′, k) ≅ H^{∗}(E_{0}/E′, k) ≅ H^{∗}(E′′, k).

The relative syzygies are the same as the syzygies for the quotient Ω_{E_{0}/E′}^{i}(k). There are natural maps Ω_{E}^{i}(k) → Ω_{E_{0}/E′}^{i}(k).

Now consider a matrix (ζ_{ij}) of elements of H^{∗}(E, k) satisfying Condition 1.10.5, and the corresponding triangle

⨁_{i} Ω^{a_{i}+1}(k) → L_{(f_{ij})} → ⨁_{j} Ω^{b_{j}}(k) ⟶^{(f̂_{ij})} ⨁_{i} Ω^{a_{i}}(k)

in stmod(kE). Since we want to replace syzygies by relative syzygies, we had better work in mod(kE) rather than stmod(kE). So consider the extension

(12.4.1)  0 → ⨁_{i} Ω^{a_{i}+1}(k) → L_{(f_{ij})} → ⨁_{j} Ω^{b_{j}}(k) → 0

corresponding to

(ζ_{ij}) ∈ Ext_{kE}^{1}(⨁_{j} Ω^{b_{j}}(k), ⨁_{i} Ω^{a_{i}+1}(k)) ≅ \underline{Hom}_{kE}(⨁_{j} Ω^{b_{j}}(k), ⨁_{i} Ω^{a_{i}}(k)).

Here, the module L_{(f_{ij})} is a particular representative of the object in stmod(kE) with the same name, and it may have projective summands at this stage. That may change when we replace syzygies by relative syzygies in the extension, as we are about to do.

First we discuss the case p = 2.

Let i index a row of the matrix (f_{ij}). Let W be a subspace of V_{E}^{rk} contained in the kernels of the functions f_{ij}(X_{1}, …, X_{r}) for 1 ≤ j ≤ n. Then the cohomology elements f_{ij}(x_{1}, …, x_{r}) (1 ≤ j ≤ n) are inflated from E_{0}/E′, and hence may be regarded as elements of relative cohomology H^{∗}(kE, kE′, k). We may make use of this under the following condition:

Condition 12.4.1. For each value of j, either f_{ij} = 0 or b_{j} = 0.

Under this condition, whenever the extension f_{ij} is non-zero, we have Ω^{b_{j}}(k) = k. So we can replace these extensions with the corresponding ones for Ω_{E_{0}/E′}^{a_{i}+1}(k) representing the elements of relative cohomology.

Similarly, let j index a column of the matrix (f_{ij}) and let W be a subspace of V_{E}^{rk} contained in the kernels of the functions f_{ij}(X_{1}, …, X_{r}) for 1 ≤ i ≤ n. Then the cohomology elements f_{ij}(x_{1}, …, x_{r}) (1 ≤ i ≤ n) are inflated from E_{0}/E′, and hence may be regarded as elements of relative cohomology H^{∗}(kE, kE′, k). We may make use of this under the following condition:

Condition 12.4.2. For each value of i, either f_{ij} = 0 or a_{i} = −1.

Under this condition, whenever the extension f_{ij} is non-zero, we have Ω^{a_{i}+1}(k) = k. So we can replace these extensions with the corresponding ones for Ω_{E_{0}/E′}^{b_{j}}(k) representing the elements of relative cohomology.

For p odd, we have to take into account the Frobenius map, because we are using f_{ij}(y_{1}, …, y_{r}) instead of f_{ij}(x_{1}, …, x_{r}). So in this case we should assume that W is a subspace of V_{E}^{rk} contained in the kernels of the functions f_{ij}(X_{1}^{p}, …, X_{r}^{p}) for 1 ≤ j ≤ n or for 1 ≤ i ≤ n in the two cases above. Apart from this, everything else works the same way.

Theorem 12.4.3. Let L′_{(ζ_{ij})} be the module obtained by the replacements described above in the short exact sequence (12.4.1). Then V_{E}^{rk}(L′_{(ζ_{ij})}) = V_{E}^{rk}(L_{(ζ_{ij})}), namely the hypersurface defined by the equation det(f_{ij}) = 0.

Proof. Outside of the set W, the computation is exactly the same as in Theorem 1.10.6. Inside W, the matrix has a zero row or a zero column, so the determinant vanishes. The map in the triangle representing the matrix is not an isomorphism because the relative syzygy restricts to a direct sum of more than one copy of the trivial module, so W ⊆ V_{E}^{rk}(L′_{(ζ_{ij})}). □

Of course, the same argument applies if we replace more than one of the summands in this manner.

12.5. Small Modules for Quadrics, p = 2

In this section, following Reid’s thesis [214], we examine small modules for an elementary abelian group of even rank E = (ℤ/2)^{r} (r = 2m) in characteristic two, with rank variety given by the irreducible quadric

(12.5.1)  Y_{1}Y_{2} + Y_{3}Y_{4} + ⋯ + Y_{2m−1}Y_{2m}.

The same discussion applies to an exterior algebra Λ on r = 2m generators in any characteristic.

Conjecture 12.5.1. If M is a Λ-module with rank variety V_{Λ}^{rk}(M) given by the hypersurface determined by the equation (12.5.1) then dim_{k}(M) ≥ 2^{m}.

Related theorems and conjectures appear in the literature on matrix factorisations. See for example [67, 178].

There are a number of ways of producing modules of dimension exactly 2^{m} with rank variety given by the hypersurface determined by (12.5.1).

The first is to use the construction given in [178]. Namely, we write down a pair A, B of 2^{m−1} × 2^{m−1} matrices of linear combinations of the Y_{i} whose product AB is a power of (12.5.1) times the identity (hence so is BA). Then the determinants of A and B are also powers of (12.5.1). Denote by A_{i} the specialisation of A with Y_{i} = 1 and Y_{j} = 0 for j ≠ i. Then the representation

g_{i} ↦ (\begin{matrix}I & 0 \\ A_{i} & I \end{matrix})

has Loewy length two and rank variety (12.5.1). To produce such matrices A and B we proceed by induction on m. For m = 2, the matrices

A = (\begin{matrix}Y_{1} & Y_{3} \\ Y_{2} & Y_{4} \end{matrix}) , B = (\begin{matrix}Y_{4} & −Y_{3} \\ −Y_{2} & Y_{1} \end{matrix} )

clearly work. For the inductive step, if A′ and B′ are 2^{m−2}×2^{m−2} matrices with this property for Y_{1}Y_{2} + ⋯ + Y_{2m−3}Y_{2m−2} then we take

A = (\begin{matrix}Y_{2m−1}I & A′ \\ B′ & −Y_{2m}I \end{matrix}) , B = (\begin{matrix}Y_{2m}I & A′ \\ B′ & −Y_{2m−1}I \end{matrix}) .

For example, if m = 3 the matrices A and B are given by

A = (\begin{matrix}Y_{5} & 0 & Y_{1} & Y_{3} \\ 0 & Y_{5} & Y_{2} & Y_{4} \\ Y_{4} & −Y_{3} & −Y_{6} & 0 \\ −Y_{2} & Y_{1} & 0 & −Y_{6} \end{matrix} ) B = (\begin{matrix}Y_{6} & 0 & Y_{1} & Y_{3} \\ 0 & Y_{6} & Y_{2} & Y_{4} \\ Y_{4} & −Y_{3} & −Y_{5} & 0 \\ −Y_{2} & Y_{1} & 0 & −Y_{5} \end{matrix} ) .

The second method for producing a module of this dimension is using the technique of Section 12.4. Consider the subgroup E′ = ⟨g_{1}, g_{3}, …, g_{2m−1}⟩ generated by the odd indexed generators of E. The corresponding subspace W of V_{E}^{rk} spanned by X_{1}, X_{3}, …, X_{2m−1} has the property that the function (12.5.1) vanishes on it. The element

y_{1}y_{2} + ⋯ + y_{2m−1}y_{2m} ∈ Ext_{kE}^{1}(Ωk, k) ≅ H^{2}(E, k)

therefore descends to an element of Ext_{kE}^{1}(Ω_{E/E′}(k), k). So we obtain a short exact sequence

0 → k → M → Ω_{E/E′}(k) → 0.

It follows from Theorem 12.4.3 that V_{E}^{rk}(M) is the hypersurface defined by the element (12.5.1). The dimension of this module is one more than the dimension of Ω_{E/E′}(k), so we have dim_{k}(M) = 2^{m}.

Here is a diagram for the case m = 3, with dim_{k}(M) = 8.

[figure: •; 2; 4; 5; •; 6; 4; 1; •; 6; 2; 3; •; 4; •; 2; •; 6; •; •]

There are an infinite number of ways of carrying out this second recipe. Namely, there is one for each choice of an isotropic subspace of dimension m for the quadratic form on V_{E}^{rk}, and none of them are isomorphic. The duals of these modules also have the same property and the same dimension, and are not isomorphic to any of these.

12.6. Small Modules for Quadrics, p Odd

It is known that given any closed homogeneous subvariety of affine r-space, there is a finitely generated kE-module with that as its rank variety. The general construction uses Carlson’s L_{ζ} technology, as in Section 1.10. But the modules produced this way are big. For example, if we were to use this method for the quadric (12.2.1), we would need an element of cohomology ζ of degree four. The corresponding module L_{ζ} is the kernel of a representative cocycle ζ̂: Ω^{4}k → k. This gives a module of dimension 6p^{3}. We shall see that by judicious modification of this technique, we can reduce the figure to 2p^{2}, but it seems hard to see how to get smaller than that.

For p odd, we can’t use Ext_{kE}^{1}(k, k) in this way for obvious reasons, so we have to use Ext_{kE}^{2}(k, k) ≅ Ext_{kE}^{1}(Ωk, k). Using the corresponding matrix of elements

(\begin{matrix}x_{1} & x_{3} \\ x_{2} & x_{4} \end{matrix}) ,

we obtain an extension of k ⊕ k by Ωk ⊕ Ωk whose variety is given by the irreducible quadric Y_{1}Y_{4} + Y_{2}Y_{3} = 0.

But we can do better than this. If a row of the matrix does not use all of the variables, we can use a relative syzygy of k instead of the absolute syzygy as in Section 12.4. The point is that the variety of the subgroup used for the relative syzygy should be contained in the variety defined by the polynomials appearing in the row or column.

To illustrate, here’s a small module with interesting variety, for p odd. We set p = 3 to give the example, but the construction works in general. Let E = (ℤ/3)^{4} = ⟨g_{1}, g_{2}, g_{3}, g_{4}⟩ and k be a field of characteristic three. Represent the action of the elements X_{1} = g_{1}−1, …, X_{4} = g_{4} − 1 by single edges, double edges, wavy edges and dotted edges respectively. Consider the following module.

[figure: •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •]

Here, the leftmost and the rightmost vertices are identified. The rank variety of this module is given by the single equation

Y_{2}Y_{3} = Y_{1}Y_{4}.

Restricting to a shifted subgroup of kE of rank three gives modules with any irreducible quadric as the variety.

Think of this module as an extension of two relative syzygies of the form Ω_{E/E′}k (with different subgroups E′) by two copies of k, corresponding to the matrix

(\begin{matrix}x_{1} & x_{3} \\ x_{2} & x_{4} \end{matrix}) ∈ Mat_{2}(H^{2}(E, k)).

The crucial properties of this matrix are: each column only involves two of the variables so that it can be regarded as an element of cohomology relative to a subgroup (ℤ/p)^{2}. Thus Condition 12.4.2 is satisfied, and so the computation of the rank variety follows from Theorem 12.4.3, giving the hypersurface determined by Y_{1}^{p}Y_{4}^{p} − Y_{2}^{p}Y_{3}^{p}. Since the coefficients are in 𝔽_{p}, this determines the same variety as Y_{1}Y_{4} − Y_{2}Y_{3}. The module M above sits in a short exact sequence

0 → k ⊕ k → M → Ω_{E/⟨g_{3},g_{4}⟩}(k) ⊕ Ω_{E/⟨g_{1},g_{2}⟩}(k) → 0.

12.7. Trying to Understand the Specht Module S^{(3^{3})}

In this section, we produce small modules with the same variety as the Specht module S^{(3^{3})} of Example 12.1.1, but whose dimension is, surprisingly, even smaller.

To do this, we apply the technique of Section 12.4 with the matrix

(\begin{matrix}x_{3} & x_{1} \\ −x_{1} & x_{3} & x_{3} \\ & x_{3} & x_{3} & x_{2} \\ & & −x_{2} & x_{3} \end{matrix})

to obtain a module M of dimension 36 for (ℤ/3)^{3} whose variety is

Y_{1}^{2}Y_{2}^{2} + Y_{1}^{2}Y_{3}^{2} + Y_{2}^{2}Y_{3}^{2} = 0

and which sits in a short exact sequence

0 → k ⊕ k ⊕ k ⊕ k → M → Ω_{E/⟨g_{2}⟩}(k) ⊕ Ω_{E/⟨g_{2}⟩}(k) ⊕ Ω_{E/⟨g_{1}⟩}(k) ⊕ Ω_{E/⟨g_{1}⟩}(k) → 0.

Comparing with the Specht module S^{(3^{3})} for Σ_{9} restricted to (ℤ/3)^{3}, the mysterious things about this module are that it is 36 dimensional instead of 42 dimensional. This is, surprisingly, even smaller than the unexpectedly small Specht module. But this module also doesn’t have enough symmetry to be the Specht module. There are more symmetric matrices with the same determinant, for example

(\begin{matrix}x_{3} & x_{1} & x_{2} \\ & & x_{3} & x_{1} \\ x_{1} & & & x_{2} \\ & x_{2} & & x_{3} \end{matrix})

but this produces a module of dimension 54, which is too big.

This matrix looks a bit more symmetric:

(\begin{matrix}x_{1} + x_{2} + x_{3} & x_{1} − x_{2} & x_{2} − x_{3} & x_{3} − x_{1} \\ x_{1} − x_{2} & & x_{2} \\ x_{2} − x_{3} & & & x_{3} \\ x_{3} − x_{1} & x_{1} \end{matrix})

but it still produces a module of dimension 36 sitting in a short exact sequence

0 → k ⊕ k ⊕ k ⊕ k → M → Ω_{E/⟨g_{1}g_{2}g_{3}⟩}(k) ⊕ Ω_{E/⟨g_{3}⟩}(k) ⊕ Ω_{E/⟨g_{1}⟩}(k) ⊕ Ω_{E/⟨g_{2}⟩}(k) → 0.

How to understand a module of dimension 42 with the correct variety is still a mystery.

12.8. Modules with Small Loewy Length

This section is based on joint work with my former Ph.D. student, Fergus Reid [54], which in turn grew out of his thesis [214].

Let E ≅ (ℤ/p)^{r} be an elementary abelian p-group of rank r and let k be an algebraically closed field of characteristic p. If M is a finite-dimensional kE-module then its rank variety V_{E}^{rk}(M) is a closed homogeneous subset of affine r-space V_{E}^{rk} = 𝔸^{r}(k) of dimension equal to the complexity of M. If M has complexity c then its Loewy length is at least (p−1)(r−c)+1. It is the purpose of this section to investigate what happens when equality holds. The case of Loewy length p and complexity r − 1 was investigated in Reid’s thesis, and this section generalises those results to arbitrary codimension. The main theorem of this section is as follows.

Theorem 12.8.1. Let E ≅ (ℤ/p)^{r}, k be an algebraically closed field of characteristic p, and M be a finite-dimensional kE-module of complexity c. Then M has Loewy length at least (p − 1)(r − c) + 1. If M has exactly this Loewy length then the following hold:

(1) If p is odd then V_{E}^{rk}(M) is a union of a finite number of linear subspaces of V_{E}^{rk} = 𝔸^{r}(k) of dimension c. Every such subvariety occurs for some M of Loewy length (p − 1)(r − c) + 1.

(2) If p = 2 then every irreducible component of V_{E}^{rk}(M) is equidimensional of dimension c. Given a closed homogeneous subvariety V ⊆ 𝔸^{r}(k) which is a finite union of set theoretic complete intersections of dimension c, there exists such a kE-module M of Loewy length r − c + 1 with V_{E}^{rk}(M) = V.

If p = 2, the group algebra kE is isomorphic to an exterior algebra on r generators. The case p = 2 of Theorem 12.8.1 generalises in an obvious way to modules in any characteristic for an exterior algebra as follows.

Theorem 12.8.2. Let Λ = Λ(X_{1}, …, X_{r}) be an exterior algebra on variables X_{1}, …, X_{r} over an algebraically closed field k, and M be a finite dimensional Λ-module of complexity c. Then M has Loewy length at least r − c + 1. If M has exactly this Loewy length then V_{Λ}^{rk}(M) is equidimensional of dimension c. Given a closed homogeneous subvariety V ⊆ 𝔸^{r}(k) which is a finite union of set theoretic complete intersections of dimension c, there exists such a Λ-module M of Loewy length r − c + 1 with V_{Λ}^{rk}(M) = V.

Remark 12.8.3. In the case p odd, Theorem 12.8.1 describes exactly which subvarieties can occur for modules with complexity c and Loewy length (p − 1)(r − c) + 1. For p = 2, and more generally for exterior algebras in any characteristic in Theorem 12.8.2 there is a gap, as described by the following series of observations.

The twisted cubic defined by the homogeneous radical ideal

(Y_{1}Y_{3} − Y_{2}^{2}, Y_{2}Y_{4} − Y_{3}^{2}, Y_{1}Y_{4} − Y_{2}Y_{3}) ⊆ k[Y_{1}, …, Y_{4}]

is not a complete intersection in the sense that this ideal is not a complete intersection; but as a variety it is a set theoretic complete intersection, because it can be expressed as an intersection (with multiplicity two) of the hypersurfaces

Y_{1}Y_{3} − Y_{2}^{2} = 0 and Y_{3}(Y_{2}Y_{4} − Y_{3}^{2}) − Y_{4}(Y_{1}Y_{4} − Y_{2}Y_{3}) = 0.

But not every equidimensional closed homogeneous subvariety of 𝔸^{r}(k) is a set theoretic complete intersection. For example, the ideal (Y_{1}, Y_{2}) ∩ (Y_{3}, Y_{4}) ⊆ k[Y_{1}, …, Y_{4}] describes the union of two 2-dimensional linear subspaces of 𝔸^{4} intersecting in the origin. It is shown in Example 9.17 of [167] that this has codimension two but arithmetic rank three, and in particular it is not a set theoretic complete intersection. Nonetheless, it is obviously a union of two complete intersections, and hence the theorem guarantees that there is a module of Loewy length three for (ℤ/2)^{4} in characteristic two (or an exterior algebra in four variables in any characteristic) with this variety.

There are also irreducible examples. Example 3.4.2 in Hartshorne [145], after homogenisation, is the closed homogeneous subset of 𝔸^{5}(k) of dimension three defined parametrically by coordinates

(s^{3}, s^{2}t, stu, su(u − s), u^{2}(u − s)).

The paper describes this over the complex numbers, but the argument given there shows that over any algebraically closed field k, this is not a set theoretic complete intersection. The argument is that set theoretic complete intersections have the property that locally, the self-intersections have codimension one. In this example, the corresponding map ℙ^{2} → ℙ^{4} is injective except for the points (1 : 0 : 0) and (1 : 0 : 1), which both go to (1 : 0 : 0 : 0 : 0), so this point is a self-intersection of codimension two. We do not know whether there is a module of Loewy length three for (ℤ/2)^{5} (or for an exterior algebra on five generators in other characteristics) with this variety.

We thank Srikanth Iyengar and Anurag Singh for referring us to the examples in [167] and Burt Totaro for pointing us to Hartshorne’s paper.

The rest of this section is devoted to the proof of Theorems 12.8.1 and 12.8.2. The following will be used in the proof of Theorem 12.8.20.

Proposition 12.8.4. Let X_{α_{1}}, …, X_{α_{c}} be a basis for a subspace U of V_{E}^{rk} of dimension r −c of V_{E}^{rk} and let kE′′ ⊆ kE be as defined in (12.1.1). Let M be a kE-module of dimension p^{r−c} such that M↓_{kE′′} is free of rank one. Then V_{E}^{rk}(M) is a linear subspace W of V_{E}^{rk} of dimension c complementary to U.

Proof. Since kE is commutative, its action on M is as kE′′-module homomorphisms. We have End_{kE′′}(kE′′) ≅ kE′′, with the isomorphism being given by multiplication by elements of kE′′. So there is a function f: {X_{1}, …, X_{r}} → kE′′ such that the action of X_{i} on M is the same as the action of f(X_{i}). The elements X_{i} − f(X_{i}) then act as zero on M. The images of these elements in J(kE)/J^{2}(kE) span a subspace W of dimension at least c intersecting U in the origin, and hence W has dimension exactly c and is a complement to U. It now follows from Lemma 6.4 of Carlson [74] that V_{E}^{rk}(M) = W. □

Lemma 12.8.5. Suppose that M is a module with Rad^{m}(M) = 0. Then for 0 ≤ i ≤ m we have Rad^{i}(M) ⊆ Soc^{m−i}(M). □

Lemma 12.8.6. If M_{1} and M_{2} are kE-modules of Loewy length ℓ_{1} and ℓ_{2} then M_{1} ⊗_{k} M_{2} has Loewy length at most ℓ_{1} + ℓ_{2} − 1.

Proof. Tensor products of semisimple kE-modules are semisimple. So defining

N_{i} = ∑_{i_{1}+i_{2}=i} J^{i_{1}}(kE)M_{1} ⊗_{k} J^{i_{2}}(kE)M_{2} ⊆ M_{1} ⊗_{k} M_{2},

it follows that N_{0} = M_{1} ⊗_{k} M_{2}, N_{i}/N_{i+1} is semisimple, and we have N_{ℓ_{1}+ℓ_{2}−1} = 0 because if i_{1} + i_{2} = ℓ_{1} + ℓ_{2} − 1 then either i_{1} ≥ ℓ_{1} or i_{2} ≥ ℓ_{2}. □

Definition 12.8.7. We say that a closed homogenous subvariety V of V_{E}^{rk} is equidimensional if all irreducible components of V have the same dimension. A (homogeneous) hypersurface is equidimensional of codimension one. Equivalently, it is defined by a single homogeneous polynomial equation.

Theorem 12.8.8. Let E ≅ (ℤ/p)^{r}, k an algebraically closed field of characteristic p and M a finite-dimensional kE-module of complexity c. Then the Loewy length of M is at least (p − 1)(r − c) + 1. If equality occurs then V_{E}^{rk}(M) is equidimensional, and for 0 ≤ i ≤ (p − 1)(r − c) + 1 we have

Rad^{i}(M) = J^{i}(kE′)M,

where kE′′ = k[X_{α_{1}}, …, X_{α_{r−c}}]/(X_{α_{1}}^{p}, …, X_{α_{r−c}}^{p}) is a flatly embedded subalgebra such that M↓_{kE′′} is free.

Proof. Since V_{E}^{rk}(M) has dimension c, there exists a linear subspace U of V_{E}^{rk} of dimension r − c whose intersection with V_{E}^{rk}(M) is exactly the origin. Let α_{1}, …, α_{r−c} be a basis for U. Then

kE′′ = k[X_{α_{1}}, …, X_{α_{r−c}}]/(X_{α_{1}}^{p}, …, X_{α_{r−c}}^{p})

is a subalgebra flatly embedded in kE, and M↓_{kE′′} is a free module. So the radical length of M↓_{kE′′} is the radical length of kE′′, which is equal to (p − 1)(r − c) + 1. It follows that the radical length of M has to be at least this big.

Now suppose that equality occurs. Then letting m = (p − 1)(r − c) + 1, by Lemma 12.8.5 we have

J^{i}(kE′′)M ⊆ Rad^{i}(M) ⊆ Soc^{m−i}(M) ⊆ J^{i}(kE′′)M

and so Rad^{i}(M) = J^{i}(kE′′)M.

If V_{E}^{rk}(M) has an irreducible component V of dimension less than c then there exists a linear subspace W of dimension r−c+2 such that V ∩W has dimension one, and W intersects other irreducible components of V_{E}^{rk}(M) in {0}. Let M(W) be the module constructed in Example 12.1.2, with dimension p^{c−2} and Loewy length (p−1)(c−2)+1. Then V_{E}^{rk}(W ⊗_{k} M) decomposes as a union of V ∩ W and another subvariety intersecting V ∩ W in {0}. So by a theorem of Carlson [75], W ⊗_{k} M has a direct summand M′ with V_{E}^{rk}(M′) = V ∩ W. The Loewy length of M′ is at most the Loewy length of W ⊗_{k} M, which by Lemma 12.8.6 is at most

((p − 1)(c − 2) + 1) + ((p − 1)(r − c) + 1) − 1 = (p − 1)(r − 2) + 1.

Since M′ has complexity one, it follows that it is a counterexample to the inequality in the first part of the theorem. This contradiction shows that V_{E}^{rk}(M) cannot have such an irreducible component. □

Next, we describe the corresponding theorem for exterior algebras. The analogue of Theorem 12.8.8 is the following, and the proof is the same. See for example Theorem 3.2 and Remark 3.4 (3) of [12].

Theorem 12.8.9. Let Λ = Λ(X_{1}, …, X_{r}) be an exterior algebra on variables X_{1}, …, X_{r} over an algebraically closed field k, and M a finite-dimensional Λ-module of complexity c. Then the Loewy length of M is at least r − c + 1. If equality occurs then V_{Λ}^{rk}(M) is equidimensional, and for 0 ≤ i ≤ r − c + 1 we have Rad^{i}(M) = J^{i}(Λ′′)M, where Λ′′ = Λ(X_{α_{1}}, …, X_{α_{r−c}}) is an exterior subalgebra of rank r − c flatly embedded in Λ in such a way that M↓_{Λ′′} is free. □

We are now ready to deal with the special case discussed in Reid’s thesis [214], namely modules of complexity c = r − 1. This depends on the following result from the theory of matrix factorisations:

Theorem 12.8.10. Let f be a homogeneous polynomial of degree d in k[Y_{1}, …, Y_{r}]. Then there exist m > 0 and m × m matrices A_{1}, …, A_{d} ∈ Mat_{m}(k[Y_{1}, …, Y_{r}]) such that each A_{i} is a matrix whose entries are either zero or polynomials of degree one, and A_{1} … A_{d} = f.I, where I is the m × m identity matrix.

Proof. This follows from Theorem 1.2 of Herzog, Ulrich and Backelin [152]. □

Remark 12.8.11. In the theorem, it follows that for 1 ≤ i ≤ d we also have

A_{i}A_{i+1} … A_{d}A_{1} … A_{i−1} = f.I.

Corollary 12.8.12. Let f be a homogeneous polynomial of degree d in k[Y_{1}, …, Y_{r}]. Then there exists m > 0 and a matrix A ∈ Mat_{m}(k[Y_{1}, …, Y_{r}]) whose entries are either zero or polynomials of degree one, such that the hypersurface determined by det(A) is the same as that determined by f.

Proof. If f is irreducible, this follows from Theorem 12.8.10 by taking A = A_{1}, since

det(A_{1}) … det(A_{d}) = det(f.I) = f^{m}

and so det(A) is a power of f. For general f it follows by taking direct sums. □

Remark 12.8.13. In the corollary, one may insist that det(A) is a power of f, but the matrices may be larger, and there is no advantage to doing this.

Theorem 12.8.14. Let k be a field and Λ = Λ(X_{1}, …, X_{r}) be an exterior algebra over k. If M is a Λ-module of Loewy length two then V_{Λ}^{rk}(M) is either a hypersurface or the whole of V_{Λ}^{rk}. Conversely, given any homogeneous hypersurface V of V_{Λ}^{rk}, there exists a Λ-module M of Loewy length two with V_{Λ}^{rk}(M) = V.

Proof. The first statement follows from Theorem 12.8.8. For the converse, we’ll make use of Corollary 12.8.12. Interpret k[Y_{1}, …, Y_{r}] as the coordinate ring of 𝔸^{r}(k). Thus Y_{1}, …, Y_{r} are dual to X_{1}, …, X_{r}. For notation, if α = (α_{1}, …, α_{r}) ∈ 𝔸^{r}(k) and f ∈ k[Y_{1}, …, Y_{r}], we write f(α) to mean f(α_{1}, …, α_{r}). We can write V ⊆ V_{Λ}^{rk} as V_{Λ}^{rk}(f), the zeros of a homogeneous polynomial f(Y_{1}, …, Y_{r}) ∈ k[Y_{1}, …, Y_{r}]. By Corollary 12.8.12, there exists a matrix A ∈ Mat_{m}(k[Y_{1}, …, Y_{r}]) whose entries are either zero or polynomials of degree one, such that the hypersurface determined by det(A) is equal to V. Let M be the module given by

X_{α} ↦ (\begin{matrix}0 & 0 \\ A(α) & 0 \end{matrix}) .

Then V_{Λ}^{rk}(M) is the set of α such that this matrix has rank less than \frac{1}{2} dim_{k}(M). The only relevant minor is the bottom left square, and so the condition is that α is a zero of det(A)(α) = f(α). Thus V_{Λ}^{rk}(M) = V. □

Corollary 12.8.15. Let k be a field of characteristic two and let E be an elementary abelian 2-group. Given any homogeneous hypersurfaces V ⊆ V_{E}^{rk}, there exists a kE-module M of Loewy length two with V_{E}^{rk}(M) = V. □

Next, we turn to odd primes. The following completes the proof of Theorem 12.8.1 in the case c = r − 1. A proof is given in Theorem 12.8.20 for c arbitrary, and it does not use this special case, so logically it is not necessary to read this proof. On the other hand it is easier, and good preparation for reading the proof of Theorem 12.8.20.

Theorem 12.8.16. Let k be an algebraically closed field of characteristic p ≥ 3 and let E be an elementary abelian p-group. Let M be a kE-module of Loewy length p with V_{E}^{rk}(M) a proper subvariety of V_{E}^{rk}. Then V_{E}^{rk}(M) is a finite union of linear hypersurfaces in V_{E}^{rk}.

Proof. Choose β ∈ V_{E}^{rk} \ V_{E}^{rk}(M). Then the action of X_{β} on M is projective, and setting kE′ = k[X_{β}]/(X_{β}^{p}), it follows from Theorem 12.8.8 that for 0 ≤ i ≤ p we have Rad^{i}(M) = X_{β}^{i}.M. So M has a basis with respect to which the matrix for X_{β} is

(\begin{matrix}0 & 0 & ⋯ & 0 & 0 \\ I & 0 & & 0 & 0 \\ 0 & I & & 0 & 0 \\ ⋮ & & ... & & ⋮ \\ 0 & 0 & ⋯ & I & 0 \end{matrix}) .

Here, there are p blocks, and 0 and I represent zero and identity square matrices of size dim_{k}(M)/p. Since Rad^{i}(M) = X_{β}^{i}.M, the generators X_{1}, …, X_{r} act as matrices of the form

(\begin{matrix}0 & 0 & ⋯ & 0 & 0 \\ ∗ & 0 & & 0 & 0 \\ ∗ & ∗ & & 0 & 0 \\ ⋮ & & ... & & ⋮ \\ ∗ & ∗ & ⋯ & ∗ & 0 \end{matrix}) .

Furthermore, since the X_{j} commute with X_{β}, this forces the matrices representing the X_{j} to take the form

(\begin{matrix}0 & 0 & ⋯ & & 0 & 0 \\ A_{j} & 0 & & & 0 & 0 \\ B_{j} & A_{j} & & & 0 & 0 \\ ⋮ & ... & ... & & & ⋮ \\ & & ... & A_{j} & 0 & 0 \\ ∗ & & ⋯ & B_{j} & A_{j} & 0 \end{matrix}) .

Since p ≥ 3, the fact that these matrices commute with each other now forces A_{1}, …, A_{r} to commute with each other. Commuting matrices over an algebraically closed field may be simultaneously lower triangularised. So after a change of basis within the blocks above, we may assume that A_{1}, …, A_{r} are lower triangular.

We are now in a position to compute V_{E}^{rk}(M). There is only one minor of size \frac{p−1}{p} dim_{k}(M) that can be non-zero, namely the bottom left part of the matrices above, obtained by removing one block from the top of each column and one block from the right of each row. If α = (λ_{1}, …, λ_{r}), then this minor for X_{α} has determinant

det(λ_{1}A_{1} + ⋯ + λ_{r}A_{r})^{p−1}.

Since A_{1}, …, A_{r} are lower triangular, this is a product of linear forms in λ_{1}, …, λ_{r}. It follows that V_{E}^{rk}(M) is the union of linear hypersurfaces determined by this product. □

Next, we handle the case of exterior algebras for arbitrary complexity, thereby completing the proof of Theorem 12.8.2 and of part (2) of Theorem 12.8.1.

Theorem 12.8.17. Let Λ = Λ(X_{1}, …, X_{r}) be an exterior algebra over an algebraically closed field k. If V ⊆ V_{Λ}^{rk} is a finite union of set theoretic complete intersections of dimension c, then there exists a Λ-module M of Loewy length r − c + 1 with V_{Λ}^{rk}(M) = V.

Proof. Since the Loewy length of M_{1} ⊕ M_{2} is equal to the larger of the Loewy lengths of M_{1} and M_{2}, using Proposition 1.9.11 (3), it suffices to prove the theorem in the case where V is a complete intersection. Let f_{1}, …, f_{r−c} be a regular sequence of homogeneous elements in k[Y_{1}, …, Y_{r}] such that the variety of the ideal (f_{1}, …, f_{r−c}) is equal to V. For each f_{i}, Theorem 12.8.14 produces a module M_{i} of Loewy length two with V_{Λ}^{rk}(M_{i}) = V (f_{i}), the hypersurface defined by f_{i}. Set M = M_{1} ⊗_{k} ⋯ ⊗_{k} M_{r−c}. Using Proposition 1.9.11 (4), we have

V_{Λ}^{rk}(M) = V_{Λ}^{rk}(M_{1}) ∩ ⋯ ∩ V_{Λ}^{rk}(M_{r−c}) = V (f_{1}) ∩ ⋯ ∩ V (f_{r−c}) = V.

Furthermore, by Lemma 12.8.6 and induction on the codimension of V, the Loewy length of M is equal to r − c + 1. □

Remark 12.8.18. If k does not have characteristic two, it is necessary to consider Λ as a graded algebra with the X_{i} in degree one, in order for it to be a Hopf algebra so that we can tensor modules together. Fortunately, the modules M_{i} in the proof of the theorem may be regarded as graded Λ-modules.

Corollary 12.8.19. Let k be an algebraically closed field of characteristic 2 and E = (ℤ/2)^{r}. If V ⊆ V_{E}^{rk} is a finite union of set theoretic complete intersections of dimension c, then there exists a kE-module M of Loewy length r − c + 1 with V_{E}^{rk}(M) = V. □

Finally, we complete the proof of Theorem 12.8.1 by proving that if p is odd and M has minimal Loewy length for its complexity, then its variety is a finite union of linear subspaces of the same dimension. The proof is a generalisation of the proof of Theorem 12.8.16, but the technical details are harder to follow.

Theorem 12.8.20. Let k be an algebraically closed field of characteristic p ≥ 3 and let E = (ℤ/p)^{r} be an elementary abelian p-group. Let M be a kE-module with complexity c and Loewy length (p − 1)(r − c) + 1. Then V_{E}^{rk}(M) is a finite union of linear subspaces of V_{E}^{rk} of dimension c. Every such subvariety occurs this way.

Proof. The fact that every such variety is realised by a module with the right Loewy length is easy to see by looking at direct sums of modules of the form M(W) described in Example 12.1.2. So we assume that M is a module with complexity c and Loewy length (p − 1)(r − c) + 1 and we examine its variety.

Let kE = k[X_{1}, …, X_{r}]/(X_{1}^{p}, …, X_{r}^{p}). As in the proof of Theorem 12.8.8, choose a linear subspace U of dimension r − c with U ∩ V_{E}^{rk}(M) = {0} and choose a basis α_{1}, …, α_{r−c} for U. Set

kE′′ = k[X_{α_{1}}, …, X_{α_{r−c}}]/(X_{α_{1}}^{p}, …, X_{α_{r−c}}^{p}).

Then M↓_{kE′′} is a free module.

Set M_{0} = J(kE′)M, which by Theorem 12.8.8 is equal to Rad(M). For 1 ≤ i ≤ r − c, set

M_{i} = X_{α_{1}}M + ⋯ + X_{α_{i}}^{2}M + ⋯ + X_{α_{r−c}}M,
M′_{i} = X_{α_{1}}M + ⋯ + X_{α_{i}}^{3}M + ⋯ + X_{α_{r−c}}M.

For 1 ≤ i < j ≤ r − c, set

M_{ij} = X_{α_{1}}M + ⋯ + X_{α_{i}}^{2}M + ⋯ + X_{α_{j}}^{2}M + ⋯ + X_{α_{r−c}}M

(just two terms with squares). Then X_{α_{i}} induces isomorphisms

M/M_{0} → M_{0}/M_{i} → M_{i}/M′_{i}.

By Theorem 12.8.8, X_{1}, …, X_{r} send M into M_{0} and M_{0} into M_{i}, and hence also induce maps

M/M_{0} → M_{0}/M_{i} → M_{i}/M′_{i}.

Since they commute with X_{α_{i}}, if we identify these spaces using X_{α_{i}} then the two maps induced by each X_{ℓ} are the same. Since the X_{ℓ} commute with each other, we may choose a basis for M/M_{0} with respect to which these maps are represented by lower triangular matrices, which we denote A_{j,ℓ}. Furthermore, we have a commutative square of isomorphisms

\xymatrix{M/M_{0} \ar[r]^{X_{α_{j′}}} \ar[d]_{X_{α_{j}}} & M_{0}/M_{j′} \ar[d]^{X_{α_{j}}} \\ M_{0}/M_{j} \ar[r]_{X_{α_{j′}}} & (M_{j} + M_{j′})/M_{j,j′}.}

The vertical maps induced by an X_{ℓ} commute with X_{α_{j′}} so they are both the same, and the horizontal maps induced by another X_{ℓ′} commute with X_{α_{j}} so they are both the same. Therefore the matrices A_{j,ℓ} and A_{j′,ℓ′} commute. It follows that we may choose a single basis for M/M_{0} so that all the matrices A_{j,ℓ} (1 ≤ j ≤ r − c, 1 ≤ ℓ ≤ r) are lower triangular. Write m_{1}, …, m_{n} for such a basis. So the image of m_{b} under A_{j,ℓ} is contained in the linear span of m_{b}, …, m_{n}. Then a basis for M is given by the elements

(12.8.1)  X_{α_{1}}^{a_{1}} … X_{α_{r−c}}^{a_{r−c}}.m_{b}

with 0 ≤ a_{i} ≤ p − 1 and 1 ≤ b ≤ n.

Next, we observe that the actions of X_{1}, …, X_{r} on M are as kE′′-module endomorphisms. We have End_{kE′′}(M) ≅ Mat_{n}(kE′′), and by Theorem 12.8.8, the actions of X_{1}, …, X_{r} are by elements of Mat_{n}(J(kE′′)). Furthermore, if x ∈ Mat_{n}(J^{2}(kE′′)) then we may write x = ∑_{i} y_{i}z_{i} with y_{i}, z_{i} ∈ Mat_{n}(J(kE′′)) and y_{i}^{p} = 0, z_{i}^{p} = 0 (to see this, it suffices to consider the case where x has only one non-zero entry, and where that entry is a product of elements of J(kE′′)). So by Proposition 2.2 of Friedlander and Pevtsova [124], if we adjust the action of X_{1}, …, X_{r} by elements of Mat_{n}(J^{2}(kE′′)) we do not change V_{E}^{rk}(M). We may make such an adjustment so that the action of X_{1}, …, X_{r} on M is determined by the equations

X_{ℓ}.m_{b} = ∑_{j} A_{j,ℓ}X_{α_{j}}.m_{b}.

These equations do determine the action on all basis elements (12.8.1) since X_{ℓ} commutes with X_{α_{1}}^{a_{1}} … X_{α_{r−c}}^{a_{r−c}}.

We make a filtration of M by submodules

M = N_{1} ⊇ ⋯ ⊇ N_{n+1} = {0}

where N_{i} is the span of the basis elements (12.8.1) of M with b ≥ i. The subquotient N_{i}/N_{i+1} (1 ≤ i ≤ n) has a basis consisting of the images of the basis elements (12.8.1) of M with b = i. It is an example of a module satisfying the hypothesis of the theorem, of dimension p^{r−c}, and where the matrices A_{j,ℓ} described above have only one entry. Proposition 12.8.4 now applies, and we deduce that V_{E}^{rk}(N_{i}/N_{i+1}) is a linear subspace W_{i} ⊆ V_{E}^{rk} of dimension c. It follows using Proposition 1.9.11 (5) that V_{E}^{rk}(M) ⊆ W_{1} ∪ ⋯ ∪ W_{n}. By Theorem 12.8.8, V_{E}^{rk}(M) is equidimensional of dimension c, so it follows that V_{E}^{rk}(M) is a union of some subset of {W_{1}, …, W_{n}}. □

12.9. Small Modules for (ℤ/p)^{2}

Let E = (ℤ/p)^{2} = ⟨g, h⟩ be an elementary abelian p-group of rank two, with p odd, and let k be an algebraically closed field of characteristic p. Set x = g − 1 and y = h − 1 in kE, so that kE = k[x, y]/(x^{p}, y^{p}).

If M is a finite-dimensional indecomposable periodic kE-module then dim_{k} M is of the form np for some n > 0. By Proposition 1.9.11 (6) (Carlson’s connectedness theorem), the rank variety of M consists of a single line through the origin. By changing the group basis for kE if necessary, we can assume that this line is a coordinate axis. In other words, we can assume that the restriction of M to k[x]/(x^{p}) is not projective, and thus x acts on M with at least one Jordan block of length less than p. Our main theorem then states that the action of x on M also has some Jordan block of length p then dim_{k} M > p^{3/2}:

Theorem 12.9.1. Let E = (ℤ/p)^{2} and k be a field of characteristic p. If there exists an indecomposable kE-module M of dimension np satisfying 0 < dim_{k} x^{p−1}M < n and dim_{k} y^{p−1}M = n then n > \sqrt{p}.

Of course, a module M satisfying the condition of the theorem is necessarily periodic. We prove in the next section that the bound is close to sharp, in the sense that for all

large enough primes p there is a module satisfying the conditions of the theorem and whose dimension satisfies

p^{\frac{3}{2}} < dim_{k}(M) < p^{\frac{3}{2}} + \sqrt{2} p^{\frac{5}{4}} .

The rest of this section is devoted to the proof of Theorem 12.9.1. So we suppose that M is an indecomposable kE-module of dimension np such that 0 < dim_{k} x^{p−1}M < n and dim_{k} y^{p−1}M = n. We also assume that n < \sqrt{p}, and that n is as small as possible subject to the existence of such an M. Our goal is to obtain a contradiction.

Notice that the condition 0 < dim_{k} x^{p−1}M < n states that the action of x on M has a mixture of Jordan blocks of length p and length less that p, while the condition dim_{k} y^{p−1}M = n states that all Jordan blocks for the action of y on M have length exactly p. Thus y^{p−1}M = Ker (y, M).

Lemma 12.9.2. If N is a non-zero submodule of M such that

dim_{k} N = p. dim_{k} y^{p−1}N and dim_{k} x^{p−1}N > 0

then N = M.

Proof. If N is a proper submodule of M of dimension mp, there are two cases to consider. If dim_{k} x^{p−1}N < m then N is a smaller counterexample to the theorem. So by minimality of n, this cannot occur. If, on the other hand, dim_{k} x^{p−1}N = m then the rank variety V_{E}^{r}(N) does not contain the line corresponding to x, and so V_{E}^{r}(N) ∩ V_{E}^{r}(M) = {0}. Thus V_{E}^{r}(M/N) = V_{E}^{r}(M) ∪ V_{E}^{r}(N), and Carlson’s connectedness theorem [75] (or Theorem 8.2 of [74]) shows that M/N is a direct sum M_{1} ⊕ M_{2} with V_{E}^{r}(M_{1}) = V_{E}^{r}(M) and V_{E}^{r}(M_{2}) = V_{E}^{r}(N). Thus Ext_{kE}^{1}(M_{1}, N) = 0, which implies that the inverse image of M_{1} in M decomposes as M′_{1} ⊕ N with M′_{1} ≅ M_{1}. Hence M decomposes as a direct sum of M′_{1} and the inverse image of M_{2} in M. Since M was assumed to be indecomposable, this is a contradiction. □

Choose v ∈ M such that x^{p−1}v ≠ 0. Replacing v by a suitable power of y times v if necessary we may assume also that yx^{p−1}v = 0. Define integers t_{0}, …, t_{p−1} between 0 and p − 1 so that y^{p−1−t_{i}}x^{i}v ≠ 0 but y^{p−t_{i}}x^{i}v = 0. Multiplying this last equation by x and using the equation yx^{p−1}v = 0, we see that

0 ≤ t_{0} ≤ t_{1} ≤ ⋯ ≤ t_{p−1} = p − 1.

Since y^{p−1}M = Ker (y, M), we may choose elements w_{1}, …, w_{p} so that

(12.9.1)  y^{t_{i}}w_{i} = x^{i}v and xw_{i} = y^{t_{i+1}−t_{i}}w_{i+1} (0 ≤ i ≤ p − 2).

To choose such elements, start with w_{p−1} and use downwards induction on i. We then have

(12.9.2)  x^{j}w_{i} = y^{t_{i+j}−t_{i}}w_{i+j},  (1 ≤ i + j ≤ p − 1).

Let N be the linear span in M of the elements y^{i}w_{j} for 0 ≤ i ≤ p − 1 and 0 ≤ j ≤ n. Since dim_{k} M = np, there is a non-trivial linear relation between these elements. Applying a power of y to this relation if necessary, we see that there is a non-trivial linear relation between the elements y^{p−1}w_{j} for 0 ≤ j ≤ n. Thus for some m ≤ n, y^{p−1}w_{m} is in the linear span of the y_{p−1}w_{j} for j < m. Since multiplication by y^{p−1} is an isomorphism from N/yN to y^{p−1}N, it follows that w_{m} is in the linear span of the remaining y^{i}w_{j} with 0 ≤ i ≤ p − 1 and 0 ≤ j ≤ m. Applying y to this relation and using induction, we see that w_{m} is in the linear span of the y^{i}w_{j} with 0 ≤ i ≤ p − 1 and 0 ≤ j ≤ m − 1. It follows that this linear span is closed under the action of x.

Thus we have shown that N is the linear span in M of the elements y^{i}w_{j} with 0 ≤ i ≤ p−1 and 0 ≤ j ≤ m − 1, and N is a kE-submodule of M of dimension mp with m ≤ n, and dim_{k} y^{p−1}N = m. Applying Lemma 12.9.2 we see that N = M.

Under the action of x, the set of y^{i}w_{j} with 0 < j ≤ m and t_{j+1} > t_{j} + i together with the y^{i}w_{0} with 0 ≤ i ≤ p − 1 forms a set of generators for M.

Next we examine the exact form of the relation expressing w_{n} in terms of the y^{i}w_{j} with 0 ≤ i ≤ p − 1 and 0 ≤ j ≤ n − 1. For this purpose we filter M by level. We define the level of y^{i}w_{j} to be ℓ(y^{i}w_{j}) = i + j − t_{i}. The level of a linear combination of these basis elements is the smallest level of a basis element appearing with non-zero coefficient. Among the basis elements appearing with non-zero coefficient in w_{n}, look at the one with the smallest value of j, and define a = n − j for this j.

Lemma 12.9.3. We have ℓ(w_{n}) = n − t_{n}.

Proof. The statement ℓ(w_{n}) = n − t_{n} is equivalent to the statement that ℓ(xw_{n−1}) = ℓ(w_{n−1}) + 1.

If ℓ(w_{n}) ≤ n − t_{n} − 1 then choose q so that a(q − 1) < p < aq. Since a ≤ n < \sqrt{p}, we have q > \sqrt{p} and so a < q. We also have ℓ(xw_{n−1}) ≤ ℓ(w_{n−1}) and so

ℓ(x^{aq}w_{n−1}) ≤ aq − q < a(q − 1) < p.

This implies that x^{aq}w_{n−1} ≠ 0, which contradicts x^{p}w_{n−1} = 0.

On the other hand, if ℓ(w_{n}) ≥ n − t_{n} + 1, write p = nq′ + r, with 0 < r < n. Thus nq′ < p < n(q′ + 1). Since n < \sqrt{p} we have q′ + 1 > \sqrt{p} > n and so q′ ≥ n. Then

ℓ(x^{p−1}v) = ℓ(x^{nq′+r−1}v) ≥ (n + 1)q′ + r = p + q′ ≥ p + n,

which contradicts x^{p−1}v ≠ 0. □

Next we do a dimension count. It is easy to check that the elements x^{k}y^{i}w_{j} with t_{j} > t_{j−1} + i and 0 ≤ k ≤ p − 1 together with the elements x^{k}y^{i}w_{0} with 0 ≤ k < n − a and 0 ≤ i ≤ p − 1 span M. The number of such elements is at most np. It follows that x^{p−1}y^{i}w_{j} ≠ 0 provided t_{j} > t_{j−1} + i and 0 ≤ k ≤ p − 1.

It now follows that for n − a ≤ j ≤ n − 1 we have t_{j+1} = t_{j} + 1, since multiples of p generate mod a. Then examining x^{p−1}w_{0} we see that t_{0} = 0, t_{j} ≤ j for 1 ≤ j ≤ n − a − 1 and t_{j} = j for n − a ≤ j ≤ n − 1.

Now dualise M (using the Lie dual rather than the group dual) to get that all the t_{j} are equal to j.

If a < n then some linear combination of the w_{j} is sent to zero by x, so generates a kE-submodule of dimension p on which y acts freely. The quotient is then a smaller counterexample to the theorem provided a < n−1. If a = n−1 then the module decomposes, again contradicting the hypotheses.

If, on the other hand, a = n then x acts freely on M, contradicting the hypotheses of the theorem.

12.10. The Bound is Close to Sharp

In this section, we show that the bound given in Theorem 12.9.1 is close to sharp. We begin with a number theoretic lemma.

Lemma 12.10.1. Given a prime p ≥ 5, there exist positive integers n with

p^{\frac{1}{2}} < n < p^{\frac{1}{2}} + \sqrt{2} p^{\frac{1}{4}}

and q < n such that

nq + q < p < nq + n.

For p large enough, n and q can also be made to satisfy

nq + \frac{n}{2} < p < nq + q + \frac{n}{2}.

Proof. If p = 5 we can take n = 3 and q = 1. So assume that p > 5. Select integers m so that

(12.10.1)  m^{2} > p > (m − 1)^{2}

and b so that

(12.10.2)  m^{2} − (b − 1)^{2} > p ≥ m^{2} − b^{2}.

Thus m^{2} > m^{2} − (b − 1)^{2} > (m − 1)^{2}, and so 1 < b < \sqrt{2m − 1} + 1. The second inequality in (12.10.2) is strict because p is a prime bigger than 5, so that b cannot equal m − 1. Set n = m + b − 1 and q = m − b. Thus nq + q = m^{2} − b^{2} < p and nq + n = m^{2} − (b − 1)^{2} > p. Finally, n = m + b − 1 < m + \sqrt{2m} < p^{\frac{1}{2}} + \sqrt{2} p^{\frac{1}{4}} .

The inequalities in the second statement follows from the fact that b < \sqrt{2m − 1} + 1, so that provided p is large, it follows that m is large compared with b. Both n and q are therefore close to m, so n/2 < q and q + n/2 > n. □

Theorem 12.10.2. For all large enough primes p, there exists a k(ℤ/p)^{2}-module M satisfying the conditions of Theorem 12.9.1, with

p^{\frac{3}{2}} < dim_{k}(M) < p^{\frac{3}{2}} + \sqrt{2}p^{\frac{5}{4}} .

Proof. Given a (large enough) prime p, and integers n and q satisfying the conditions of the lemma, we build a module M of dimension np as follows. Set α = n + nq − p and β = p − nq − q. The inequalities in the lemma show that we have 0 < α < \frac{n}{2} and 0 < β < \frac{n}{2}.

Take two generators v and w, with relations:

y^{β}w = x^{n−α}v x^{α}w = y^{n+1−β}v.

Using these relations we have x^{n}v = x^{α}y^{β}w = y^{β}y^{n+1−β}v = y^{n+1}v. So

x^{p−1}w = x^{n−2α−1}(x^{n})^{q}x^{α}w = x^{n−2α−1}(x^{n})^{q}y^{n+1−β}v
= x^{n−2α−1}y^{n−β+1}(y^{n+1})^{q}v = x^{n−2α−1}y^{n−2β+1}y^{p}v = 0.

We also have

x^{p−1}v = x^{n−α−1}(x^{n})^{q}v = x^{n−α−1}(y^{n+1})^{q}v = x^{n−α−1}y^{p−β}v ≠ 0

while x^{p}v = x^{n−α}y^{p−β}v = y^{p−β}y^{β}v = 0. So writing w_{i} = x^{i}v for 0 ≤ i < n − α and w_{i} = x^{i−(n−α)}w for n − α ≤ i < n, the module M has a basis consisting of the elements y^{j}w_{i} with 0 ≤ i < n and 0 ≤ j < p.

To see that M is indecomposable, we argue as follows. Since x^{p−1}v ≠ 0 and x^{p−1}w = 0, any endomorphism sends w to a multiple of w plus an element of Rad(M). So if M decomposes then without loss of generality one of the summands is generated by w. But then the other summand is generated by v plus a multiple of w. But x^{n−α}v is in the submodule generated by w, so the summands have non-zero intersection, which is a contradiction. □

APPENDIX A Modules for ℤ/p

Here are tables of the tensor products of modules for ℤ/p in characteristic p given by the formulas in Section 2.3 for small values of p. In these tables we use bracket notation for Jordan types as explained in Section 4.4, so that for example [3]^{2}[2] is the notation for J_{3} ⊕ J_{3} ⊕ J_{2}.

\begin{matrix}p = 3 & [1] & [2] & [3] \\ [1] & [1] & [2] & [3] \\ [2] & [2] & [3][1] & [3]^{2} \\ [3] & [3] & [3]^{2} & [3]^{3} \end{matrix}  \begin{matrix}p = 5 & [1] & [2] & [3] & [4] & [5] \\ [1] & [1] & [2] & [3] & [4] & [5] \\ [2] & [2] & [3][1] & [4][2] & [5][3] & [5]^{2} \\ [3] & [3] & [4][2] & [5][3][1] & [5]^{2}[2] & [5]^{3} \\ [4] & [4] & [5][3] & [5]^{2}[2] & [5]^{3}[1] & [5]^{4} \\ [5] & [5] & [5]^{2} & [5]^{3} & [5]^{4} & [5]^{5} \end{matrix}

\begin{matrix}p = 7 & [1] & [2] & [3] & [4] & [5] & [6] & [7] \\ [1] & [1] & [2] & [3] & [4] & [5] & [6] & [7] \\ [2] & [2] & [3][1] & [4][2] & [5][3] & [6][4] & [7][5] & [7]^{2} \\ [3] & [3] & [4][2] & [5][3][1] & [6][4][2] & [7][5][3] & [7]^{2}[4] & [7]^{3} \\ [4] & [4] & [5][3] & [6][4][2] & [7][5][3][1] & [7]^{2}[4][2] & [7]^{3}[3] & [7]^{4} \\ [5] & [5] & [6][4] & [7][5][3] & [7]^{2}[4][2] & [7]^{3}[3][1] & [7]^{4}[2] & [7]^{5} \\ [6] & [6] & [7][5] & [7]^{2}[4] & [7]^{3}[3] & [7]^{4}[2] & [7]^{5}[1] & [7]^{6} \\ [7] & [7] & [7]^{2} & [7]^{3} & [7]^{4} & [7]^{5} & [7]^{6} & [7]^{7} \end{matrix}

For p = 11, 13 we abbreviate the table by omitting the Jordan blocks of length p. The general pattern and its symmetries should be apparent from the formulas and these tables.

\begin{matrix}p = 11 & [1] & [2] & [3] & [4] & [5] & [6] & [7] & [8] & [9] & [10] \\ [1] & [1] & [2] & [3] & [4] & [5] & [6] & [7] & [8] & [9] & [10] \\ [2] & [2] & [3][1] & [4][2] & [5][3] & [6][4] & [7][5] & [8][6] & [9][7] & [10][8] & [9] \\ [3] & [3] & [4][2] & [5][3][1] & [6][4][2] & [7][5][3] & [8][6][4] & [9][7][5] & [10][8][6] & [9][7] & [8] \\ [4] & [4] & [5][3] & [6][4][2] & [7][5][3][1] & [8][6][4][2] & [9][7][5][3] & [10][8][6][4] & [9][7][5] & [8][6] & [7] \\ [5] & [5] & [6][4] & [7][5][3] & [8][6][4][2] & [9][7][5][3][1] & [10][8][6][4][2] & [9][7][5][3] & [8][6][4] & [7][5] & [6] \\ [6] & [6] & [7][5] & [8][6][4] & [9][7][5][3] & [10][8][6][4][2] & [9][7][5][3][1] & [8][6][4][2] & [7][5][3] & [6][4] & [5] \\ [7] & [7] & [8][6] & [9][7][5] & [10][8][6][4] & [9][7][5][3] & [8][6][4][2] & [7][5][3][1] & [6][4][2] & [5][3] & [4] \\ [8] & [8] & [9][7] & [10][8][6] & [9][7][5] & [8][6][4] & [7][5][3] & [6][4][2] & [5][3][1] & [4][2] & [3] \\ [9] & [9] & [10][8] & [9][7] & [8][6] & [7][5] & [6][4] & [5][3] & [4][2] & [3][1] & [2] \\ [10] & [10] & [9] & [8] & [7] & [6] & [5] & [4] & [3] & [2] & [1] \end{matrix}

\begin{matrix}p = 13 & [1] & [2] & [3] & [4] & [5] & [6] & [7] & [8] & [9] & [10] & [11] & [12] \\ [1] & [1] & [2] & [3] & [4] & [5] & [6] & [7] & [8] & [9] & [10] & [11] & [12] \\ [2] & [2] & [3][1] & [4][2] & [5][3] & [6][4] & [7][5] & [8][6] & [9][7] & [10][8] & [11][9] & [12][10] & [11] \\ [3] & [3] & [4][2] & [5][3][1] & [6][4][2] & [7][5][3] & [8][6][4] & [9][7][5] & [10][8][6] & [11][9][7] & [12][10][8] & [11][9] & [10] \\ [4] & [4] & [5][3] & [6][4][2] & [7][5][3][1] & [8][6][4][2] & [9][7][5][3] & [10][8][6][4] & [11][9][7][5] & [12][10][8][6] & [11][9][7] & [10][8] & [9] \\ [5] & [5] & [6][4] & [7][5][3] & [8][6][4][2] & [9][7][5][3][1] & [10][8][6][4][2] & [11][9][7][5][3] & [12][10][8][6][4] & [11][9][7][5] & [10][8][6] & [9][7] & [8] \\ [6] & [6] & [7][5] & [8][6][4] & [9][7][5][3] & [10][8][6][4][2] & [11][9][7][5][3][1] & [12][10][8][6][4][2] & [11][9][7][5][3] & [10][8][6][4] & [9][7][5] & [8][6] & [7] \\ [7] & [7] & [8][6] & [9][7][5] & [10][8][6][4] & [11][9][7][5][3] & [12][10][8][6][4][2] & [11][9][7][5][3][1] & [10][8][6][4][2] & [9][7][5][3] & [8][6][4] & [7][5] & [6] \\ [8] & [8] & [9][7] & [10][8][6] & [11][9][7][5] & [12][10][8][6][4] & [11][9][7][5][3] & [10][8][6][4][2] & [9][7][5][3][1] & [8][6][4][2] & [7][5][3] & [6][4] & [5] \\ [9] & [9] & [10][8] & [11][9][7] & [12][10][8][6] & [11][9][7][5] & [10][8][6][4] & [9][7][5][3] & [8][6][4][2] & [7][5][3][1] & [6][4][2] & [5][3] & [4] \\ [10] & [10] & [11][9] & [12][10][8] & [11][9][7] & [10][8][6] & [9][7][5] & [8][6][4] & [7][5][3] & [6][4][2] & [5][3][1] & [4][2] & [3] \\ [11] & [11] & [12][10] & [11][9] & [10][8] & [9][7] & [8][6] & [7][5] & [6][4] & [5][3] & [4][2] & [3][1] & [2] \\ [12] & [12] & [11] & [10] & [9] & [8] & [7] & [6] & [5] & [4] & [3] & [2] & [1] \end{matrix}

Here are some lists of exterior powers Λ^{d} and symmetric powers S^{d} of modules for ℤ/p in characteristic p for d < p. These were calculated using the method described in Section 2.8.

S^{2}[1] ≅ Λ^{2}[2] ≅ [1]
S^{2}[2] ≅ Λ^{2}[3] ≅ [3] p ≥ 3
S^{2}[3] ≅ \begin{cases}[3]^{2} & p = 3 \\ Λ^{2}[4] ≅ [5][1] & p ≥ 5 \end{cases}

S^{2}[4] ≅ Λ^{2}[5] ≅ \begin{cases}[5]^{2} & p = 5 \\ [7][3] & p ≥ 7 \end{cases}

S^{2}[5] ≅ \begin{cases}[5]^{3}  p = 5 \\ Λ^{2}[6] ≅ \begin{cases}[7]^{2}[1] & p = 7 \\ [9][5][1] & p ≥ 11 \end{cases} \end{cases}

S^{2}[6] ≅ Λ^{2}[7] ≅ \begin{cases}[7]^{3} & p = 7 \\ [11][7][3] & p ≥ 11 \end{cases}

S^{2}[7] ≅ \begin{cases}[7]^{4}  p = 7 \\ Λ^{2}[8] ≅ \begin{cases}[11]^{2}[5][1] & p = 11 \\ [13][9][5][1] & p ≥ 13 \end{cases} \end{cases}

S^{2}[n] ≅ Λ^{2}[n + 1] ≅ \begin{cases}[p]^{n−(p−1)/2}[2p − 2n − 3][2p − 2n − 7][2p − 2n − 11] ⋯ & p ≤ 2n − 3 \\ [2n − 1][2n − 5][2n − 9][2n − 13] ⋯ & p ≥ 2n − 1 \end{cases}

S^{3}[1] ≅ Λ^{3}[3] ≅ [1]  p ≥ 5
S^{3}[2] ≅ Λ^{3}[4] ≅ [4]  p ≥ 5
S^{3}[3] ≅ Λ^{3}[5] ≅ Λ^{2}[5] p ≥ 5

S^{3}[4] ≅ \begin{cases}[5]^{4}  p = 5 \\ Λ^{3}[6] ≅ \begin{cases}[7]^{2}[6] & p = 7 \\ [10][6][4] & p ≥ 11 \end{cases} \end{cases}

S^{3}[5] ≅ \begin{cases}[5]^{7}  p = 5 \\ Λ^{3}[7] ≅ \begin{cases}[7]^{5} & p = 7 \\ [11]^{2}[7][5][1] & p = 11 \\ [13][9][7][5][1] & p ≥ 13 \end{cases} \end{cases}

S^{3}[6] ≅ \begin{cases}[7]^{8}  p = 7 \\ Λ^{3}[8] ≅ \begin{cases}[11]^{4}[8][4] & p = 11 \\ [13]^{2}[12][8][6][4] & p = 13 \\ [16][12][10][8][6][4] & p ≥ 17 \end{cases} \end{cases}

S^{3}[7] ≅ \begin{cases}[7]^{12}  p = 7 \\ Λ^{3}[9] ≅ \begin{cases}[11]^{7}[7] & p = 11 \\ [13]^{5}[9][7][3] & p = 13 \\ [17]^{2}[13][11][9][7]^{2}[3] & p = 17 \\ [19][15][13][11][9][7]^{2}[3] & p ≥ 19 \end{cases} \end{cases}

S^{4}[1] ≅ Λ^{4}[4] ≅ [1] p ≥ 5
S^{4}[2] ≅ Λ^{4}[5] ≅ [5] p ≥ 5
S^{4}[3] ≅ S^{2}[5] p ≥ 5
Λ^{4}[6] ≅ S^{2}[5] p ≥ 7
S^{4}[4] ≅ S^{3}[5] p ≥ 5
Λ^{4}[7] ≅ S^{3}[5] p ≥ 7

S^{4}[5] ≅ \begin{cases}[5]^{14}  p = 5 \\ [7]^{10}  p = 7 \\ Λ^{4}[8] ≅ \begin{cases}[11]^{5}[9][5][1] & p = 11 \\ [13]^{3}[11][9][5]^{2}[1] & p = 13 \\ [17][13][11][9]^{2}[5]^{2}[1] & p ≥ 17 \end{cases} \end{cases}

S^{4}[6] ≅ \begin{cases}[7]^{18}  p = 7 \\ Λ^{4}[9] ≅ \begin{cases}[11]^{11}[5] & p = 11 \\ [13]^{8}[9][7][5][1] & p = 13 \\ [17]^{3}[15][13][11][9]^{2}[7][5]^{2}[1] & p = 17 \\ [19]^{2}[15][13]^{2}[11][9]^{2}[7][5]^{2}[1] & p = 19 \\ [21][17][15][13]^{2}[11][9]^{2}[7][5]^{2}[1] & p ≥ 23 \end{cases} \end{cases}

S^{5}[1] ≅ Λ^{5}[5] ≅ [1]  p ≥ 7
S^{5}[2] ≅ Λ^{5}[6] ≅ [6]  p ≥ 7
S^{5}[3] ≅ Λ^{5}[7] ≅ S^{2}[6] p ≥ 7
S^{5}[4] ≅ S^{3}[6] p ≥ 7
Λ^{5}[8] ≅ S^{3}[6] p ≥ 11
S^{5}[5] ≅ S^{4}[6] p ≥ 7
Λ^{5}[9] ≅ S^{4}[6] p ≥ 11

S^{5}[6] ≅ \begin{cases}[7]^{36}  p = 7 \\ Λ^{5}[10] ≅ \begin{cases}[11]^{22}[10] & p = 11 \\ [13]^{18}[10][6][2] & p = 13 \\ [17]^{10}[14][12][10]^{3}[8][6]^{2}[4][2] & p = 17 \\ [19]^{6}[18][16][14]^{2}[12][10]^{3}[8]^{2}[6]^{2}[4][2] & p = 19 \\ [23]^{2}[22][18]^{2}[16]^{2}[14]^{2}[12]^{2}[10]^{3}[8]^{2}[6]^{2}[4][2] & p = 23 \\ [26][22][20][18]^{2}[16]^{2}[14]^{2}[12]^{2}[10]^{3}[8]^{2}[6]^{2}[4][2] & p ≥ 29 \end{cases} \end{cases}

Next we list the effects of Schur functors S^{λ} on modules for ℤ/p in characteristic p, where λ is a partition of d < p. These were calculated using the method described in Section 2.9.

S^{[2][1]}[2] ≅ [2]  p ≥ 5
S^{[2][1]}[3] ≅ [5][3] p ≥ 5

S^{[2][1]}[4] ≅ \begin{cases}[5]^{4} & p = 5 \\ [7]^{2}[4][2] & p = 7 \\ [8][6][4][2] & p ≥ 11 \end{cases}

S^{[2][1]}[5] ≅ \begin{cases}[5]^{8} & p = 5 \\ [7]^{5}[5] & p = 7 \\ [11][9][7][5]^{2}[3] & p ≥ 11 \end{cases}^{S^{[2][1]}[6] ≅ \begin{cases}[7]^{10} & p = 7 \\ [11]^{4}[8][6]^{2}[4][2] & p = 11 \\ [13]^{2}[10][8]^{2}[6]^{2}[4][2] & p = 13 \\ [14][12][10][8]^{2}[6]^{2}[4][2] & p ≥ 17 \end{cases}}^{S^{[3][1]}[2] ≅ S^{[2][1]^{2}}[3] ≅ [3] p ≥ 5}^{S^{[3][1]}[3] ≅ S^{[2][1]^{2}}[4] ≅ \begin{cases}[5]^{3} & p = 5 \\ [7][5][3] & p ≥ 7 \end{cases}}^{S^{[3][1]}[4] ≅ S^{[2][1]^{2}}[5] ≅ \begin{cases}[5]^{9} & p = 5 \\ [7]^{6}[3] & p = 7 \\ [11][9][7]^{2}[5][3]^{2} & p ≥ 11 \end{cases}}^{S^{[3][1]}[5] ≅ \begin{cases}[5]^{21}  p = 5 \\ S^{[2][1]^{2}}[6] ≅ \begin{cases}[7]^{15} & p = 7 \\ [11]^{6}[9][7]^{2}[5]^{2}[3]^{2} & p = 11 \\ [13]^{3}[11][9]^{2}[7]^{3}[5]^{2}[3]^{2} & p = 13 \\ [15][13][11]^{2}[9]^{2}[7]^{3}[5]^{2}[3]^{2} & p ≥ 17 \end{cases} \end{cases}}^{S^{[2]^{2}}[2] ≅ [1]  p ≥ 5}^{S^{[2]^{2}}[3] ≅ [5][1] p ≥ 5}^{S^{[2]^{2}}[4] ≅ \begin{cases}[5]^{4} & p = 5 \\ [7]^{2}[5][1] & p = 7 \\ [9][5]^{2}[1] & p ≥ 11 \end{cases}}^{S^{[2]^{2}}[5] ≅ \begin{cases}[5]^{10} & p = 5 \\ [7]^{7}[1] & p = 7 \\ [11]^{2}[9][7][5]^{2}[1]^{2} & p = 11 \\ [13][9]^{2}[7][5]^{2}[1]^{2} & p ≥ 13 \end{cases}}^{S^{[2]^{2}}[6] ≅ \begin{cases}[7]^{15} & p = 7 \\ [11]^{7}[9][7][5]^{2}[1]^{2} & p = 11 \\ [13]^{4}[11][9]^{2}[7][5]^{3}[1]^{2} & p = 13 \\ [17][13]^{2}[11][9]^{3}[7][5]^{3}[1]^{2} & p ≥ 17 \end{cases}}

267

S^{[4][1]}[2] ≅ S^{[2][1]^{3}}[4] ≅ [4] p ≥ 7^{S^{[4][1]}[3] ≅ S^{[2][1]^{3}]}[5] ≅ \begin{cases}[9][7][5][3] & p ≥ 11 \\ [7]^{3}[3] & p = 7 \end{cases}}^{S^{[4][1]}[4] ≅ S^{[2][1]^{3}}[6] ≅ \begin{cases}[7]^{12} & p = 7 \\ [11]^{4}[10][8][6]^{2}[4]^{2}[2] & p = 11 \\ [13]^{2}[10]^{2}[8]^{2}[6]^{2}[4]^{2}[2] & p = 13 \\ [14][12][10]^{2}[8]^{2}[6]^{2}[4]^{2}[2] & p ≥ 17 \end{cases}}^{S^{[4][1]}[5] ≅ S^{[2][1]^{3}}[7] ≅ \begin{cases}[7]^{32} & p = 7 \\ [11]^{17}[9][7]^{2}[5]^{2}[3][1] & p = 11 \\ [13]^{11}[11][9]^{3}[7]^{3}[5]^{3}[3]^{2}[1] & p = 13 \\ [17]^{3}[15][13]^{3}[11]^{3}[9]^{4}[7]^{4}[5]^{3}[3]^{2}[1] & p = 17 \\ [19][17][15]^{2}[13]^{3}[11]^{3}[9]^{4}[7]^{4}[5]^{3}[3]^{2}[1] & p ≥ 19 \end{cases}}^{S^{[3][2]}[2] ≅ [2]  p ≥ 7}^{S^{[3][2]}[3] ≅ [7][5][3] p ≥ 7}^{S^{[3][2]}[4] ≅ \begin{cases}[7]^{8}[4] & p = 7 \\ [11]^{2}[8]^{2}[6]^{2}[4]^{2}[2] & p = 11 \\ [12][10][8]^{2}[6]^{2}[4]^{2}[2] & p ≥ 13 \end{cases}}^{S^{[3][2]}[5] ≅ \begin{cases}[7]^{25} & p = 7 \\ [11]^{11}[9]^{2}[7]^{2}[5]^{3}[3]^{2}[1] & p = 11 \\ [13]^{6}[11]^{2}[9]^{3}[7]^{3}[5]^{4}[3]^{2}[1] & p = 13 \\ [17][15][13]^{2}[11]^{3}[9]^{4}[7]^{3}[5]^{4}[3]^{2}[1] & p ≥ 17 \end{cases}}^{S^{[3][1]^{2}}[3] ≅ [5][1]  p ≥ 7}^{S^{[3][1]^{2}}[4] ≅ \begin{cases}[7]^{4}[6][2] & p = 7 \\ [10][8][6]^{2}[4][2] & p ≥ 11 \end{cases}}^{S^{[2]^{2}[1]}[3] ≅ [3] p ≥ 7}^{S^{[2]^{2}[1]}[4] ≅ \begin{cases}[7]^{2}[4][2] & p = 7 \\ [8][6][4][2] & p ≥ 11 \end{cases}}^{S^{[2]^{2}[1]}[5] ≅ \begin{cases}[7]^{10}[5] & p = 7 \\ [11]^{3}[9][7]^{2}[5]^{3}[3][1] & p = 11 \\ [13][11][9]^{2}[7]^{2}[5]^{3}[3][1] & p ≥ 13 \end{cases}}

268

APPENDIX B Problems

This appendix consists of a selection of open problems connected with the material of this volume. Throughout this chapter, let p be a prime number, let E = ⟨g_{1}, …, g_{r}⟩ ≅ (ℤ/p)^{r} be an elementary abelian p-group of rank r ≥ 2, and let k be an algebraically closed field of characteristic p, and let M be a finitely generated kE-module. Write X_{i} for the element g_{i} − 1 ∈ kE. We begin with a couple of questions of David Eisenbud that are peripherally connected with the main subject of this text.

Problem 1. Let M be a module of complexity r. Then

∑_{n=0}^{∞} dim_{k} Ext_{kE}^{n}(M, k)t^{n} = f(t)/(1 − t^{2})^{r}

with f(t) a polynomial with integer coefficients and f(1) > 0. Eisenbud asks whether it is always true that f(1) ≥ 2^{r}.

Problem 2. Greenlees [139] has constructed a local cohomology spectral sequence

H_{𝔪}^{∗∗} Ex̂t_{kE}^{∗}(M, k) ⇒ 0.

Eisenbud asks whether it can ever happen that the local cohomology of Tate cohomology is non-zero in degree one.

Problem 3. Given the prime p and the rank r ≥ 2, what are the possible Jordan types of a kE-module of constant Jordan type? A slightly less ambitious but nonetheless extremely difficult problem is to find the possible stable Jordan types.

A great deal of partial information on this problem is given in Chapters 5 and 10. In particular, the simplest stable Jordan type for which we do not know the answer is type [4] [1] with p = 11, r = 2, see Section 5.14.

Subproblems of Problem 3 include the following.

Problem 4. Find a proof or a counterexample to Rickard’s Conjecture 5.13.1. This states that if a kE-module of constant Jordan type has no Jordan blocks of length j then the total number of Jordan blocks of length larger than j is divisible by p.

Investigate the stronger form of Rickard’s conjecture, stating under the same hypotheses that the number of blocks of length larger than j is divisible by p^{r−1}.

The cases j = 1 and j = p − 1 of Rickard’s conjecture are proved in Section 5.13, but the remaining cases are open. The analogue of Rickard’s conjecture in characteristic zero is the following, which asks whether Conjecture 5.14.7 is true.

Problem 5. Let k be an algebraically closed field of characteristic zero, and let A, B be commuting d × d matrices over k with d ≥ 1, such that for all λ, μ ∈ k, not both zero, λA + μB has the same Jordan canonical form (in particular this implies that A and B are nilpotent). Is it true that if there is a Jordan block of some given size then there are also Jordan blocks of all smaller sizes?

In particular, this asks whether there have to be Jordan blocks of size 1. This special case is proved in Theorem 5.14.8.

Problem 6. Find a proof or a counterexample to Suslin’s Conjecture 5.15.1. This states that if a kE-module of constant Jordan type has a Jordan block of length j with 2 ≤ j ≤ p−1 then there is either a Jordan block of length j + 1 or one of length j − 1.

Both Rickard’s conjecture and Suslin’s conjecture fail for nilvarieties instead of kEmodules, so information from the theory of vector bundles and Chern numbers alone cannot hope to prove these conjectures.

Problem 7. Find a proof or a counterexample to Conjecture 5.15.2 of Carlson, Friedlander and Pevtsova. This states that if p ≥ 5 and a kE-module has stable constant Jordan type [2] [1]^{j} then j ≥ r − 1.

Corollary 10.4.2 provides partial information for this problem using Chern numbers.

Problem 8. Let p ≥ 5 and E = (ℤ/p)^{2}. Do there exist kE-modules of stable constant Jordan type [2] [1] that are not of the form Ω^{2n}(kE/J^{2}(kE)) or Ω^{2n}(Soc^{2}(kE))? More generally, if p ≥ 5 and E = (ℤ/p)^{r}, do there exist kE-modules of stable constant Jordan type [2] [1]^{r−1} that are not of the form Ω^{2n}(kE/J^{2}(kE)) or Ω^{2n}(Soc^{2}(kE))?

Problem 9. Suppose that M has constant Jordan type, and let ϕ: kE → kE be an automorphism sending each X_{i} to X_{i} plus an element of J^{2}(kE), and let ϕ^{∗}(M) be the module obtained by letting kE act on M through ϕ. Is it true that there is an isomorphism of vector bundles ℱ_{j}(ϕ^{∗}(M)) ≅ ℱ_{j}(M) for each j with 1 ≤ j ≤ p?

Problem 10. Let M be a kE-module, and if necessary assume that M has constant Jordan type. Find module theoretic interpretations for expressions such as H^{1}(ℙ^{r−1}, Ker (θ_{M})), H^{0}(ℙ^{r−1}, M̃/Ker (θ_{M})), H^{i}(ℙ^{r−1}, ℱ_{j}(M)(ℓ)), etc.

Note that H^{0}(ℙ^{r−1}, Ker (θ_{M})) = Soc(M).

Problem 11. Let p be an odd prime. Given a vector bundle ℱ on ℙ^{r−1} does there exist a kE-module M of constant Jordan type and an integer i with 1 ≤ i ≤ p such that ℱ_{i}(M) ≅ ℱ?

For partial information, see the Realisation Theorem 8.1.1.

Problem 12. Let p be an odd prime, let r ≥ 3, and let M be a kE-module of constant Jordan type. If ℱ_{i}(M) ≅ 𝒪(n) for some 1 ≤ i ≤ p − 1 and some n ∈ ℤ, what are the restrictions on the possible values of i and n modulo p?

Problem 13. Let p = 2 and r be odd. For what values of n ≥ 0 do there exist kEmodules of constant Jordan type [2]^{n}[1]^{r−2}?

Partial information is given in Lemma 10.5.3. This question is closely related to the question of what Chern numbers can occur for rank r −2 vector bundles on the even dimensional projective space ℙ^{r−1}.

Problem 14. Let M be a kE-module such that for all shifted subgroups E′ ⊆ kE of rank two with X_{1} ∈ kE′, M↓_{E′} is stably isomorphic to a direct sum of copies of k. Does it follow that M is stably isomorphic to a direct sum of copies of k?

Compare with Theorem 3.2.1 in Okonek, Schneider and Spindler [202].

Problem 15. Find a proof or a counterexample to the Babylonian Tower Conjecture 10.11.3. This states that an infinitely stably extendable kE-module of constant Jordan type with non-projective part of dimension at most p − 2 has to be a direct sum of syzygies of the trivial module.

Problem 16. What is the smallest dimension of a kE-module with a given rank variety?

A lower bound is given by Carlson’s Theorem 12.1.3, and an upper bound is given by taking tensor products of Carlson modules L_{ζ}, see Section 1.5. These bounds are not at all close to each other.

Problem 17. For p = 2 the lower bound on dimension obtained by Carlson using Bézout’s theorem, see Theorem 12.1.3, is often sharp. When p is odd, it seems to be quite far from sharp. What is a good lower bound for p odd?

Problem 18. Let Λ be an exterior algebra on r generators. If M is a Λ-module of complexity c and Loewy length r − c + 1, what are the possibilites for V_{E}^{rk}(M)?

In Section 12.8 it is shown that V_{E}^{rk}(M) must be equidimensional, and that every equidimensional variety expressible as a finite union of set theoretic complete intersections occurs this way, but that leaves a gap in our knowledge.

Problem 19. Let Λ = Λ(X_{1}, …, X_{2m}) be an exterior algebra on 2m generators. What is the smallest dimension of a Λ-module whose rank variety is the quadric

Y_{1}Y_{2} + ⋯ + Y_{2m−1}Y_{2m} = 0 ?

There are several ways to produce a module of dimension 2^{m}, see Section 12.5. Is this the smallest possible?

Problem 20. Fløystad [120] investigated monads on projective space of the form

𝒪(−1)^{a} → 𝒪^{b} → 𝒪(1)^{c}.

Investigate the consequences of his results for modules of Loewy length three for exterior algebras and for elementary abelian p-groups.

References

[1] K. Akin, D. A. Buchsbaum, and J. Weyman, Schur functors and Schur complexes, Adv. in Math. 44 (1982), 207–278.

[2] G. Almkvist, The number of nonfree components in the decomposition of symmetric powers in characteristic p, Pacific J. Math. 77 (1978), 293–301.

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Index

𝔸^{r}, 𝔸^{r}(k), 47 abelian category, 7 additive category, 86 admissible

epimorphism, 86

monomorphism, 86 affine

patches, 50

scheme, 55

space, 3, 4, 47

variety, 47 Almkvist and Fossum, 43 almost split sequence, 83, 86 antipode, 22 automorphisms of the

Horrocks–Mumford bundle, 190

Babylonian towers, 116, 221 Bernoulli numbers, 146 Bernstein–Gelfand–Gelfand correspondence, 149,

169 Bézout’s theorem, 242 BGG, 169 bidirectional Koszul complex, 230 block idempotent, 30 Bockstein, 11, 237 Bott’s formula, 122, 124, 125 bounded derived category, 7 bundle

canonical, 122

cotangent, 117, 133

Higgs, 118

homogeneous, 117

Horrocks–Mumford, 115, 187, 194 instanton, 144, 183, 210 n-homogeneous, 84 null correlation, 84, 119, 180 Parent, 115 restriction, 134 Schwarzenberger, 185 simple, 112 tangent, 116, 133, 179 Tango’s examples ℱ_{W}, 120, 186, 210 Tango’s rank two, 115, 197 topological, 112 uniform, 118 vector, 111

calculus of finite differences, 142 canonical bundle, 122 Carlson module, 6, 271 Cauchy’s integral formula, 145, 214 Čech complex, 60 chain complex, 7 charge, 183 Chern

character, 130, 133, 136, 147 numbers, 130, 133, 147, 205 of dual bundle, 137 of Frobenius, 146 of twists, 136

polynomial, 130–133, 136, 147, 153 roots, 137, 147 Chouinard’s theorem, 13 Chow ring, 130 circle group, 40 cJt(kE), 87, 154 closed

point, 48–50

subvariety of 𝔸^{r}, 47 closure of GL_{n}(k)-orbit, 69 coherent

𝒪_{V} -module, 56 modules, 7, 112 sheaves, 129, 137

cohomology of a monad, 119 of groups, 5 of 𝒪(j), 121 of Ω^{m}(j), 122 relative, 245

complete intersection, 224, 250

set theoretic, 251 symmetric function, 25

complexity, 12, 15, 250 congruence, 205 conjugate partition, 25, 26, 68 constant

image property, 91, 152 Jordan type, ix, x, 11, 77

with one non-projective block, 102 kernel property, 97, 153 rank, 77, 112, 151, 203

content, 38 contour integral, 145 coordinate ring, 47, 48 cotangent bundle, 117, 133, 179 cyclic

group, 33

shifted subgroup, x, 3, 13 Sylow subgroups, 1

D(r), 49 Dade’s lemma, 13, 80, 178, 208, 210–214 decomposable vector bundle, 111 derivations, sheaf of, 116 derived category, 7, 165, 170 determinantal form, 25, 27 diagonal map, 20 diagrams for modules, 17 differential graded modules, 170 differentials, sheaf of, 117 dihedral group, 2 direct image sheaf, 134 divided power, 24 dominance order, 68, 69 dual

bundle, 111

Chern numbers of, 137 group theoretic, 22 module, ℱ_{i} of, 158 restricted Lie, 22

elementary abelian group, 3 endotrivial module, x, 87, 179 enough injectives, 59 equation, Markov, 118 equidimensional variety, 251 Euler

characteristic, 143

sequence, 117 exact category, 86, 154 examples of Tango, 120, 186, 210 exterior algebra, 2, 214, 250 powers, 23, 122 extraspecial group, 190

ℱ_{i}(M), 152

ℱ_{i}(M ⊗_{k} N), 167 ℱ_{i}(M^{∗}), 158 ℱ_{i}(Ω(M)), 157 fibre, 55 first Gambelli formula, 25 fl(R), 58 flat

homomorphism, 5

module, 13 forgetful functor, 33 free

module, 13 Frobenius

functor, 27

map, 11, 146, 149 morphism, 149 function field, 48

𝒢, 167

Gambelli formulas, 25

Γ(X, ℱ), 53

Gaussian polynomial, 36, 44 general linear group, 117 generalised

Gaussian polynomial, 37, 44 quaternion group, 2, 13

generic image, 99 Jordan type, 72, 73 kernel, 94, 103 point, 48, 50 properties, 48 rank, 73 germ, 53 Gerstenhaber’s theorem, 66 GL(r, k), 117 gluing condition, 52 graded module, 2, 9, 56 Green, J. A., 35 Grothendieck, 115 group, 87 –Riemann–Roch theorem, 146

group algebra, 2 elementary abelian, 3 extraspecial, 190 general linear, 117 scheme, 5 symmetric 𝔖_{9}, 241 theoretic dual, 22 theoretic tensor product, 21

h_{d}, 25 H^{2}(GL(r, k), k^{×}), 117 Hilbert series, 129 Hilbert’s

basis theorem, 126 Nullstellensatz, 47 syzygy theorem, 125, 134, 144

Hirzebruch–Riemann–Roch theorem, 143, 210,

212, 213, 215, 216, 218 Hochschild resolution, 126 homogeneous

ideal, 49 module, 81 polynomial, 49 vector bundle, 117

hook formula, 37, 45 length, 37

Horrocks parent bundle, 197 Horrocks–Mumford bundle, 115, 187, 194 hypersurface, 224, 252

ℑ(M), 99 image, generic, 99 indecomposable

module, 33 vector bundle, 111

injective 𝒪-module, 59 resolution, 59

injective module, 13 instanton bundle, 144, 183, 210 inverse image sheaf, 135 irreducible

closed set, 47 vector bundle, 112

irrelevant ideal, 49, 50

Jacobi–Trudi identity, 25 Jacobson radical J(kE), 3 Jordan type, 69 JType, 87

𝔎(M), 94 k(x), 53 K_{0}(cJt(kE)), 87 Kempf’s splitting theorem, 116 kernel, generic, 94 Koszul

complex, 122, 125, 177, 188

factorisation, 231 Kronecker’s classification, 2 Krull–Schmidt theorem, 81 Künneth theorem, 168

L^{λ}, 27 L_{ζ}, 6 L_{V,a}, 51 λ-ideal, 41 λ-ring, 39 last survivor, 169 Laurent

polynomial, 129, 130, 140, 147 power series, 129

left

derived functor, 135

total, 135 exact, 54

line bundle, 111 linear space, 51 locally

closed, 50

free 𝒪-module, 111

ringed space, 55

split sequence, 85 Loewy

layers, 186

length, 250

Macaulay2, 12, 236 Magma, 12, 191, 193, 241 mapping cone, 7 Markov equation, 118 Massey product, 11 matrix factorisation, 225 maximal, 70

Cohen–Macaulay module, 224 Jordan set, 73 Jordan type, 72 rank, 74, 95 shifted subgroup, 3

MCM(A), 225 \underline{MCM}(A), 225 MF(f), 225 \underline{MF}(f), 226 mod(R), 58 module category, 5 moduli space

of instantons, 183 monad, 118, 180, 183, 185, 186, 189, 271 morphism of presheaves, 52

Nakayama’s lemma, 29, 94, 113 Newton’s identity, 139 nilpotent

endomorphisms, 14

Jordan type, 69 nilvariety, 31, 64, 178, 219

of constant Jordan type, 77, 214, 217 Noetherian scheme, 55, 112 null correlation bundle, 84, 119, 180

𝒪_{X}-module, 55 Ω = Ω_{ℙ^{r−1}}, 117, 179 Ω(M), 6

ℱ_{i} of, 157 orbit closure, 69 Orlov’s correspondence, 223

P_{n}, 40 P_{m,n}, 40 ℙ^{r−1}, ℙ^{r−1}(k), 49 p-λ-ideal, 41 Parent bundle, 115 partition, 24, 25, 68 perfect complex, 7, 170 phenomenology, 241 π-points, 5 Picard group, 111 Poincaré series, 129 polynomial ring, 11 power sums, 138 presheaf, 52 prime ideal, 47 problems, 269 projective

module, 13

space, 49, 115

spectrum, 50

variety, 50 properties, generic, 48

qfl(R), 58 qmod(R), 58 quadratic reciprocity, 209 quantum

complete intersection, 30

number, 183 quasi-coherent 𝒪_{X}-module, 56 quasi-finite length module, 58 quasi-finitely generated module, 58 quasi-isomorphism, 7 quasi-projective variety, 50, 51 quaternion group Q_{8}, 30 Quillen, 86 quiver with relations, 89, 118 quotient

bundle, 114 sheaf, 54

ℛ, 8 radical

ideal, 47

layers of kE, 27 rank, 129

maximal, 65

of a locally free sheaf, 111

of a nilvariety, 64 variety, 4, 14

rational functions, 48 map, 50 reduced

matrix factorisation, 226

ring, 55

scheme, 55, 112 reduction of dimension, 223 regular

map, 50

sequence, 125 relative

cohomology, 245 syzygy, 248

representation type, 1

restricted Lie dual, 22 tensor product, 21

restriction maps, 52 of vector bundles, 134

Rickard’s

conjecture, 103 Riemann–Roch theorem, 143 right derived functor, 134 ringed space, 52 \underline{RMF}(f), 226

S^{λ}, 26

Schanuel’s lemma, 126 scheme, 54, 55, 112 Schur

function, 25, 27, 37

functor, 26, 44, 75, 79, 80, 106, 266 ’s lemma, 112 Schwarzenberger’s bundles, 185 theorem, 143

second Gambelli formula, 26 semidihedral group, 2 Serre

quotient, 59

subcategory, 59 set theoretic complete intersection, 251 sheaf, 52

direct image, 134

inverse image, 135

of derivations, 116

of differentials, 117

restriction, 134 sheafification, 54 shift of a graded module, 56 shifted subgroup, 3 short exact sequences, 86 simple vector bundle, 112 singularity category, 9, 223, 226 SL(2, 5), 191 SL(5, k), 190 SL(r, k), 117 SL(2, ℂ), representations of, 40 small module, variety of, 241 space

tangent, 65 Specht module, 241 special

λ-ring, 40

linear group, 117 spectrum, 48 splitting

principle, 41, 138

type, 118 stable module category, 5, 225 stably extends, 221 stalk, 53 structure sheaf, 52, 54 subbundle, 114 Suslin’s conjecture, 108 symmetric

group 𝔖_{9}, 241

powers, 23, 121 symplectic form, 83, 119 syzygy

Hilbert’s — theorem, 125, 134, 144 relative, 248

T_{V,a}, 51 𝒯, 116, 179 tangent

bundle, 116, 133, 179

space, 51, 65 Tango’s

bundles ℱ_{W}, 120, 186, 210

rank two bundle, ix, 115, 197 Tate

cohomology, 6, 86

duality, 6 Td(ℱ), 145 tensor product, 20, 75

ℱ_{i} of, 167 for ℤ/p, 35 group theoretic, 21 of 𝒪_{X}-modules, 55 of sheaves, 54 restricted Lie, 21

θ_{M}, 150

Todd class, 145

topological vector bundle, 112 total left derived functor, 135 transfer, 6, 168 triangulated category, 6, 224 trichotomy theorem, 1 truncated polynomial ring, 4 twist, twisting sheaf, 56 twisted

cubic, 251

version of kE, 30, 239

U_{max}(M), 73

uniform vector bundle, 118

V (I), 49 V (r), 49 V_{E}^{rk}(M), 4 Vandermonde determinant, 25 variety of a small module, 241 Vec(ℙ^{r−1}), 154 vector bundle, see also bundle, 111

Weyl module, 27

wild representation type, 1, 88 W -modules, 100, 103, 107 W_{n,d}, 100

X_{i}, 3, 150

Y_{i}, 150

Yang–Mills equations, 183

(ℤ/p)^{r}, 3

Zariski topology, 47–50