Some results on modules of constant Jordan type

for elementary abelian p-groups

A thesis presented for the degree of Doctor of Philosophy at the University of Aberdeen

Shawn Baland BA (U. Colorado, Boulder)

Declaration

I declare that I have composed this thesis myself, that
it has not been accepted in any previous application for
a degree, and that the work contained herein is my own.
I also declare that all quotations have been
distinguished by quotation marks, and that all sources
of information have been specifically acknowledged.

Acknowledgements

I would first like to thank my supervisor, Dave Benson, for his expertise and generosity with his time. It was an honour to be your student. I would also like to thank the members of the Institute of Mathematics for welcoming me into their department and allowing me to be part of their working family.

There were several people in the department with whom I had discussions that contributed to this work or my overall knowledge. In particular, I am grateful to Markus Linckelmann, Radha Kessar, Michał Stolorz and Aaron Chan. In addition, I would like to thank Michael Weiss and Filipp Levikov for electing to eat lunch at a reasonable hour. My warmest thanks also go to anybody who bought me coffee at the Hub on any occasion.

I was the recipient of some rather non-trivial funding whilst at Aberdeen. I am eternally grateful to Geoffrey Robinson for his profound generosity. My appreciation also goes to the Colorado Alpha Chapter of Phi Beta Kappa for partially funding my postgraduate research through the Crisp Fellowship.

Finally, I would like to thank Richard Green in Boulder for fostering my love of algebra, and for much encouragement throughout my undergraduate and postgraduate years.

Dedication

To Lyndsie and Stoney Pants, with all my love.

Summary

Let E be an elementary abelian p-group of rank r and let k be an algebraically closed field of characteristic p. We investigate finitely generated kE-modules of stable constant Jordan type [a][b] with 1 ≤ a, b ≤ p − 1 using the functors F_{i} from finitely generated kEmodules to vector bundles on the projective space ℙ^{r−1} constructed by Benson and Pevtsova. In particular, we study relations on the Chern numbers of the trivial bundle M̃ to obtain restrictions on a and b for sufficiently large ranks and primes.

We then study kE-modules with the constant image property and define the constant image layers of a module with respect to its maximal submodule having the constant image property. We prove that almost all such subquotients are semisimple. Focusing on the class of W -modules in rank two, we also calculate the vector bundles F_{i}(M) for all W -modules M.

For E of rank two, we derive a duality formula for kE-modules M of constant Jordan type and their generic kernels 𝔎(M). We use this to answer a question of Carlson, Friedlander and Suslin regarding whether or not the submodules J^{−i}𝔎(M) also have constant Jordan type for all i ≥ 0. We show that this question has an affirmative answer whenever p = 3 or J^{2}𝔎(M) = 0. We also show that it has a negative answer in general by constructing a kE-module M of constant Jordan type for p ≥ 5 such that J^{−1}𝔎(M) does not have constant Jordan type.

Finally, we recall the proof of a theorem of Benson to show that if M is a kE-module of constant Jordan type containing no Jordan blocks of length one, then there always exist certain submodules of J^{−1}𝔎(M)/J^{2}𝔎(M) having a particularly nice structure.

Contents

Chapter 1. Introduction  1
Chapter 2. Preliminaries  5
2.1. Nilpotent matrices and Jordan types  5
2.2. Elementary abelian p-group algebras  9
2.3. Representation type  11
2.4. Diagrams for modules  12
2.5. The generic Jordan type  15
2.6. Modules of constant Jordan type  19
2.7. Examples of modules of constant Jordan type  21
2.8. Some questions and conjectures  26
Chapter 3. Restrictions on Jordan types from vector bundles on projective space 28
3.1. Vector bundles on projective space  28
3.2. Trivial sheaves for kE-modules and the operator θ_{M}  30
3.3. The functors F_{i,j} and F_{i}  33
3.4. Chern classes for vector bundles on ℙ^{r−1}  35
3.5. Benson’s theorem on small Jordan types  36
3.6. Restrictions on the stable Jordan type [a][b], a ≠ b  37
3.7. Restrictions on the stable Jordan type [a]^{2}  40
3.8. Non-realisation of the stable Jordan type [3][2][1] in certain cases  42
3.9. Discussion  43
Chapter 4. Constant image layers and vector bundles for W -modules  45
4.1. The constant image property  45
4.2. Constant image layers  47
4.3. W -modules and the constant image property in rank two  49
4.4. Vector bundles for W -modules  51
Chapter 5. Generic kernels and a question of Carlson, Friedlander and Suslin 53
5.1. Generic kernels  53
5.2. Generic kernels for modules of constant rank  55
5.3. The generic kernel filtration and the main question  56
5.4. Generic images  58
5.5. The constant kernel property  60
5.6. Duality for subquotients in the generic kernel filtration  62
5.7. Blocks of small length in certain subquotients  63
5.8. Question 5.10 in the cases p = 3 and J^{2}𝔎(M) = 0  66
5.9. Some counterexamples to Question 5.10  68

Chapter 6. On the structure of the subquotient J^{−1}𝔎(M)/J^{2}𝔎(M)  75
6.1. Benson’s theorem on Rickard’s conjecture  75
6.2. Diamond shaped submodules of J^{−1}𝔎(M)/J^{2}𝔎(M)  78
Appendix A. Generic points in algebraic geometry  81
A.1. Generic points for affine varieties  81
A.2. Generic points in the language of schemes  83
Appendix B. Modules for ℤ/p  85
Appendix C. A brief introduction to rank varieties  88
Bibliography  90

CHAPTER 1 Introduction

Elementary abelian p-groups have played an essential role in modular representation theory for over four decades now. This can be traced back to Quillen [33, 34], who showed for a finite group G and field k of characteristic p that the ring H^{∗}(G, k) modulo its nilpotent elements has Krull dimension equal to the maximal rank of any elementary abelian p-subgroup of G. A few years later, Dade [20, 21] classified the endotrivial modules for all finite abelian p-groups by using information coming from elementary abelian p-group algebras. Further motivation in this area came via Chouinard [18], who showed that if M is a module for a finite group G, then M is projective if and only if its restriction M ↓_{E} to every elementary abelian subgroup E of G is projective.

Perhaps the greatest contribution in this area was that of Carlson [12, 13], who defined support varieties for modules over finite groups and, for an elementary abelian p-group E, constructed the rank variety V_{E}^{♯}(M) of a kE-module M. Avrunin and Scott [2] later proved Carlson’s conjecture that the rank variety and support variety coincide for kE-modules, effectively placing elementary abelian p-group representations at the centre of the rapidly emerging area of support theory.

The objects of interest in this thesis, namely modules of constant Jordan type, constitute a relatively new area in the modular representation of elementary abelian p-groups. That said, their definition arises naturally from the classical set-up.

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. Choosing generators g_{1}, …, g_{r} of E, we set X_{i} = g_{i} − 1 ∈ kE, and for each non-zero point α = (λ_{1}, …, λ_{r}) ∈ 𝔸^{r}(k) we define the element X_{α} = λ_{1}X_{1} + ⋯ + λ_{r}X_{r} ∈ kE. Because k has characteristic p, it follows that X_{α}^{p} = 0, hence X_{α} acts on a finite dimensional kE-module M via a matrix whose Jordan canonical form contains Jordan blocks of length at most p. The Jordan type of X_{α} on M is the partition [p]^{a_{p}}[p − 1]^{a_{p−1}} … [1]^{a_{1}} of dim_{k}(M), where X_{α} acts on M via a_{j} Jordan blocks of length j. We say that M has constant Jordan type if the Jordan type of X_{α} on M is the same for all non-zero α ∈ 𝔸^{r}(k). In this case, we call the list [p − 1]^{a_{p−1}} … [1]^{a_{1}} the stable Jordan type of M.

Chapter 1.0 Page 2

Modules of constant Jordan type were defined by Carlson, Friedlander and Pevtsova [16] in the context of finite group schemes and have been investigated more recently in [5, 6, 7, 9, 17] for elementary abelian p-groups. One of the primary objectives in studying these modules is to determine which stable Jordan types are realised as the stable Jordan types of kE-modules having constant Jordan type. Several conjectures have been made with the above goal in mind, although there has been little progress towards a full classification of realisable Jordan types.

An interesting development towards understanding modules of constant Jordan type was due to Benson and Pevtsova [9], who defined the functors F_{i} for 1 ≤ i ≤ p from finitely generated kE-modules to coherent sheaves on the projective space ℙ^{r−1}. They also showed that a module M has constant Jordan type if and only if F_{i}(M) is an algebraic vector bundle on ℙ^{r−1} for all i. Benson [7] later used information coming from the Chern classes of the bundles F_{i}(M) to prove that if a kE-module has stable constant Jordan type [p−1]^{a_{p−1}} … [1]^{a_{1}} and ∑_{i=1}^{p−1} i·a_{i} ≤ min(r−1, p−2), then a_{i} = 0 for all 2 ≤ i ≤ p − 1. Roughly speaking, this statement says that if a stable Jordan type contains some block of length greater than one and is ‘small’ relative to p and r, then it cannot be realised by a kE-module of constant Jordan type.

More recently, Carlson, Friedlander and Suslin [17] have investigated modules having the constant image property, i.e., those kE-modules M for which the image of X_{α} on M is independent of the choice of non-zero α ∈ 𝔸^{r}(k). One of the key facts about the constant image property is that it implies the constant Jordan type property, although the converse does not hold.

In the case where E has rank two, the authors of [17] also defined the generic kernel 𝔎(M) of a kE-module M, which can be characterised as the largest submodule of M having the constant image property. One of the striking features of the generic kernel is that, whenever M has constant rank (i.e., the rank of X_{α} on M is independent of the choice of non-zero α ∈ 𝔸^{2}(k)), the kernel of the action of X_{α} on M is contained in 𝔎(M). Benson [6] used this property to prove that if a module (for any rank r) has constant Jordan type containing no blocks of length one, then the total number of Jordan blocks is divisible by p. This statement proves a special case of an unpublished conjecture of Rickard.

The results in this thesis are roughly divided into two parts. The first part expands on Benson and Pevtsova’s vector bundle approach to obtain new restrictions on small Jordan types, while the second investigates properties of the generic kernel for modules of constant rank over elementary abelian p-groups of rank two. The specific organisation is as follows.

In Chapter 2 we introduce the necessary background material for elementary abelian p-group algebras. We also provide a detailed exposition on generic Jordan types and then use this theory to give complete proofs of the main theorems in [16] concerning the behaviour of modules of constant Jordan type under common module theoretic operations. We then give some important examples of modules of constant Jordan type. Some of these appeared in [16], but we also provide a new family of examples in rank two whose constant Jordan types contain only blocks of length two and three. We close Chapter 2 with a discussion of some current problems in the area.

Chapter 3 begins with an overview of vector bundles on projective space. We then introduce the functors F_{i} defined by Benson and Pevtsova and discuss their general properties. We also recall the theory of Chern classes for vector bundles on projective space and record a congruence formula for Chern polynomials of twists due to Benson and Pevtsova, which can be thought of as an analog of Fermat’s little theorem. This will allow us to give a complete proof of Benson’s theorem on small Jordan types. We then use the functors F_{i} to obtain restrictions on the values of a and b for which there exists a kE-module of stable constant Jordan type [a][b] for sufficiently large ranks and primes. In several cases, we show that these restrictions provide new bounds on p and r for which there does not exist a kE-module of stable constant Jordan type [a][b]. Finally, we use similar techniques to obtain restrictions on p and r for which there exists a kE-module of stable constant Jordan type [3][2][1].

In Chapter 4 we transition to a discussion of modules with the constant image property. We show that every kE-module contains a unique submodule that is maximal with respect to the constant image property, and then define the constant image layers of a kE-module with respect to this submodule, analogously to how one defines the socle layers of modules over arbitrary rings. We prove for every kE-module that almost all of the constant image layers are semisimple. Restricting our attention to the case in which E has rank two, we then examine a special class of kE-modules with the constant image property called W -modules, which were also defined in [17]. After proving a general lemma about the behaviour of the functors F_{i} with respect to modules with the constant image property, we then calculate the bundles F_{i}(M) for all W -modules M. As a corollary, we show that the functor F_{1} from modules of constant Jordan type to vector bundles on ℙ^{1} is essentially surjective.

Chapter 5 seeks to answer Question 5.10, which was posed by Carlson, Friedlander and Suslin [17]. It asks the following: If a k(ℤ/p)^{2}-module M has constant Jordan type, is it true that J^{−i}𝔎(M) has constant Jordan type for all i ≥ 0?

Here J = J(kE) denotes the Jacobson radical of kE, and J^{−i}𝔎(M) denotes the set of elements m ∈ M for which J^{i}m ⊆ 𝔎(M). We should point out that the original statement of Question 5.10 was given in terms of the submodules X_{1}^{−i}𝔎(M), but as we shall see in Proposition 5.8, these two formulations are equivalent. In addition to the above statement having been true in all prior known examples, motivation for Question 5.10 came from the observation that if M has constant Jordan type, then there exists a filtration

0 = J^{p}𝔎(M) ⊆ J^{p−1}𝔎(M) ⊆ ⋯ ⊆ J^{−p+2}𝔎(M) ⊆ J^{−p+1}𝔎(M) = M

of M in which J^{i}𝔎(M) has constant Jordan type for all i ≥ 0 and for i = −p + 1.

Our results will show that Question 5.10 has an affirmative answer whenever p = 3 or J^{2}𝔎(M) = 0. In proving this, we derive a particularly nice duality formula for the subquotients J^{a}𝔎(M)/J^{b}𝔎(M), a ≤ b, in the case where M has constant rank. We take special care to show that this result is independent of the choice of Hopf algebra structure on kE. Our approach will also lead us to investigate the number of blocks having small length in the Jordan type of X_{α} on J^{−i}𝔎(M)/J^{j}𝔎(M), where α is a non-zero point in 𝔸^{2}(k), i ≥ 1 and j ≥ 2. Finally, we show that Question 5.10 has a negative answer in general by constructing a k(ℤ/p)^{2}-module such that J^{−1}𝔎(M) does not have constant Jordan type in the first interesting case, namely that in which p ≥ 5, J^{2}𝔎(M) ≠ 0 and J^{3}𝔎(M) = 0.

In Chapter 6 we recount the proof of the main theorem in Benson [6], which deals with modules of constant Jordan type containing no blocks of length one. In the rank two case, we use ideas from that proof to show for such kE-modules M that the subquotient J^{−1}𝔎(M)/J^{2}𝔎(M) always contains submodules of constant Jordan type isomorphic to the examples introduced in Chapter 2.

In summary, the author’s original results presented in this thesis are Theorems 2.22, 3.13 (this was due to Benson; we provide a new proof here), 3.14, 3.21, 3.23, Lemma 4.11, Theorem 4.12, Lemma 4.17, Theorems 4.18, 5.23, Lemma 5.24, Theorems 5.25, 5.26, 5.27, Lemma 5.28, Theorem 5.29, Example 5.30 and Theorem 6.2.

CHAPTER 2 Preliminaries

This chapter is dedicated to introducing background material from the modular representation theory of elementary abelian p-groups and modules of constant Jordan type. Our goal is to give a detailed summary of the main results of Carlson, Friedlander and Pevtsova [16]. For the representation theory of arbitrary finite groups, we refer the reader to Dornhoff [22, 23], Benson [3, 4] and Carlson [14]. A very nice treatment of modules over commutative rings can be found in Atiyah and MacDonald [1]. We close this chapter with a few interesting examples of modules of constant Jordan type and a discussion of some current problems in the area.

2.1. Nilpotent matrices and Jordan types

While it would be tempting to delve directly into the structures on which this thesis is based, we should note that modules of constant Jordan type have their roots in the theory of nilpotent matrices. Many of the notions we shall encounter when dealing with elementary abelian p-group representations are best introduced with respect to nilpotent matrices, and what’s more, modules of constant Jordan type may only be defined intrinsically when one views them in this broader context. We begin with a discussion of orderings on nilpotent matrices.

If R is a ring, we denote by Mat_{n}(R) the ring of n × n matrices over R. An element A ∈ Mat_{n}(R) is nilpotent if there exists an integer d > 0 such that A^{d} = 0. The nilpotent elements of Mat_{n}(R) may be partially ordered via the relation

A ⪰ B ⟺ rank(A^{s}) ≥ rank(B^{s}) for all s ≥ 0.  (2.1)

We obtain an equivalence relation from this ordering by setting A ∼ B if A ⪰ B and B ⪰ A.

We shall focus mainly on the case where R is an algebra over an algebraically closed field k, e.g., the function field k(t_{1}, …, t_{r}) in the algebraically independent variables t_{1}, …, t_{r} over k. For nilpotent matrices over any field, the partial ordering ⪰ can be described in terms of Jordan canonical forms.

Let k be a field and let A ∈ Mat_{n}(k). Recall that if k contains all eigenvalues of A, then A is conjugate in Mat_{n}(k) to a block diagonal matrix

A′ = (\begin{matrix}J_{1} \\ & J_{2} \\ & & . & . & . \\ & & & & & J_{t} \end{matrix})

where each block J_{i} is of the form

J_{i} = (\begin{matrix}λ & 1 \\ & λ & 1 \\ & & ... & ... \\ & & & λ & 1 \\ & & & & λ \end{matrix}) ,

λ being an eigenvalue of A. (Here we have omitted entries equal to zero.) The matrix A′ is called the Jordan canonical form of A, and each block J_{i} is called a Jordan block of A′. Note that the Jordan canonical form of A is unique up to permutation of its Jordan blocks. Given a Jordan block J_{i}, we refer to the number of rows in J_{i} as its length.

Definition 2.1. If A ∈ Mat_{n}(k) is nilpotent, then all of the eigenvalues of A are zero, hence the Jordan canonical form of A is completely determined by the lengths of its Jordan blocks. Listing the lengths of the Jordan blocks of A in descending order d_{1} ≥ d_{2} ≥ ⋯ ≥ d_{t}, we obtain a partition [d_{1}][d_{2}] … [d_{t}] of n. We call this partition the Jordan type of A.

Partitions of n are partially ordered via the dominance ordering. Specifically, if \underline{d} = d_{1} ≥ d_{2} ≥ ⋯ ≥ d_{t} and \underline{e} = e_{1} ≥ e_{2} ≥ ⋯ ≥ e_{t} are partitions of n, then \underline{d} is said to dominate \underline{e}, written \underline{d} ≥ \underline{e}, if

∑_{j=1}^{i} d_{j} ≥ ∑_{j=1}^{i} e_{j} for all 1 ≤ i ≤ t.

This allows us to partially order the nilpotent elements of Mat_{n}(k) by defining A ⪰ B if the Jordan type of A dominates the Jordan type of B. It is well known that this ordering is equal to that defined by the condition (2.1). (For example, see Lemma 6.2.2 of [19].) In this case, we have A ∼ B if and only if A and B have the same Jordan canonical form, i.e., if and only if A and B are conjugate.

Our immediate goal is to use the above ordering to obtain some maximality criteria for Jordan types using the theory of generic points for affine varieties. (See Appendix A for a discussion of generic points and generic properties.) The following is a variation of Lemma 1.2 of [26].

Proposition 2.2. If k is an algebraically closed field, then the following are equivalent for matrices A, A_{1}, …, A_{r} ∈ Mat_{n}(k).

(i) If k(t_{1}, …, t_{r}) is the function field in the algebraically independent variables t_{1}, …, t_{r} over k, then

rank(A) = rank(A + t_{1}A_{1} + ⋯ + t_{r}A_{r})

as matrices in Mat_{n}(k(t_{1}, …, t_{r})). (ii) For every field extension K/k and all (λ_{1}, …, λ_{r}) ∈ 𝔸^{r}(K),

rank(A) ≥ rank(A + λ_{1}A_{1} + ⋯ + λ_{r}A_{r})

as matrices in Mat_{n}(K). (iii) For all (λ_{1}, …, λ_{r}) ∈ 𝔸^{r}(k),

rank(A) ≥ rank(A + λ_{1}A_{1} + ⋯ + λ_{r}A_{r}).
(iv) The points (λ_{1}, …, λ_{r}) for which
rank(A) = rank(A + λ_{1}A_{1} + ⋯ + λ_{r}A_{r})

form a dense open subset of 𝔸^{r}(k).

Proof. We may think of (t_{1}, …, t_{r}) ∈ 𝔸^{r}(k(t_{1}, …, t_{r})) as the generic point of 𝔸^{r}(k) since, for any polynomial f ∈ k[t_{1}, …, t_{r}], we have

f(λ_{1}, …, λ_{r}) = 0 for all (λ_{1}, …, λ_{r}) ∈ 𝔸^{r}(k)

if and only if f = 0. Setting s = rank(A), the condition

rank(A + t_{1}A_{1} + ⋯ + t_{r}A_{r}) = s  (2.2)

is characterised in terms of the matrix minors of A + t_{1}A_{1} + ⋯ + t_{r}A_{r}, which are homogeneous polynomials in k[t_{1}, …, t_{r}] by the formula for determinants. Specifically, (2.2) holds if and inly if every (s + 1) × (s + 1) minor of A + t_{1}A_{1} + ⋯ + t_{r}A_{r} is the zero polynomial, and some s × s minor of A + t_{1}A_{1} + ⋯ + t_{r}A_{r} is non-zero. The proof now follows by applying Proposition A.2 to the set T ⊆ k[t_{1}, …, t_{r}] of s × s minors of A + t_{1}A_{1} + ⋯ + t_{r}A_{r}. Note that the open subset in (iv) is non-empty since it contains the point (0, …, 0) ∈ 𝔸^{r}(k). □

The proof of Proposition 2.2 can be easily applied to the matrix minors of powers of A ∈ Mat_{n}(k). The following appeared as Proposition 1.3 of [26].

Proposition 2.3. If k is an algebraically closed field, then the following are equivalent for commuting nilpotent matrices A, A_{1}, …, A_{r} ∈ Mat_{n}(k).

(i) If k(t_{1}, …, t_{r}) is the function field in the algebraically independent variables t_{1}, …, t_{r} over k, then

A ∼ A + t_{1}A_{1} + ⋯ + t_{r}A_{r}

as matrices in Mat_{n}(k(t_{1}, …, t_{r})). (ii) For every field extension K/k and all (λ_{1}, …, λ_{r}) ∈ 𝔸^{r}(K),

A ⪰ A + λ_{1}A_{1} + ⋯ + λ_{r}A_{r}

as matrices in Mat_{n}(K). (iii) For all (λ_{1}, …, λ_{r}) ∈ 𝔸^{r}(k),

A ⪰ A + λ_{1}A_{1} + ⋯ + λ_{r}A_{r}.
(iv) The points (λ_{1}, …, λ_{r}) for which
A ∼ A + λ_{1}A_{1} + ⋯ + λ_{r}A_{r}

form a dense open subset of 𝔸^{r}(k).

Motivated by the previous two propositions, the following appeared as Definition 1.4 of [26].

Definition 2.4. Let A, A_{1}, …, A_{r} ∈ Mat_{n}(k) be commuting nilpotent matrices over an algebraically closed field k. We say that A is rank maximal with respect to A_{1}, …, A_{r} if A, A_{1}, …, A_{r} satisfy the equivalent conditions of Proposition 2.2. Similarly, we say that A is maximal with respect to A_{1}, …, A_{r} if A, A_{1}, …, A_{r} satisfy the equivalent conditions of Proposition 2.3.

The following two theorems, which appear as Theorems 1.9 and 1.12 of [26], respectively, are the main results concerning rank maximality and maximality that will be needed in the sequel. The second will be especially important when proving the intrinsic nature of the constant Jordan type property in Section 2.6.

Theorem 2.5. If k is an algebraically closed field and A, A_{1}, …, A_{r}, B_{1}, …, B_{r} ∈ Mat_{n}(k) are commuting nilpotent matrices such that A is rank maximal with respect to A_{1}, …, A_{r}, then A is also rank maximal with respect to A_{1}, …, A_{r}, A_{1}B_{1}, …, A_{r}B_{r}. Moreover, we have rank(A) = rank(A + A_{1}B_{1} + ⋯ + A_{r}B_{r}).

Theorem 2.6. If k is an algebraically closed field and A, A_{1}, …, A_{r}, B_{1}, …, B_{r} ∈ Mat_{n}(k) are commuting nilpotent matrices such that A is maximal with respect to A_{1}, …, A_{r}, then A is also maximal with respect to A_{1}, …, A_{r}, A_{1}B_{1}, …, A_{r}B_{r}. Moreover, we have A ∼ A + A_{1}B_{1} + ⋯ + A_{r}B_{r}.

2.2. Elementary abelian p-group algebras

Let p be a prime number and k an algebraically closed field of characteristic p. Throughout the rest of this thesis we consider an elementary abelian p-group E of rank r, that is,

E = ⟨g_{1}, …, g_{r}⟩ ≅ (ℤ/p)^{r},

where the g_{i} denote a fixed set of pairwise commuting generators of order p. To study finite dimensional kE-modules, it will be convenient to consider the elements X_{i} = g_{i} − 1 ∈ kE. Because k has characteristic p, we have X_{i}^{p} = g_{i}^{p} − 1^{p} = 0 for all i. It follows that kE is isomorphic to a truncated polynomial ring

kE ≅ k[t_{1}, …, t_{r}]/(t_{1}^{p}, …, t_{r}^{p}),

where the k-algebra isomorphism maps each X_{i} to the coset containing t_{i}.

Note that any element x ∈ kE can be written in the form λ + f(X_{1}, …, X_{r}), where λ ∈ k and f ∈ k[t_{1}, …, t_{r}] is a polynomial with zero constant term. Since each X_{i} has pth power zero, we have f(X_{1}, …, X_{r})^{p} = 0, hence x^{p} = λ^{p}. It follows that x is a zero divisor if and only if λ = 0. This shows that kE is a local ring with unique maximal ideal

J = J(kE) = (X_{1}, …, X_{r}).

In particular, every projective kE-module is free, thus the regular representation kE is the unique projective indecomposable kE-module up to isomorphism. By the usual correspondence between projective indecomposable and simple modules, this implies that the trivial module k ≅ kE/J is the unique simple kE-module up to isomorphism.

Whenever we consider a kE-module throughout this thesis, it will always be implied that the module in question is finitely generated, i.e., finite dimensional. As a matter of notation, if M is a kE-module and x ∈ kE, we denote by Ker(x, M), Im(x, M) and rank(x, M) the kernel, image and rank of x when considered as a vector space endomorphism M → M. Assuming we have chosen a basis of M, we denote by x_{M} the image of x under the matrix representation kE → Mat_{dim_{k}(M)}(k) affording M.

By the above remarks, if M is a kE-module and x ∈ J, then x_{M} is a nilpotent matrix whose Jordan canonical form contains Jordan blocks of length at most p. We call the Jordan type of x_{M} the Jordan type of x on M (see Definition 2.1) and denote this by

JType(x, M) = [p]^{a_{p}}[p − 1]^{a_{p−1}} … [1]^{a_{1}}

where, for 1 ≤ j ≤ p, x_{M} contains a_{j} Jordan blocks of length j. It is clear from the definition that JType(x, M) is a partition of dim_{k}(M).

We now introduce a special family of elements in J that are parameterised by points in the affine space 𝔸^{r}(k). Specifically, for each α = (λ_{1}, …, λ_{r}) ∈ 𝔸^{r}(k) we define

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

Under the assignment α ↦ X_{α} + J^{2}, we may identify 𝔸^{r}(k) with J/J^{2} as k-vector spaces. This relationship will be of particular importance when we study projective (r−1)-space in the context of kE-modules. We shall often find it necessary to restrict our attention to the elements X_{α} for which α ≠ 0. For such a point α, we define the element g_{α} = 1 + X_{α} ∈ kE. Because X_{α}^{p} = 0 and X_{α} ≠ 0, we have g_{α}^{p} = 1 and g_{α}^{i} ≠ 1 for all i < p. This shows that g_{α} has order p in the multiplicative group kE^{×}. We call ⟨g_{α}⟩ ≅ ℤ/p a cyclic shifted subgroup of kE.

A significant amount of the theory we investigate takes advantage of various k-Hopf algebra structures on kE. Recall that every counit for kE is necessarily a k-algebra homomorphism kE → k. We have a vector space decomposition kE = k ⊕ J, and all elements in J are nilpotent. This shows that there exists a unique counit for kE, namely the projection of kE onto the first summand. One readily verifies that this coincides with the augmentation map ε: kE → k given by g ↦ 1 for all g ∈ E.

We draw particular attention to two common k-Hopf algebra structures on kE. The group theoretic Hopf algebra structure has comultiplication Δ: kE → kE ⊗_{k} kE given by g ↦ g ⊗ g for all g ∈ E, and antipode σ: kE → kE given by g ↦ g^{−1} for all g ∈ E. The p-restricted Lie Hopf algebra structure on kE has comultiplication Δ̃: kE → kE ⊗_{k} kE given by g − 1 ↦ (g − 1) ⊗ 1 + 1 ⊗ (g − 1) for all g ∈ E, and antipode σ̃: kE → kE given by g − 1 → −(g − 1) for all g ∈ E.

The above Hopf algebra structures yield two, generally different tensor product and dual kE-module structures. If M and N are finite dimensional kE-modules, we denote by M ˜⊗_{k} N the k-vector space M ⊗_{k} N equipped with the kE-module structure

(g − 1).(m ⊗ n) = (g − 1)m ⊗ n + m ⊗ (g − 1)n for all m ∈ M, n ∈ N, g ∈ E

induced by the Lie theoretic comultiplication Δ̃, reserving the notation M ⊗_{k} N for the kE-module structure

g.(m ⊗ n) = gm ⊗ gn for all m ∈ M, n ∈ N, g ∈ E

induced by the group theoretic comultiplication Δ.

Similarly, if M is a finite dimensional kE-module, we denote by M ˜∗ the k-linear dual Hom_{k}(M, k) equipped with the kE-module structure

((g − 1).f)(m) = −f((g − 1)m) for all f ∈ Hom_{k}(M, k), g ∈ E, m ∈ M

induced by the Lie theoretic antipode σ̃, reserving the notation M^{∗} for the dual kE-module structure

(g.f)(m) = f(g^{−1}m) for all f ∈ Hom_{k}(M, k), g ∈ E, m ∈ M

induced by the group theoretic antipode σ.

The main distinction we wish to draw between the above module structures is that, in the Lie theoretic case, restriction to cyclic shifted subgroups commutes with taking tensor products, i.e., given kE-modules M and N we have

(M ˜⊗_{k} N)↓_{⟨g_{α}⟩} ≅ M ↓_{⟨g_{α}⟩} ˜⊗_{k} N ↓_{⟨g_{α}⟩} for all non-zero α ∈ 𝔸^{r}(k).

This follows from the fact that kE, equipped with the p-restricted Lie Hopf algebra structure, is isomorphic to the universal enveloping algebra of a p-restricted Lie algebra with r generators, zero bracket and zero pth power map. In this case, the inclusion of a cyclic shifted subgroup into kE is induced by an inclusion of Lie algebras, hence is an inclusion of Hopf algebras.

2.3. Representation type

If k is an algebraically closed field, then by the trichotomy theorem of Drozd [24], any finite dimensional k-algebra A satisfies one of the following mutually exclusive conditions.

(1) Finite representation type: There are only finitely many isomorphism classes of finite dimensional indecomposable left A-modules.

(2) Tame representation type: There are infinitely many isomorphism classes of finite dimensional indecomposable left A-modules, but in any dimension they come in one parameter subfamilies, with finitely many exceptions.

(3) Wild representation type: There exists a finite dimensional A-k⟨X, Y ⟩-bi-module M that is free as a right k⟨X, Y ⟩-module, such that the functor M ⊗_{k⟨X,Y ⟩} − taking finite dimensional right k⟨X, Y ⟩-modules to finite dimensional left A-modules preserves indecomposability and isomorphism classes. In this case, classifying the finite dimensional indecomposable A-modules would involve classifying pairs of square matrices under simultaneous conjugation, which is thought to be impossible.

For group algebras we have the following theorem of Bondarenko and Drozd [10].

Theorem 2.7. Let G be a finite group and k an algebraically closed field of 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 gener-

alised quaternion, then kG has tame representation type. (iii) In all other cases, kG has wild representation type.

In terms of elementary abelian p-group algebras, this means that kE has finite representation type when r = 1, tame representation type when p = r = 2, and wild representation type in all other cases. (For the classification of the indecomposable k(ℤ/p)-modules, see Appendix B.) This means that in almost all cases we cannot hope to find a classification of the finite dimensional indecomposable kE-modules.

This obstruction to fully understanding the category mod(kE) often forces us to restrict our attention to certain subclasses of kE-modules that we do hope to classify, e.g., endotrivial modules. (See Dade [21].) Another common approach is to identify an invariant of a kE-module, and then try to study the behaviour of the invariant with respect to that of the module. A particularly fruitful example of this is Carlson’s rank variety. (See Appendix C for a brief introduction.) As we shall see in the following sections, both of these ideas motivate the study of modules of constant Jordan type in the sense that we restrict our attention to a special subclass of kE-modules for which there is an associated invariant, namely the Jordan type of any element of J \ J^{2} acting the module.

2.4. Diagrams for modules

The most useful tools for gaining intuition about representations of kE in low ranks are diagrams for modules. These were defined rigorously by Benson and Carlson [8], although the use of similar objects in the literature preceded their definition. For our purposes, a module diagram for a kE-module M consists of the following data.

(1) A finite directed graph with no loops whose vertex set is a a fixed k-basis v_{1}, …, v_{d} of M.

(2) There are r types of edges, r being the rank of E. An edge of type i has source u_{j} and target v_{l} if and only if X_{i}v_{j} = v_{l} in M.

(3) If there is no arrow of type i with source v_{j}, then X_{i}v_{j} = 0.

It follows from the definition that a kE-module is indecomposable if and only if its associated diagram is a connected graph. We note that most kE-modules do not have corresponding module diagrams, and even when such diagrams exists, they may not be easy to draw.

The diagrams we work with will almost exclusively be for modules in the case where E has rank two. An instructive example of such a diagram is that for the regular representation kE for p = 3.

[figure: 1; •; •; •; •; •; •; •; •]

Here X_{1} acts via single edges downwards to the left and X_{2} acts via double edges downwards to the right. For example, the middle vertex in the above diagram represents the element X_{1}X_{2} ∈ kE. The submodule Rad(kE) is represented by the diagram

[figure: •; •; •; •; •; •; •; •]

and Rad(kE)/Rad^{4}(kE) is represented by the diagram

[figure: •; •; •; •; •; •; •]

Note that the Jordan type of X_{1} on the above modules can be read directly from the respective diagrams by counting the number of vertices in each path containing only X_{1}-edges. For example, X_{1} has Jordan type [3]^{3} on kE, [3]^{2}[2] on Rad(kE) and [3][2]^{2} on Rad(kE)/Rad^{4}(kE). Similar statements hold for X_{2}. In fact, we shall see in Section 2.7 that each of the above modules has constant Jordan type.

To see an example of a module diagram in rank three, we present the diagram for the regular representation kE for p = 2.

[figure: •; •; 1; •; •; •; •; •]

Here X_{1} acts downwards to the left, X_{2} acts downwards to the right and X_{3} acts upwards. In particular, the socle of kE is spanned by the middle vertex in the third row. An attempt to draw a similar diagram for p > 2 quickly illustrates the limitations of diagrammatic techniques.

Returning to the rank two case, we now present a diagram representing the module M = kE/(X_{1}^{2} − X_{2}) for p = 5.

1

•

[figure]

•

[figure]

•

[figure]

•

Here X_{1} acts downwards via single edges and X_{2} acts downwards via double edges. This module originally appeared as Example 2.3 of [26]. The authors of that paper observed that, although X_{1}^{2} − X_{2} ≡ X_{2} (mod J^{2}), X_{1}^{2} − X_{2} and X_{2} have different Jordan types on M. In particular, X_{1}^{2} − X_{2} acts with Jordan type [1]^{5}, whereas X_{2} acts with Jordan type [3][2]. (Note that this can be read directly from the diagram.)

One should compare this with the statement of Theorem 2.8, which shows that the above behaviour does not happen ‘generically’.

2.5. The generic Jordan type

In this section we introduce the concept of the generic Jordan type for a kE-module. This work was originally pioneered by Wheeler [35], and later strengthened by Friedlander, Pevtsova and Suslin [26], who showed that a module’s generic Jordan type is independent of the choice of generators of E. The material presented here will be essential in Section 2.6 when proving first properties of modules of constant Jordan type. The following appeared as Theorem 2.7 of [26].

Theorem 2.8. Let M be a finite dimensional kE-module.

(i) There exists a unique maximal Jordan type JType(x, M) with respect to the dominance ordering on partitions of dim_{k}(M), where x ranges over all elements of J.

(ii) If x ∈ J achieves the maximal Jordan type on M and x − y ∈ J^{2}, then y also

achieves the maximal Jordan type on M. (iii) If M is not semisimple, then no element of J^{2} achieves the maximal Jordan

type on M.

Proof. If M is semisimple, then JM = 0 so that every element of J has the same Jordan type on M. So assume M is not semisimple. Let

(t_{1}, …, t_{r}) ∈ 𝔸^{r}(k(t_{1}, …, t_{r}))

be the generic point of 𝔸^{r}(k) and define

X_{gen} = t_{1}X_{1} + ⋯ + t_{r}X_{r} ∈ kE ⊗_{k} k(t_{1}, …, t_{r}).

Writing M̄ = M ⊗_{k} k(t_{1}, …, t_{r}), the matrix minors of (X_{gen})M̄ are polynomials in k[t_{1}, …, t_{r}], hence by Proposition A.2 there exists a dense open subset U ⊆ 𝔸^{r}(k) such that α ∈ U if and only if (X_{α})M̄ ∼ (X_{gen})M̄. Fixing α ∈ U, we have

(X_{α})M̄ ∼ (X_{gen})M̄ ⪰ (X_{α+β})M̄ = (X_{α})M̄ + (X_{β})M̄

for all β ∈ 𝔸^{r}(k). It follows that (X_{α})_{M} ⪰ (X_{α})_{M} + (X_{β})_{M} for all β ∈ 𝔸^{r}(k), thus (X_{α})_{M} is maximal with respect to the family of matrices (X_{1})_{M}, …, (X_{r})_{M}. Next observe that any x ∈ J is of the form x = X_{β} + X_{1}y_{1} + ⋯ + X_{r}y_{r} with β ∈ 𝔸^{r}(k) and y_{1}, …, y_{r} ∈ J, hence each y_{i} acts nilpotently on M. Applying Theorem 2.6 to the family of nilpotent matrices (X_{1})_{M}, …, (X_{r})_{M}, (y_{1})_{M}, …, (y_{r})_{M}, we see that

(X_{α})_{M} ⪰ (X_{β} + X_{1}y_{1} + ⋯ + X_{r}y_{r})_{M} = x_{M},

thus JType(X_{α}, M) dominates JType(x, M) for all x ∈ J. This establishes (i).

To prove (ii), note that if x−y ∈ J^{2}, then as above we have y = x+X_{1}y_{1} +⋯+X_{r}y_{r} where y_{1}, …, y_{r} ∈ J act nilpotently on M. Because x achieves the maximal Jordan type on M, we also have x_{M} ⪰ x_{M} + (X_{β})_{M} for all β ∈ 𝔸^{r}(k), hence x_{M} is maximal with respect to (X_{1})_{M}, …, (X_{r})_{M}. It now follows from the second part of Theorem 2.6 that

x_{M} ∼ (x + X_{1}y_{1} + ⋯ + X_{r}y_{r})_{M} = y_{M}

so that JType(x, M) = JType(y, M).

(iii) now follows from (ii) by noting that if x ∈ J^{2} achieves the maximal Jordan type on M, then this Jordan type is equal to that of 0 ∈ kE, hence M must be semisimple. □

Definition 2.9 (Friedlander, Pevtsova and Suslin [26]). If M is a kE-module, then in light of Theorem 2.8 (i) we define the generic Jordan type of M to be the maximal Jordan type achieved by any element of J acting on M. (This terminology is motivated by the fact that the maximal Jordan type on M is equal to that of the generic element X_{gen} on M̄.) By the proof of the same theorem, the points α ∈ 𝔸^{r}(k) for which X_{α} has the generic Jordan type on M form a dense open subset of 𝔸^{r}(k). We denote this subset by U_{gen}(M).

The next proposition shows that the generic Jordan type is well behaved with respect to field extensions.

Proposition 2.10. Let M be a kE-module and let K/k be any field extension. If α ∈ U_{gen}(M), then (X_{α})_{M⊗_{k}K} ⪰ (X_{β})_{M⊗_{k}K} for all β ∈ 𝔸^{r}(K).

Proof. We have (X_{α})_{M} ⪰ (X_{α+β})_{M} = (X_{α})_{M} + (X_{β})_{M} for all β ∈ 𝔸^{r}(k). By condition (ii) of Proposition 2.3, this implies that

(X_{α})_{M⊗_{k}K} ⪰ (X_{α})_{M⊗_{k}K} + (X_{β})_{M⊗_{k}K} for all β ∈ 𝔸^{r}(K).

Thus if γ ∈ 𝔸^{r}(K), then (X_{α})_{M⊗_{k}K} ⪰ (X_{α})_{M⊗_{k}K} + (X_{γ−α})_{M⊗_{k}K} = (X_{γ})_{M⊗_{k}K}. □

The following proposition relating the generic Jordan type to dualisation appeared as Proposition 5.2 of [16].

Proposition 2.11. If M is a kE-module, then the generic Jordan type of M^{∗} is equal to the generic Jordan type of M, and we have U_{gen}(M) = U_{gen}(M^{∗}).

Proof. If σ: kE → kE is the group theoretic antipode of kE (see Section 2.2), then for each 1 ≤ i ≤ r we have

X_{i} + σ(X_{i}) = (g_{i} − 1) + (g_{i}^{−1} − 1) = −(g_{i} − 1)(g_{i}^{−1} − 1) ∈ J^{2},

hence X_{α} + σ(X_{α}) ∈ J^{2} for all α ∈ 𝔸^{r}(k). If X_{α} has the generic Jordan type on M, then it follows from the second part of Theorem 2.6 that (X_{α})_{M} ∼ σ(X_{α})_{M}. We then have (X_{α})_{M} ∼ (X_{α})_{M^{∗}} after noting that the representation affording M^{∗} is obtained from the representation affording M by precomposing with σ and then composing with the transpose map. It now follows that X_{α} has the generic Jordan type on M^{∗}, for otherwise we would have (X_{β})_{M^{∗}} ≻ (X_{α})_{M^{∗}} for any β ∈ U_{gen}(M^{∗}), yielding

(X_{β})_{M} ∼ (X_{β})_{M^{∗}} ≻ (X_{α})_{M^{∗}} ∼ (X_{α})_{M},

a contradiction. □

We shall also be interested in the behaviour of generic Jordan types with respect to direct sums. The following appeared as part of Theorem 4.7 of [26].

Proposition 2.12. If M and N are kE-modules, then the generic Jordan type of M ⊕ N is equal to the union of the generic Jordan types of M and N as partitions. We also have U_{gen}(M ⊕ N) = U_{gen}(M) ∩ U_{gen}(N).

Proof. Since 𝔸^{r}(k) is irreducible, U_{gen}(M) ∩ U_{gen}(N) ∩ U_{gen}(M ⊕ N) is nonempty, being an intersection of dense open subsets of 𝔸^{r}(k). Choosing a point α in this intersection, the equation (X_{α})_{M⊕N} = (X_{α})_{M} ⊕ (X_{α})_{N} proves the first statement of the theorem, and the containment ⊇ in the second statement.

To prove the other containment, let α ∈ U_{gen}(M ⊕ N) and suppose without loss of generality that X_{α} does not achieve the generic Jordan type on M. Choosing β ∈ U_{gen}(M) ∩ U_{gen}(N), we then have (X_{β})_{M} ≻ (X_{α})_{M}, hence there exists an integer s > 0 such that rank(X_{β}^{s}, M) > rank(X_{α}^{s}, M). We also have (X_{β})_{N} ⪰ (X_{α})_{N} so that rank(X_{β}^{s}, N) ≥ rank(X_{α}^{s}, N). It follows that

rank(X_{α}^{s}, M ⊕ N) = rank(X_{α}^{s}, M) + rank(X_{α}^{s}, N) < rank(X_{β}^{s}, M) + rank(X_{β}^{s}, N)
= rank(X_{β}^{s}, M ⊕ N),

contradicting the fact that (X_{α})_{M⊕N} ∼ (X_{β})_{M⊕N}. □

The final result of this section relates the generic Jordan types of two kE-modules to that of their tensor product. It appeared as Proposition 4.4 of [26].

Proposition 2.13. If M and N are kE-modules and α ∈ U_{gen}(M) ∩ U_{gen}(N), then

(M ⊗_{k} N)↓_{⟨g_{α}⟩} ≅ M ↓_{⟨g_{α}⟩} ⊗_{k} N ↓_{⟨g_{α}⟩}

as k⟨g_{α}⟩-modules, and α ∈ U_{gen}(M ⊗_{k} N). In other words, the generic Jordan type of a tensor product is the tensor product of the respective generic Jordan types.

Proof. We first recall the Lie theoretic tensor produce M ˜⊗_{k} N introduced in Section 2.2. Choosing a point α ∈ U_{gen}(M) ∩ U_{gen}(N) ∩ U_{gen}(M ⊗_{k} N), we have

(M ˜⊗_{k} N)↓_{⟨g_{α}⟩} ≅ M ↓_{⟨g_{α}⟩} ˜⊗_{k} N ↓_{⟨g_{α}⟩} ≅ M ↓_{⟨g_{α}⟩} ⊗_{k} N ↓_{⟨g_{α}⟩}  (2.3)

by Corollary B.4.

Next recall from the proof of Theorem 2.8 that since X_{α} has the generic Jordan type on M, (X_{α})_{M} is maximal with respect to the family of matrices (X_{1})_{M}, …, (X_{r})_{M}. By Proposition 2.3, this means that

(X_{α})_{M} ∼ (X_{α})_{M} + t_{1}(X_{1})_{M} + ⋯ + t_{r}(X_{r})_{M}  (2.4)

for algebraically independent variables t_{1}, …, t_{r} over k. Similarly, we have

(X_{α})_{N} ∼ (X_{α})_{N} + s_{1}(X_{1})_{N} + ⋯ + s_{r}(X_{r})_{N}  (2.5)

for another set of algebraically independent variables s_{1}, …, s_{r} over k. It follows that

\begin{aligned}(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} \\ & + t_{1}(X_{1})_{M} ⊗ I_{N} + ⋯ + t_{r}(X_{r})_{M} ⊗ I_{N} \\ & + I_{M} ⊗ s_{1}(X_{1})_{N} + ⋯ + I_{M} ⊗ s_{r}(X_{r})_{N}. \end{aligned}  (2.6)

Indeed, if conjugation by A makes the matrices in (2.4) similar and conjugation by B makes the matrices in (2.5) similar, then conjugation by A ⊗ B makes the matrices in (2.6) similar. This shows that (X_{α})_{M ˜⊗_{k}N} is maximal with respect to the family of matrices

(X_{1})_{M} ⊗ I_{N}, …, (X_{r})_{M} ⊗ I_{N}, I_{M} ⊗ (X_{1})_{N}, …, I_{M} ⊗ (X_{r})_{N}.

Now notice for each 1 ≤ i ≤ r that

Δ(X_{i}) = X_{i} ⊗ 1 + 1 ⊗ X_{i} + X_{i} ⊗ X_{i} = Δ̃(X_{i}) + X_{i} ⊗ X_{i}.

Writing α = (λ_{1}, …, λ_{r}) with each λ_{i} ∈ k, it follows from the above equation and Theorem 2.6 that

(X_{α})_{M⊗_{k}N} = (X_{α})_{M ˜⊗_{k}N} + ∑_{i=1}^{r} λ_{i}((X_{i})_{M} ⊗ I_{N})(I_{M} ⊗ (X_{i})_{N}) ∼ (X_{α})_{M ˜⊗_{k}N}.

Combining this with (2.3) completes the proof. □

2.6. Modules of constant Jordan type

By the discussion in Appendix B, if ⟨g⟩ ≅ ℤ/p is a cyclic group of order p and M is a finite dimensional k⟨g⟩-module, then the isomorphism type of M is completely determined by the Jordan type of X = g − 1 on M.

It follows that if M is a finite dimensional kE-module and α is a non-zero point in 𝔸^{r}(k), then since X_{α} = g_{α} − 1, the partition JType(X_{α}, M) completely determines the isomorphism type of M ↓_{⟨g_{α}⟩} as a k⟨g_{α}⟩-module. In the general setting, this isomorphism type depends greatly on the choice of α. Viewed in this way, modules of constant Jordan type are precisely those kE-modules M for which M ↓_{⟨g_{α}⟩} does not depend on α. They were first defined by Carlson, Friedlander and Pevtsova [16].

Definition 2.14. A finite dimensional kE-module M has constant Jordan type if JType(X_{α}, M) is independent of the choice of non-zero α ∈ 𝔸^{r}(k), i.e., if and only if rank(X_{α}^{s}, M) is independent of the choice of non-zero α ∈ 𝔸^{r}(k) for all s ≥ 0.

If M has constant Jordan type and JType(X_{α}, M) = [p]^{a_{p}} … [1]^{a_{1}} for all non-zero α ∈ 𝔸^{r}(k), we call the partition [p]^{a_{p}} … [1]^{a_{1}} the constant Jordan type of M. In this case, the list [p − 1]^{a_{p−1}} … [1]^{a_{1}} is called the stable constant Jordan type of M. It is obtained by calculating the Jordan type of X_{α} on the non-projective part of M ↓_{⟨g_{α}⟩} for any non-zero α ∈ 𝔸^{r}(k).

Remark 2.15. The proof of Theorem 2.8 (i) shows that if M is not semisimple, then M has constant Jordan type if and only if U_{gen}(M) = 𝔸^{r}(k) \ {0}.

It is not clear from Definition 2.14 that the constant Jordan type property is independent of the choice of generators g_{1}, …, g_{r} of E. The following proposition shows that this is indeed the case.

Proposition 2.16. A finite dimensional kE-module M has constant Jordan type if and only if JType(x, M) is independent of x ∈ J \ J^{2}.

Proof. If x ∈ J \ J^{2}, then x ≡ X_{α} (mod J^{2}) for some non-zero α ∈ 𝔸^{2}(k). If M has constant Jordan type, then by the proof of Theorem 2.8 (i), X_{α} must have the generic Jordan type on M. It then follows by part (ii) of the same theorem that x_{M} ∼ (X_{α})_{M}. This establishes the ‘only if’ direction, and the reverse implication is obvious. □

Definition 2.17. A short exact sequence 0 → M_{1} → M_{2} → M_{3} → 0 of kE-modules is locally split if 0 → M_{1} ↓_{⟨g_{α}⟩} → M_{2} ↓_{⟨g_{α}⟩} → M_{3} ↓_{⟨g_{α}⟩} → 0 splits as a short exact sequence of k⟨g_{α}⟩-modules for all non-zero α ∈ 𝔸^{r}(k).

Carlson and Friedlander [15] have shown that the class of kE-modules of constant Jordan type forms a Quillen exact category cJt(kE), where the short exact sequences in cJt(kE) are defined to be the locally split short exact sequences in mod(kE). For a thorough discussion of exact categories, see Bühler [11].

The following theorem is due to Carlson, Friedlander and Pevtsova [16]. It shows that the constant Jordan type property remains invariant under many common operations on modules for Hopf algebras. Recall that if M is a kE-module, then the Heller shift Ω(M) is defined to be the kernel of the projective cover P_{M} → M.

Theorem 2.18.

(i) If M is a kE-module of constant Jordan type, then Ω(M) and M^{∗} have constant Jordan type.

(ii) If M and N are kE-modules, then M ⊕ N has constant Jordan type if and

only if M and N have constant Jordan type. (iii) If M and N are kE-modules of constant Jordan type, then M ⊗_{k} N has

constant Jordan type.

Proof. The second part (i) follows from Remark 2.15 and Proposition 2.11. Similarly, part (ii) follows from Remark 2.15 and Proposition 2.12. Part (iii) follows from Remark 2.15 and Proposition 2.13. We are therefore left to prove the first part of (i). Let α be a non-zero point in 𝔸^{r}(k) and consider the restriction

0 ⟶ Ω(M)↓_{⟨g_{α}⟩} ⟶ P_{M} ↓_{⟨g_{α}⟩} ⟶ M ↓_{⟨g_{α}⟩} ⟶ 0

of the short exact sequence defining Ω(M). Letting

0 ⟶ Ω(M ↓_{⟨g_{α}⟩}) ⟶ P_{M↓_{⟨gα⟩}} ⟶ M ↓_{⟨g_{α}⟩} ⟶ 0

be the short exact sequence defining Ω(M ↓_{⟨g_{α}⟩}), we have

Ω(M)↓_{⟨g_{α}⟩} ⊕ P_{M↓_{⟨gα⟩}} ≅ Ω(M ↓_{⟨g_{α}⟩}) ⊕ P_{M} ↓_{⟨g_{α}⟩}

by Schanuel’s lemma. Shaving off the projective summand P_{M↓_{⟨gα⟩}} using the Krull-Schmidt theorem, we see that Ω(M) ↓_{⟨g_{α}⟩} is independent of α since M ↓_{⟨g_{α}⟩} is independent of α. □

It is easy to deduce from Propositions 2.11, 2.12 and 2.13 which constant Jordan types we obtain from the above operations. Observe that if M has constant Jordan type [p]^{a_{p}}[p − 1]^{a_{p−1}} … [i]^{a_{i}} … [1]^{a_{1}}, then Ω(M) has constant Jordan type

[p]^{b_{p}}[p − 1]^{a_{1}} … [i]^{a_{p−i}} … [1]^{a_{p−1}}

for some value of b_{p}. This follows from the way one computes Heller shifts of cyclic modules for k(ℤ/p).

2.7. Examples of modules of constant Jordan type

We now present some fairly naturally occurring examples of modules of constant Jordan type. Further examples will appear throughout Chapter 5.

First note by Dade’s lemma (see Theorem C.1) that a kE-module is projective if and only if its restriction to every cyclic shifted subgroup is a projective k(ℤ/p)-module. It follows that the projective kE-modules are precisely those having constant Jordan type [p]^{n} for some n ≥ 0, and in this case we have p^{r−1} | n.

Closely related to the projective kE-modules, our next family of examples appeared in Proposition 2.1 of [16].

Proposition 2.19. The subquotients J^{a}/J^{b} of kE have constant Jordan type for all a ≤ b.

Proof. It suffices to show for every non-zero α ∈ 𝔸^{r}(k) that

JType(X_{α}, J^{a}/J^{b}) = JType(X_{1}, J^{a}/J^{b}).

Let ϕ: kE → kE be an automorphism of kE induced from a matrix in GL(r, k) sending the r-tuple (1, 0, …, 0) to α, i.e., such that ϕ(X_{1}) = X_{α}. Under this automorphism we have (J^{a}/J^{b})↓_{⟨g_{α}⟩} = ϕ(J^{a}/J^{b})↓_{⟨ϕ(1+X_{1})⟩} ≅ (J^{a}/J^{b})↓_{⟨1+X_{1}⟩}. □

Another natural family of examples were investigated in Section 5 of [16]. Recall that if G is a finite group, then a kG-module is endotrivial if End_{k}(M) ≅ k ⊕ (projective). Dade [21] classified the endotrivial modules for finite abelian p-groups. The following appeared as Theorem 10.1 of [21].

Theorem 2.20. A module for a finite abelian p-group is endotrivial if and only if it is stably isomorphic to Ω^{n}(k) for some n ∈ ℤ.

Carlson, Friedlander and Pevtsova have provided an alternative characterisation of endotrivial modules for elementary abelian p-groups in terms or their stable Jordan types. The following appeared as Theorem 5.6 of [16].

Theorem 2.21. A kE module is endotrivial if and only if it has stable constant Jordan type [1] or [p − 1].

Proof. If M is endotrivial, then M ≅ Ω^{n}(k) for some n by Theorem 2.20. Theorem 2.18 (i) and its proof show that M has stable constant Jordan type [1] for p = 2 or n even, and [p − 1] for n odd. (This can also be shown without using Dade’s classification, as was done in [16].)

Conversely, if M has stable constant Jordan type [1], then M^{∗} has stable constant Jordan type [1] so that End_{k}(M) ≅ M^{∗} ⊗_{k} M has stable constant Jordan type [1]. Consider the short exact sequence

0 ⟶ N ⟶ End_{k}(M) ⟶^{Tr} k ⟶ 0

where the map on the right is the matrix trace map and N is its kernel. Since dim_{k}(M) is not divisible by p, the map k → End_{k}(M) given by inclusion of scalar matrices is a splitting of the above sequence, hence End_{k}(M) ≅ k ⊕ N. Since End_{k}(M) has stable constant Jordan type [1], it follows from Theorem 2.18 (ii) that N has constant Jordan type containing only blocks of length p. By Dade’s lemma, this shows that N is projective, thus M is endotrivial.

If M has stable constant Jordan type [p − 1], then the restriction of M^{∗} ⊗_{k} M to a cyclic shifted subgroup is stably isomorphic to J_{p−1} ⊗_{k} J_{p−1} as a k(ℤ/p)-module. (See Appendix B for an explanation of this notation.) We have J_{p−1} ≅ Ω(k), and Ω(k) ⊗_{k} Ω(k) is stably isomorphic to Ω^{2}(k) ≅ k. It follows that End_{k}(M) has stable constant Jordan type [1]. The proof that M is endotrivial now follows exactly as above. □

We end this section with a family of modules in rank two having constant Jordan type [3]^{a}[2]^{b} for certain a, b > 0. These modules can only be described efficiently using module diagrams, and what’s more, the verification of their constant Jordan types serves as an instructive exercise in the use of diagrammatic techniques.

We begin by fixing an alphabet

A = {A, V, D_{n} | n ≥ 2}.

For each abstract word W over A, we construct a kE-module M(W) of radical length at most three in the following way. We associate to the letter A the graph

[figure: •; •; •]

we associate to the letter V the graph

[figure: •; •; •]

and for n ≥ 2 we associate to the letter D_{n} the graph

[figure: •; •; •; •; •; •; ⋯; •; •; •; •; •]

where the number of middle vertices is equal to n. We then associate to the word W the graph obtained by connecting the graphs corresponding to adjacent letters appearing in W by identifying the very right vertex of the left letter with the very left vertex of the right letter. For example, the graph corresponding to the word W = V D_{3}A is

[figure: •; •; •; •; •; •; •; •; •; •; •]

Without risk of confusion, we shall use W to denote both a word over A and its corresponding graph.

We now define M(W) to be the kE-module with k-basis consisting of the vertices of W whose kE-module action is given as in Section 2.4, i.e., such that X_{1} maps a vertex v to a vertex w if there is a single edge downwards to the left from v to w, and X_{2} maps v to w if there is a double edge downwards to the right from v to w. Our main result regarding such modules is the following.

Theorem 2.22. Let W be a word over A satisfying the following conditions.

(i) W begins and ends with letters of the form D_{n}.

(ii) If D_{n} appears in W, then p | n.

(iii) W contains no strings of the form AA, V V, AV, V A or D_{m}D_{n}.

If the letters of the form D_{n} in W are (from left to right) D_{n_{1}}, …, D_{n_{t}}, then the corresponding module M(W) has constant Jordan type [3]^{a}[2]^{b} where

a = ∑_{i=1}^{t}(n_{i} − 1) − 1.

and b = t + 1.

Proof. For convenience write M = M(W). It is clear from inspecting the graph W that X_{2} acts on M with Jordan type [3]^{a}[2]^{b}. In order to prove that M has constant Jordan type, it therefore suffices to prove that rank(X_{1} + λX_{2}, M) = 2a + b and rank((X_{1} + λX_{2})^{2}, M) = a for all λ ∈ k.

The fact that rank(X_{1} + λX_{2}, M) = 2a + b can be verified by noting that the map

X_{1} + λX_{2} : M/Rad(M) ⟶ Rad(M)/Rad^{2}(M)

is injective, and the map

X_{1} + λX_{2} : Rad(M)/Rad^{2}(M) ⟶ Rad^{2}(M)

is surjective. (To see this, consider the proof of Proposition 4.14 and its dual.) In other words, rank(X_{1} + λX_{2}, M) is the number of top vertices in W plus the number of bottom vertices in W.

To calculate rank((X_{1} + λX_{2})^{2}, M), we label the top and bottom vertices of W from left to right by v_{1}, v_{2}, … and w_{1}, w_{2}, …, respectively, increasing the index on v_{i} by one after passing a subgraph of the form V and increasing the index on w_{i} by one after passing a subgraph of the form A. This labeling scheme is illustrated in the following diagrams.

[figure: v_{jp−1}; v_{jp+1}; ⋯; •; •; •; •; ⋯; w_{jp−1}; w_{jp}; w_{jp+1}]

[figure: v_{jp−1}; v_{jp}; v_{jp+1}; ⋯; •; •; •; •; ⋯; w_{jp−1}; w_{jp+1}]

We now consider an element ∑_{i} a_{i}v_{i} ∈ Ker((X_{1} + λX_{2})^{2}, M), where a_{i} ∈ k. For convenience we set a_{i} = 0 when v_{i} does not exist. Examining the coefficients on w_{i} in (X_{1} + λX_{2})^{2} ∑_{j} a_{j}v_{j}, which must all equal zero, we obtain the recursion relations

λ^{2}a_{i−1} + 2λa_{i} + a_{i+1} = 0 if w_{i} exists.  (2.7)

Let C_{1}, …, C_{s} be the maximal subwords of W (from left to right) that do not contain the letter A so that W = C_{1}AC_{2}A … AC_{s}. For 1 ≤ l ≤ s, let j(l) ≥ 0 be the integer such that v_{j(l)p+1} is the leftmost top vertex in C_{l}. We claim that any solution to the recurrence relations (2.7) is obtained by independently choosing the coefficients a_{j(l)p+1}. From this it will follow that the kernel of

(X_{1} + λX_{2})^{2} : M/Rad(M) → Rad^{2}(M)

has dimension s, showing that the rank of (X_{1} + λX_{2})^{2} on M is

(number of top vertices in W) − s = a

as required. In order to prove the claim, we will show for 1 ≤ l ≤ s that a_{j(l)p+1} completely determines the coefficients a_{i} for j(l)p + 1 ≤ i ≤ j(l + 1)p, proceeding by induction on l.

To prove the claim for l = 1, we first show that

a_{i} = (−λ)^{i−1}ia_{1} for all 1 ≤ i ≤ j(2)p,

using induction on i. This clearly holds for i = 1. For i > 1, it follows from the (i − 1)st relation and the inductive hypothesis that

a_{i} = −2λa_{i−1} − λ^{2}a_{i−2} = −2λ(−λ)^{i−2}(i − 1)a_{1} − λ^{2}(−λ)^{i−3}(i − 2)a_{1} = (−λ)^{i−1}ia_{1}

as required. (Note that if p | i, then a_{i} = 0 since k has characteristic p, hence this calculation is consistent with our convention that a_{i} = 0 if v_{i} does not exist.) This shows that a_{1} determines a_{i} for all 1 ≤ i ≤ j(2)p. In particular, we have a_{j(2)p} = 0. Note that v_{j(2)} is the top vertex in the subgraph A directly following C_{1} in W, hence there is no vertex w_{j(2)p}. Referring to the recurrence relations (2.7), because there is no j(2)pth relation, this shows that none of the coefficients a_{i} for i ≥ j(2)p + 1 depends on a_{1}. This establishes the claim for l = 1.

For l > 1, we assume by induction that a_{j(l)p+1} does not depend on any of the previous a_{i}. The proof then follows by starting at the (j(l)p + 1)st relation and using the same argument as in the case l = 1. □

Remark 2.23. It follows from the above proof that if M is the kE-module having diagram

[figure: •; •; •; •; •; •; ⋯; •; •; •; •; •]

then M has constant Jordan type if and only if the number of middle vertices is divisible by p. This illustrates the necessity of condition (ii) in Theorem 2.22.

2.8. Some questions and conjectures

Having defined modules of constant Jordan type and examined some of their properties, it is now natural to consider what kind of information we would like to know about these modules, and which techniques will be effective in obtaining that information.

One might first be tempted to investigate modules of constant Jordan type in terms of their rank varieties. (See Appendix C.) The problem with this approach is that a module of constant Jordan type is either (a) projective, in which case its rank variety contains only the zero point, or (b) non-projective, in which case its restriction to any cyclic shifted subgroup is non-projective as a k(ℤ/p)-module, hence its rank variety is all of 𝔸^{r}(k). In other words, rank varieties are only capable of distinguishing two modules of constant Jordan type if one is projective and the other is non-projective.

Having established that rank varieties are too coarse to yield meaningful information about modules of constant Jordan type, one is left to search for finer invariants that are suitable for this purpose. Such a natural invariant would be the constant Jordan type itself. Having this observation in mind, Carlson, Friedlander and Pevtsova [16] posed the following general question: For which Jordan types [p]^{a_{p}} … [1]^{a_{1}} does there exist a kE-module of constant Jordan type [p]^{a_{p}} … [1]^{a_{1}}? This is sometimes called the realisation problem for modules of constant Jordan type.

We now record some conjectures that are related to the above question. The first is due to Suslin and appeared as Question 9.6 of [16].

Conjecture 2.24 (Suslin’s conjecture). Let r ≥ 2. If a kE-module has constant Jordan type [p]^{a_{p}} … [1]^{a_{1}} and a_{i} ≠ 0 for some 2 ≤ i ≤ p − 1, then either a_{i−1} ≠ 0 or a_{i+1} ≠ 0.

The second conjecture we record was made by Jeremy Rickard at MSRI in 2008.

Conjecture 2.25 (Rickard’s conjecture). Let r ≥ 2. If a kE-module has constant Jordan type [p]^{a_{p}} … [1]^{a_{1}} and a_{i} = 0 for some 1 ≤ i ≤ p−1, then ∑_{j=i+1}^{p} a_{j} is divisible by p.

Note that neither of these conjectures implies the other. For example, Suslin’s conjecture fails to rule out the realisation of the Jordan type [3][2], whereas Rickard’s conjecture fails to rule out the realisation of the Jordan type [2]^{2}. Nonetheless, both imply the following, which appeared as Conjecture 9.5 of [16].

Conjecture 2.26. If p ≥ 5 and r ≥ 2, then there does not exist a kE-module of stable constant Jordan type [2].

While Suslin and Rickard’s conjectures remain open, Conjecture 2.26 has been answered in the affirmative by the following, more general result of Benson [5].

Theorem 2.27. If r ≥ 2, then there does not exist a kE-module of stable constant Jordan type [a] with 2 ≤ a ≤ p − 2.

We remark that the proof of Theorem 2.27 is highly specialised, using exterior and symmetric powers of a module to derive contradictions modulo p on its dimension. In Chapter 3 we prove a slightly weaker version of this theorem using the theory of Chern classes for vector bundles on projective space. Benson’s Theorem 6.1 related to Rickard’s conjecture provides an alternative proof of Theorem 2.27.

The final conjecture we record appeared as Conjecture 9.7 of [16].

Conjecture 2.28. If p ≥ 5 and a kE-module has stable constant Jordan type [2][1]^{n}, then n ≥ r − 1.

CHAPTER 3 Restrictions on Jordan types from vector bundles on projective space

The goal of this chapter is to study kE-modules of stable constant Jordan type [a][b] with 1 ≤ a, b ≤ p − 1 using Chern classes for vector bundles on projective space. Our approach is motivated by a theorem of Benson [7]. We begin by recalling some basic concepts from coherent sheaves on projective space. The primary reference for this material is Hartshorne [30]. A valuable reference for the theory of vector bundles on projective space is Okonek, Schneider and Spindler [32].

3.1. Vector bundles on projective space

We observed in Section 2.2 that the elements X_{1}, …, X_{r} ∈ J may be viewed as the coordinates of the affine r-space 𝔸^{r}(k). As in Appendix A, we consider the corresponding dual basis Y_{1}, …, Y_{r}, where each Y_{i} acts as a coordinate function on 𝔸^{r}(k) via the relation Y_{i}(X_{j}) = δ_{ij}. Recall that k[Y_{1}, …, Y_{r}] is an ℕ-graded ring, where the homogeneous elements of degree n are defined to be the k-linear combinations of monomials of degree n in Y_{1}, …, Y_{r}.

If M = ⨁_{i∈ℤ} M_{i} is a ℤ-graded k[Y_{1}, …, Y_{r}]-module and 𝔭 is a homogeneous prime ideal of k[Y_{1}, …, Y_{r}], we denote by M_{(𝔭)} the set of degree zero elements in the localisation S^{−1}M, where S is the set of homogeneous elements in k[Y_{1}, …, Y_{r}] that do not lie in 𝔭. If n is any integer, we denote by M(n) the module M with shifted grading given by M(n)_{i} = M_{i+n} for all i ∈ ℤ.

We now consider the projective space ℙ^{r−1} = Proj k[Y_{1}, …, Y_{r}]. In other words, ℙ^{r−1} is the set of homogeneous prime ideals in k[Y_{1}, …, Y_{r}] that do not contain the ideal (Y_{1}, …, Y_{r}). The topology on ℙ^{r−1} is defined by taking the closed sets to be those of the form

V (𝔞) = {𝔭 ∈ ℙ^{r−1} | 𝔞 ⊆ 𝔭},

where 𝔞 is a homogeneous ideal of k[Y_{1}, …, Y_{r}].

The structure sheaf O = O_{ℙ^{r−1}} on ℙ^{r−1} is defined in the following way. For each open set U ⊆ ℙ^{r−1}, we let O(U) be the ring of functions

s: U ⟶ ∐_{𝔭∈U} k[Y_{1}, …, Y_{r}]_{(𝔭)}

such that s(𝔭) ∈ k[Y_{1}, …, Y_{r}]_{(𝔭)} for all 𝔭 ∈ U, and such that s is locally a homogeneous rational function of degree zero in Y_{1}, …, Y_{r}. Specifically, for each 𝔭 ∈ U there exists a neighbourhood V of 𝔭 contained in U and homogeneous elements a, f ∈ k[Y_{1}, …, Y_{r}] of the same degree such that, for all 𝔮 ∈ V, f ∉ 𝔮 and s(𝔮) = a/f ∈ k[Y_{1}, …, Y_{r}]_{(𝔮)}.

Similarly, if M is any ℤ-graded k[Y_{1}, …, Y_{r}]-module, we construct the sheaf M̃ of O-modules on ℙ^{r−1} in the following way. For each open set U ⊆ ℙ^{r−1}, we let M̃(U) be the O(U)-module whose elements are functions s: U → ∐_{𝔭∈U} M_{(𝔭)} such that, for all 𝔭 ∈ U, s(𝔭) ∈ M_{(𝔭)} and there exists a neighbourhood V of 𝔭 contained in U and homogeneous elements m ∈ M, f ∈ k[Y_{1}, …, Y_{r}] of the same degree such that, for all 𝔮 ∈ V, f ∉ 𝔮 and s(𝔮) = m/f ∈ M_{(𝔮)}. If M is finitely generated, then it follows from Proposition 5.11 (c) of [30, Chapter II] that M̃ is a coherent sheaf on ℙ^{r−1}.

An important class of O-modules constructed in the above way are the twists of the structure sheaf O. Specifically, for n ∈ ℤ, the nth twist of O is the coherent sheaf O(n) = k[Y_{1}, …, Y_{r}](n)^{∼}. It is apparent from the above construction that sections of O(n) are given locally by homogenous rational functions of degree n in Y_{1}, …, Y_{r}. For any sheaf F on ℙ^{r−1}, we define F(n) to be the tensor product sheaf F ⊗_{O} O(n). By Proposition 5.12 (b) of [30, Chapter II], we have O(m) ⊗_{O} O(n) ≅ O(m + n) for all m, n ∈ ℤ.

A coherent sheaf F on ℙ^{r−1} is locally free if there exists an open covering {U_{i}}_{i∈I} of ℙ^{r−1} such that for each i ∈ I, the restriction F|_{U_{i}} is a free O|_{U_{i}}-module, i.e., a direct sum of copies of O|_{U_{i}}. Since ℙ^{r−1} is irreducible, we have U_{i} ∩ U_{j} ≠ ∅ for all i, j ∈ I, hence the free rank of F|_{U_{i}} is independent of U_{i}. We therefore define the rank of F to be the free rank of its restriction to any U_{i}. A vector bundle on ℙ^{r−1} is a locally free sheaf of finite rank. A vector bundle of rank one is called a line bundle.

Proposition 3.1. The twisted sheaves O(n) for n ∈ ℤ are line bundles on ℙ^{r−1}.

Proof. Consider the open sets U_{i} = D(Y_{i}) = ℙ^{r−1} \ V ((Y_{i})) for 1 ≤ i ≤ r. These form an affine open covering of ℙ^{r−1} since each homogeneous prime ideal in ℙ^{r−1} does not contain some Y_{i}. If s is a section in O(U_{i}) and x ∈ U_{i}, then s(x) is a rational function of degree n in Y_{1}, …, Y_{r}. Letting s/Y_{i}^{n}: U_{i} → ∐_{x∈U_{i}} O_{x} denote the function mapping each x ∈ U_{i} to the rational function s(x)/Y_{i}^{n} of degree zero, the assignment s ↦ s/Y_{i}^{n} induces an isomorphism O(n)|_{U_{i}} ≅ O|_{U_{i}}. This shows that O(n) is a vector bundle of rank one on ℙ^{r−1}. □

Remark 3.2. The converse is also true, that is, every line bundle on ℙ^{r−1} is isomorphic to O(n) for some n ∈ ℤ. (See Corollary 6.17 of [30, Chapter II].)

Recall that the residue field of a point x ∈ ℙ^{r−1} is the field k(x) = O_{x}/𝔪_{x}, where 𝔪_{x} is the unique maximal ideal of the local ring O_{x}. If F is any sheaf on ℙ^{r−1}, then the fibre of F at x is defined to be the k(x)-vector space F_{x} ⊗_{O_{x}} k(x). The following proposition is a special case of Exercise 5.8 of [30, Chapter II]. It gives an essential criterion for a coherent sheaf on ℙ^{r−1} to be a vector bundle in terms of the respective dimensions of its fibres.

Proposition 3.3. If F is a coherent sheaf on ℙ^{r−1}, then F is a vector bundle if and only if dim_{k(x)} F_{x} ⊗_{O_{x}} k(x) is independent of the point x ∈ ℙ^{r−1}.

An immediate consequence of Proposition 3.3 is that injectivity and surjectivity for maps of vector bundles can be detected on restriction to fibres.

Proposition 3.4. If ϕ: F → F′ is a map of vector bundles on ℙ^{r−1}, then ϕ is a monomorphism (resp. epimorphism, isomorphism) if and only the induced map of fibres

ϕ_{x}: F ⊗_{O_{x}} k(k) ⟶ F′ ⊗_{O_{x}} k(x)

is a monomorphism (resp. epimorphism, isomorphism) for all x ∈ ℙ^{r−1}.

3.2. Trivial sheaves for kE-modules and the operator θ_{M}

Let M be a kE-module and set d = dim_{k}(M). We define the trivial sheaf associated to M to be the coherent sheaf M̃ = M ⊗_{k} O on ℙ^{r−1} whose sections on an open set U ⊆ ℙ^{r−1} are M ⊗_{k} O(U).^{1} Choosing a k-basis m_{1}, …, m_{d} of M, we may identify M̃(U) with the k-vector space O^{⊕d}(U) via the relation

∑_{i=1}^{d} λ_{i}m_{i} ⊗ s ⟼ (λ_{1}s, …, λ_{d}s) ∈ O^{⊕d}(U).

It follows that M̃ ≅ O^{⊕d} as O-modules. Passing through this isomorphism, we obtain

M̃(n) = (M ⊗_{k} O) ⊗_{O} O(n) ≅ M ⊗_{k} (O ⊗_{O} O(n)) ≅ M ⊗_{k} O(n).

1 The trivial sheaf associated to a kE-module is denoted by a wide tilde in order to distinguish it from the sheaf constructed from a graded k[Y_{1}, …, Y_{r}]-module in Section 3.1.

If U is an open subset of ℙ^{r−1} and s is a section in O(n)(U), then for 1 ≤ i ≤ r we define Y_{i}s to be the function U → ∐_{x∈U} O(n + 1)_{x} given by x ↦ Y_{i}s(x). Since s(x) is a rational function of degree n in the coordinates Y_{1}, …, Y_{r}, we see that Y_{i}s(x) is a rational function of degree n + 1. The assignment s ↦ Y_{i}s induces a sheaf morphism O(n) → O(n + 1). In fact, it follows from Theorem 5.1 of [30, Chapter III] that

Hom_{O}(O(n), O(n + 1)) ≅ H^{0}(ℙ^{r−1}, O(1)) ≅ span_{k}(Y_{1}, …, Y_{r}).

Combining the above morphisms with the action of X_{1}, …, X_{r} ∈ kE on M, we define θ_{M}: M̃(n) → M̃(n + 1) to be the operator given on each open set U ⊆ ℙ^{r−1} by

θ_{M,U}(m ⊗ s) = ∑_{i=1}^{r} X_{i}m ⊗ Y_{i}s for all m ∈ M, s ∈ O(n)(U).

The above construction is due to Friedlander and Pevtsova [25].

Note that the action of θ_{M} can be visualised in terms of matrices of linear forms. Specifically, if ρ_{M}: kE → Mat_{d}(k) is the matrix representation affording M, then the matrix representing θ_{M} is given locally by

∑_{i=1}^{r} Y_{i}ρ_{M}(X_{i}) ∈ Mat_{d}(k[Y_{1}, …, Y_{r}]).

Now let ϕ: M → N be a homomorphism of finitely generated kE-modules. We define the sheaf morphism ϕ̃: M̃ → Ñ to be the map defined on each open subset U ⊆ ℙ^{r−1} by

ϕ̃_{U}(m ⊗ s) = ϕ(m) ⊗ s ∈ Ñ(U) for all m ∈ M, s ∈ O(U).

Note for each m ∈ M and s ∈ O(U) that we have

\begin{aligned}ϕ̃_{U}(θ_{M,U}(m ⊗ s)) & = ϕ̃_{U} (∑_{i=1}^{r} X_{i}m ⊗ Y_{i}s) = ∑_{i=1}^{r} ϕ(X_{i}m) ⊗ Y_{i}s = ∑_{i=1}^{r} X_{i}ϕ(m) ⊗ Y_{i}s \\ & = θ_{N,U}(ϕ̃_{U}(m ⊗ s)), \end{aligned}

hence ϕ̃ ∘ θ_{M} = θ_{N} ∘ ϕ̃.

Our immediate goal is to show that the kE-module structure of M is encoded in the fibres of the operator θ_{M} at points in ℙ^{r−1}. To illustrate this, we first recall the method of calculating the fibres of a sheaf F on an arbitrary scheme (X, O_{X}). Since the fibre of F at a point x ∈ X is equal to the fibre of its restriction to any affine open set containing x, we may assume that X = Spec A for some commutative ring A. If 𝔭 is any prime ideal of A, then k(𝔭) is an A-module under the composition of ring homomorphisms

A ⟶ A_{𝔭} ⟶ A_{𝔭}/𝔭A_{𝔭}

and we have

F_{𝔭} ⊗_{O_{X,𝔭}} k(𝔭) ≅ Γ(F) ⊗_{A} k(𝔭).

Applying this to the sheaf M̃(n) on ℙ^{r−1}, recall that if x ∈ ℙ^{r−1} then there exists a coordinate function Y_{i} such that x ∈ U_{i} = ℙ^{r−1} \ V ((Y_{i})). Because O(n)|U_{i} ≅ O|_{U_{i}} by the proof of Proposition 3.1, it follows that

M̃(n)_{x} ⊗_{O_{x}} k(x) ≅ M ⊗_{k} O(U_{i}) ⊗_{O(U_{i})} k(x) ≅ M ⊗_{k} k(x).

To calculate the fibre of the morphism θ_{M}: M̃(n) → M̃(n + 1) at x ∈ ℙ^{r−1}, note that O(U_{i}) is isomorphic to the degree zero elements in the localised ring k[Y_{1}, …, Y_{r}]_{Y_{i}} by Proposition 2.5 of [30, Chapter II], and this in turn is equal to k[Y_{1}Y_{i}^{−1}, …, Y_{r}Y_{i}^{−1}]. We extend k to k(x) by forming the fibred product

ℙ_{k(x)}^{r−1} = ℙ^{r−1} ×_{Spec k} Spec k(x) ≅ Proj k(x)[Y_{1}, …, Y_{r}].

If y ∈ ℙ_{k(x)}^{r−1} is the generic point of x (see Section A.2), then x is the image of y under the projection ℙ_{k(x)}^{r−1} → ℙ^{r−1}, and the fibre of θ_{M} at x is equal to the fibre of θ_{M} at y. To calculate the latter, it suffices to restrict θ_{M} to the affine open set

U_{i,k(x)} = U_{i} ×_{Spec k} Spec k(x) ≅ Spec k(x)[Y_{1}Y_{i}^{−1}, …, Y_{r}Y_{i}^{−1}]

containing y. Because y is closed in ℙ_{k(x)}^{r−1}, it corresponds to a point

ᾱ = [λ_{1} : ⋯: λ_{r}]

in the projective variety ℙ^{r−1}(k(x)) whose ith homogeneous coordinate is non-zero. The k(x)[Y_{1}Y_{i}^{−1}, …, Y_{r}Y_{i}^{−1}]-module structure on k(x) is then given via the specialisation

\begin{matrix}k(x)[Y_{1}Y_{i}^{−1}, …, Y_{r}Y_{i}^{−1}] & ⟶ k(x) \\ f & ⟼ f(α), \end{matrix}

where α = (λ_{1}, …, λ_{r}) ∈ 𝔸^{r}(k(x)) is any point lying above ᾱ. It follows that the fibre of θ_{M} at y maps the element

m ⊗ 1 ⊗ 1 ∈ M ⊗_{k} O(U_{i,k(x)}) ⊗_{O(U_{i,k(x)})} k(x)

to

∑_{j=1}^{r} X_{j}m ⊗ Y_{j} ⊗ 1 = ∑_{j=1}^{r} X_{j}m ⊗ 1 ⊗ λ_{j}λ_{i}^{−1} ∈ M ⊗_{k} k(x).

This establishes the following, which appeared implicitly in [25].

Proposition 3.5. Let M be a finitely generated kE-module and x ∈ ℙ^{r−1}. Then the fibre of M̃(n) at x is isomorphic to the extension M ⊗_{k} k(x) as a k(x)-vector space. If ᾱ = [λ_{1} : ⋯: λ_{r}] ∈ ℙ^{r−1}(k(x)) corresponds to the generic point of x in ℙ_{k(x)}^{r−1} and α = (λ_{1}, …, λ_{r}) ∈ 𝔸^{r}(k(x)) is any point lying above ᾱ, then up to a scalar in k(x), the fibre of θ_{M}: M̃(n) → M̃(n + 1) at x is given by the action of X_{α} = ∑_{j} X_{j} ⊗ λ_{j} on M ⊗_{k} k(x) as a (kE ⊗_{k} k(x))-module.

3.3. The functors F_{i,j} and F_{i}

Let M be a finitely generated kE-module. For 1 ≤ i ≤ p and 0 ≤ j < i, Benson and Pevtsova [9] have constructed the subquotient

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})}

of M̃ where, for each exponent l in the above expression, Ker θ_{M}^{l} denotes the kernel of the sheaf morphism θ_{M}: M̃ → M̃(l), and Im θ_{M}^{l} denotes the image of the sheaf morphism θ_{M}: M̃(−l) → M̃. In the case j = 0, the authors of [9] also define

F_{i}(M) = F_{i,0}(M) = \frac{Ker θ_{M} ∩ Im θ_{M}^{i−1}}{Ker θ_{M} ∩ Im θ_{M}^{i}} .

Note that F_{i,j}(M) is a coherent sheaf of O-modules on ℙ^{r−1} by Proposition 5.7 of [30, Chapter II]. Moreover, if ϕ: M → N is any kE-module homomorphism, then there is an obvious sheaf morphism

F_{i,j}(ϕ): F_{i,j}(M) ⟶ F_{i,j}(N)

factoring through ϕ̃: M̃ → Ñ. This map is well defined since ϕ̃ ∘ θ_{M} = θ_{N} ∘ ϕ̃. (See Section 3.2 for further details.) It follows for each 0 ≤ j < i ≤ p that F_{i,j} is a functor from the category mod(kE) of finitely generated kE-modules to the category Coh(ℙ^{r−1}) of coherent sheaves on ℙ^{r−1}. The following proposition, which appeared as Proposition 2.1 of [9], shows that the functors F_{i} map modules of constant Jordan type to vector bundles.

Proposition 3.6. A kE-module M has constant Jordan type [p]^{a_{p}} … [1]^{a_{1}} if and only if F_{i}(M) is a vector bundle of rank a_{i} on ℙ^{r−1} for all 1 ≤ i ≤ p. Hence the assignment M ↦ F_{i}(M) defines a functor from the category cJt(kE) of kE-modules of constant Jordan type (see Section 2.6) to the category Vec(ℙ^{r−1}) of vector bundles on ℙ^{r−1}.

Sketch of proof. Recall from Proposition 3.5 that the fibre of θ_{M} at a point x ∈ ℙ^{r−1} is given by the action of X_{α} ∈ kE ⊗_{k} k(x) on M ⊗_{k} k(x), where

ᾱ = [λ_{1} : ⋯: λ_{r}] ∈ ℙ^{r−1}(k(x))

corresponds to the generic point of x in the scheme ℙ_{k(x)}^{r−1}. The fibre of F_{i}(M) at x is then equal to

\frac{Ker(X_{α}, M ⊗_{k} k(x)) ∩ Im(X_{α}^{i−1}, M ⊗_{k} k(x))}{Ker(X_{α}, M ⊗_{k} k(x)) ∩ Im(X_{α}^{i}, M ⊗_{k} k(x))}.

If M has constant Jordan type [p]^{a_{p}} … [1]^{a_{1}}, it follows from Proposition 2.10 that the rank of X_{α}^{s} on M ⊗_{k} k(x) is independent of x for each s ≥ 0. This means that the dimension of the above expression is independent of x ∈ ℙ^{r−1} and equal to a_{i}. By Proposition 3.3, this shows that F_{i}(M) is a vector bundle of rank a_{i} on ℙ^{r−1}. The converse follows by a similar argument. □

Our next proposition appeared as Lemma 2.2 of [9].

Proposition 3.7. If M is a finitely generated kE-module, then M̃ has a filtration with filtered quotients F_{i,j}(M) for all 0 ≤ j < i ≤ p.

Proof. Consider the kernel and image filtrations of M̃:

0 = Ker θ_{M}^{0} ⊆ Ker θ_{M} ⊆ ⋯ ⊆ Ker θ_{M}^{p} = M̃,
0 = Im θ_{M}^{p} ⊆ Im θ_{M}^{p−1} ⊆ ⋯ ⊆ Im θ_{M}^{0} = M̃.

For ease of notation, set K_{j} = Ker θ_{M}^{j} and I_{l} = Im θ_{M}^{p−l}. Mimicking the proof of the Jordan-Hölder theorem, we refine the kernel filtration by

K_{j} ⊆ (K_{j+1} ∩ I_{1}) + K_{j} ⊆ ⋯ ⊆ (K_{j+1} ∩ I_{p−1}) + K_{j} ⊆ K_{j+1}.

If F, G and H are subsheaves of M̃ with G ⊆ F, then by the second isomorphism theorem and the modular law we have

\frac{F + H}{G + H} = \frac{F + (G + H)}{G + H} ≅ \frac{F}{F ∩ (G + H)} = \frac{F}{G + (F ∩ H)}.

Applying this to the above refined filtration, we obtain

\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 l = p − i + j, the right hand expression is equal to F_{i,j}(M). □

The following proposition appeared as Lemma 2.3 of [9]. In particular, it shows for each kE-module M that F_{i,j}(M) ≅ F_{i}(M)(j).

Proposition 3.8. There is a natural isomorphism of functors F_{i,j} ≅ F_{i}(−)(j) for all 0 ≤ j < i ≤ p.

Proof. The map θ_{M}: M̃ → M̃(1) induces an isomorphism

F_{i,j}(M) ⟶^{≅} F_{i,j−1}(M)(1).

We obtain F_{i,j}(M) ≅ F_{i}(M)(j) by induction on j, noting that F_{i}(M) = F_{i,0}(M). Naturality can be checked at each step of the induction. □

The final result of this section describes how the functors F_{i} behave with respect to taking duals in mod(kE). Recall that there is also a concept of dualisation for vector bundles. Specifically, for a vector bundle F on ℙ^{r−1}, we define F^{∨} = Hom_{O}(F, O). Here, Hom_{O} denotes the operation of taking the sheaf hom of two sheaves on ℙ^{r−1}. (See Exercise 1.15 of [30, Chapter II].) It is routine to verify that F^{∨} is also a vector bundle on ℙ^{r−1}, and that if G is any sheaf on ℙ^{r−1}, then there is an isomorphism

Hom_{O}(F, G) ≅ F^{∨} ⊗_{O} G.

The following appeared as Theorem 3.6 of [9].

Proposition 3.9. If M is a kE-module, then

F_{i}(M^{∗}) ≅ F_{i}(M)^{∨}(−i + 1)

for all 1 ≤ i ≤ p.

3.4. Chern classes for vector bundles on ℙ^{r−1}

In this section we recall some basic notions from the theory of Chern classes for vector bundles on ℙ^{r−1}. The standard reference for this material is Fulton [27]. We conclude by presenting a lemma due to Benson and Pevtsova concerning congruences modulo p for Chern classes that will be essential in the following sections.

Recall that the Chow ring of ℙ^{r−1} is the graded commutative ring

A^{∗}(ℙ^{r−1}) = ⨁_{i≥0} A^{i}(ℙ^{r−1}),

where A^{i}(ℙ^{r−1}) is the free abelian group generated by the closed irreducible subvarieties of codimension i in ℙ^{r−1}, modulo rational equivalence. By the discussion at the beginning of [27, Section 8.4], we have A^{∗}(ℙ^{r−1}) ≅ ℤ[h]/(h^{r}). If F is a vector bundle on ℙ^{r−1}, then there exists a well defined polynomial class

c(F) = 1 + c_{1}(F)h + c_{2}(F)h^{2} + ⋯ + c_{r−1}(F)h^{r−1} ∈ A^{∗}(ℙ^{r−1})

characterised by the following properties.

(1) c_{i}(F) = 0 for all i ≥ rank(F).

(2) If 0 → F_{1} → F_{2} → F_{3} → 0 is a short exact sequence of vector bundles on ℙ^{r−1}, then c(F_{2}) = c(F_{1})c(F_{3}).

(3) c(O(n)) = 1 + nh for all n ∈ ℤ.

The equivalence class c(F) ∈ A^{∗}(ℙ^{r−1}) is called the Chern polynomial of F, and the integers c_{i}(F) are called the Chern numbers of F. We record the following formula for the Chern numbers of twists, which follows from Example 3.2.2 of [27].

Lemma 3.10. If F is a vector bundle on ℙ^{r−1}, then for all n ∈ ℤ, the ith Chern number of F(n) is given by

c_{i}(F(n)) = ∑_{j=0}^{i} n^{j}\binom{rank(F) − i + j}{j}c_{i−j}(F).

The following result, which appeared as Lemma 6.2 of Benson and Pevtsova [9], may be thought of as a vector bundle analog of Fermat’s little theorem. Used in conjunction with Proposition 3.7, it will provide an essential tool for obtaining congruences modulo p on the Chern numbers of the trivial sheaf M̃, where M is a kE-module of constant Jordan type.

Lemma 3.11. If F is a vector bundle on ℙ^{r−1}, then

c(F)c(F(1)) ⋯ c(F(p − 1)) ≡ 1 − rank(F)h^{p−1} (mod (p, h^{p})).

3.5. Benson’s theorem on small Jordan types

We now present the main result of Benson [7], along with its proof. It can be interpreted as saying the following: Given a stable Jordan type containing some block of length greater than one, there always exist p and r sufficiently large such that there does not exist a kE-module of constant Jordan type realising the stable Jordan type. The goal throughout the remainder of the chapter will be to tighten the bounds on p and r provided by the following theorem in a few special cases.

Theorem 3.12. If a kE-module has constant Jordan type [p]^{a_{p}} … [1]^{a_{1}} and

∑_{i=1}^{p−1} i · a_{i} ≤ min(r − 1, p − 2),

then a_{i} = 0 for all 2 ≤ i ≤ p − 1.

Proof. Let M be a kE-module of constant Jordan type satisfying the hypothesis of the theorem and consider the trivial sheaf M̃ associated to M on ℙ^{r−1}, which is isomorphic to a direct sum of copies of the structure sheaf O. By Proposition 3.7, M̃ has a filtration with filtered quotients F_{i}(M)(j) for all 0 ≤ j < i ≤ p. Since Chern polynomials are multiplicative over short exact sequences, it follows by Lemma 3.11 that

1 = c(M̃) = ∏_{0≤j<i≤p} c(F_{i}(M)(j)) ≡ ∏_{0≤j<i≤p−1} c(F_{i}(M)(j)) (mod (p, h^{p−1}, h^{r})).

Letting a = ∑_{i=1}^{p−1} i · a_{i}, the polynomial on the right hand side has degree a, because each F_{i}(M) has rank a_{i}. Since a ≤ r − 1 and a ≤ p − 2, the above congruence can be viewed as an equality in 𝔽_{p}[h]. The only units in 𝔽_{p}[h] are the constant polynomials, so all of the above factors must be equal to 1 ∈ 𝔽_{p}[h]. In particular, for i ≥ 2, this is true of c(F_{i}(M)) and c(F_{i}(M)(1)). For such i, this implies that p divides c_{1}(F_{i}(M)) and c_{1}(F_{i}(M)(1)). We have

c_{1}(F_{i}(M)(1)) = c_{1}(F_{i}(M)) + rank(F_{i}(M))

by Lemma 3.10, so p also divides the rank of F_{i}(M). Now observe that

rank(F_{i}(M)) = a_{i} ≤ a < p

by Proposition 3.6. It follows that rank(F_{i}(M)) = 0 so that a_{i} = 0 as required. □

3.6. Restrictions on the stable Jordan type [a][b], a ≠ b

Although Theorem 3.12 does provide restrictions on p and r for which there exists a kE-module of constant Jordan type realising a given stable Jordan type having some block of length greater than one, it is natural to ask whether or not these restrictions are sharp. As we shall see, this question has a negative answer in general.

The technique we use in obtaining sharper bounds is similar to that used in proving Theorem 3.12, but rather than focusing only on the first Chern numbers of the first twists of the bundles F_{i}(M), we exploit even higher Chern numbers of these bundles, and do so simultaneously to solve congruences modulo p on the first few Chern numbers of the trivial sheaf M̃. Unfortunately, the calculations required in this process become increasingly difficult for larger stable Jordan types, and we shall restrict or attention to kE-modules whose stable constant Jordan types contain only two blocks. To illustrate the set-up, we first use our technique to prove a slightly weaker version of Theorem 2.27.

Theorem 3.13. If p ≥ 5 and r ≥ 3, then there does not exist a kE-module of stable constant Jordan type [a] with 2 ≤ a ≤ p − 2.

Proof. Suppose M has stable constant Jordan type [a]. Let α = c_{1}(F_{a}(M)) be the first Chern number of F_{a}(M), which is a rank one vector bundle by Proposition 3.6. It follows by the properties of Chern polynomials that all higher Chern numbers of F_{a}(M) are zero. Using Proposition 3.7 and Lemmas 3.10 and 3.11, we have

1 = c(M̃) ≡ (1 + αh)(1 + (α + 1)h) ⋯ (1 + (α + a − 1)h) (mod (p, h^{p−1}, h^{r})).

Since p ≥ 5 and r ≥ 3, the coefficients on h and h^{2} in this polynomial must be divisible by p. For convenience, we normalise by setting α′ = α + \frac{1}{2}(a − 1). The coefficient on h is then aα′ so that α′ ≡ 0 (mod p). One then calculates the coefficient on h^{2} to be

∑_{− \frac{a−1}{2} ≤n<m≤ \frac{a−1}{2}}(α′ + n)(α′ + m) = −\frac{1}{24}a(a + 1)(a − 1)

after substituting α′ ≡ 0. The latter expression must also be congruent to 0 (mod p), contradicting our assumption that 2 ≤ a ≤ p − 2. □

We now use the above technique to obtain restrictions on the values of a and b for which there exists a kE-module of stable constant Jordan type [a][b] with a ≠ b, provided p and r are sufficiently large. The following is the main theorem of this section.

Theorem 3.14. Suppose p ≥ 5 and r ≥ 4. If a kE-module has stable constant Jordan type [a][b] with a ≠ b, then one of the following holds.

(i) a = p − b. (ii) a = p − b ± 1. (iii) a^{2} + b^{2} − ab − 1 ≡ 0 (mod p).

Proof. Suppose M has stable constant Jordan type [a][b] and that a ≠ p − b. Since F_{a}(M) and F_{b}(M) are vector bundles of rank one, their Chern polynomials are of the form c(F_{a}(M)) = 1 + αh and c(F_{b}(M)) = 1 + βh, respectively. As in the proof of Theorem 3.13, we have

1 = c(M̃) ≡ ∏_{n=0}^{a−1}(1 + (α + n)h) ∏_{m=0}^{b−1}(1 + (β + m)h) (mod (p, h^{p−1}, h^{r})).

Because p ≥ 5 and r ≥ 4, the coefficients on h, h^{2} and h^{3} must all be divisible by p. For convenience, we normalise by setting α′ = α + \frac{1}{2}(a − 1) and β′ = β + \frac{1}{2}(b − 1). The coefficient on h is aα′ + bβ′ so that

β′ ≡ −\frac{a}{b}α′ (mod p).  (3.1)

The coefficient on h^{2} is

\begin{aligned}& ∑_{− \frac{a−1}{2} ≤n<m≤ \frac{a−1}{2}}(α′ + n)(α′ + m) + ∑_{\begin{matrix}− \frac{a−1}{2} ≤n≤ \frac{a−1}{2} \\ − \frac{b−1}{2} ≤m≤ \frac{b−1}{2} \end{matrix}}(α′ + n)(β′ + m) \\ + & ∑_{− \frac{b−1}{2} ≤n<m≤ \frac{b−1}{2}}(β′ + n)(β′ + m) = −\frac{a(a + b)}{2b}α^{′2} − \frac{(a + b)(a^{2} + b^{2} − ab − 1)}{24} \end{aligned}

after substituting (3.1). Since a ≠ p − b, this implies

α^{′2} ≡ −\frac{b(a^{2} + b^{2} − ab − 1)}{12a} (mod p).  (3.2)

The coefficient on h^{3} is

∑_{− \frac{a−1}{2} ≤n<m<l≤ \frac{a−1}{2}}(α′ + n)(α′ + m)(α′ + l) + ∑_{− \frac{b−1}{2} ≤n<m<l≤ \frac{b−1}{2}}(β′ + n)(β′ + m)(β′ + l)

+ ∑_{\begin{matrix}− \frac{a−1}{2} ≤n<m≤ \frac{a−1}{2} \\ − \frac{b−1}{2} ≤l≤ \frac{b−1}{2} \end{matrix}}(α′ + n)(α′ + m)(β′ + l) + ∑_{\begin{matrix}− \frac{b−1}{2} ≤n<m≤ \frac{b−1}{2} \\ − \frac{a−1}{2} ≤l≤ \frac{a−1}{2} \end{matrix}}(α′ + l)(β′ + n)(β′ + m)

= − \frac{a(a + b)(a − b)}{3b^{2}}α^{′3} + \frac{a(a + b)(a − b)}{12}α′

after substituting (3.1). Since a ≠ b and a ≠ p − b, we have

α^{′3} − \frac{1}{4}b^{2}α′ ≡ 0 (mod p)

so that α′ ≡ 0 or ±\frac{1}{2}b (mod p). Substituting α′ ≡ 0 into (3.2) yields

a^{2} + b^{2} − ab − 1 ≡ 0 (mod p),

and substituting α′ ≡ ±\frac{1}{2}b into (3.2) yields

a^{2} + b^{2} + 2ab − 1 ≡ 0 (mod p).

The last congruence has solutions b ≡ −a ± 1 (mod p), completing the proof. □

Corollary 3.15. Let p ≥ 5 and r ≥ 4. If a kE-module has stable constant Jordan type [a][1] with 2 ≤ a ≤ p − 1, then a = p − 1 or a = p − 2.

Proof. Conditions (i) and (ii) of Theorem 3.14 imply the result. If condition (iii) holds, then a^{2} − a ≡ 0 (mod p), forcing a = 1 or a = p, a contradiction. □

Corollary 3.16. If p ≥ 5, r ≥ 4 and 2 ≤ a ≤ p − 3, then there does not exist a kE-module of stable constant Jordan type [a][1].

Remark 3.17. Note that Theorem 3.12 guarantees the stable Jordan type [a][1] is not realised provided r ≥ a + 2 and p ≥ a + 3, whereas Corollary 3.16 allows us to deduce the same statement whenever r ≥ 4. Hence our technique provides new information in the cases where 4 ≤ r ≤ a + 1.

Example 3.18. If p ≥ 7 and r ≥ 4, then there does not exist a kE-module of stable constant Jordan type [3][1].

The above restriction technique can be applied to many other stable Jordan types [a][b]. For example we have the following.

Corollary 3.19. If p ≥ 7 and r ≥ 4, then there does not exist a kE-module of stable constant Jordan type [3][2].

Proof. Since p ≥ 7, conditions (i) and (ii) of Theorem 3.14 cannot hold. Hence if such a module exists, then condition (iii) must hold so that 6 ≡ 0 (mod p). It follows that p = 2 or 3, a contradiction. □

We end this section by calculating which stable Jordan types [a][b] would be excluded from being realised by Suslin’s Conjecture 2.24 in light of the restrictions given by Theorem 3.14.

Proposition 3.20. Suppose p ≥ 5, r ≥ 4 and that Conjecture 2.24 holds. If a kEmodule has stable constant Jordan type [a][b] with a ≠ b, then this stable Jordan type is either [p − 1][1] or [\frac{p+1}{2}] [\frac{p−1}{2}].

Proof. Suppose there exists a module of stable constant Jordan type [a][b] and that this stable Jordan type is not [p − 1][1] or [\frac{p+1}{2}] [\frac{p−1}{2}]. By Conjecture 2.24, we must have b = a ± 1. Without loss of generality, assume 2 ≤ a ≤ p − 2. Since p ≥ 5, conditions (i) and (ii) of Theorem 3.14 cannot hold. So assume condition (iii) holds, that is,

a^{2} + b^{2} − ab − 1 ≡ 0 (mod p).

Then

0 ≡ a^{2} + (a ± 1)^{2} − a(a ± 1) − 1 = a^{2} ± a

so that a ≡ ±1 (mod p), a contradiction. □

3.7. Restrictions on the stable Jordan type [a]^{2}

We now use the technique from Section 3.6 to place restrictions on the value of a for which there exists a kE-module of stable constant Jordan type [a]^{2} with 2 ≤ a ≤ p−2 provided p and r are sufficiently large.

Theorem 3.21. Suppose p ≥ 7 and r ≥ 5. If a kE-module has stable constant Jordan type [a]^{2} with 2 ≤ a ≤ p − 2, then a = \frac{p−1}{2} or a = \frac{p+1}{2}.

Proof. Suppose M has stable constant Jordan type [a]^{2}. Letting u = c_{1}(F_{a}(M)) and v = c_{2}(F_{a}(M)), we have

c_{1}(F_{a}(M)(n)) = u + 2n and c_{2}(F_{a}(M)(n)) = v + un + n^{2}

by Lemma 3.10. As in the proof of Theorem 3.14, we then obtain

1 = c(M̃) ≡ ∏_{n=0}^{a−1}(1 + (u + 2n)h + (v + un + n^{2})h^{2}) (mod (p, h^{p−1})).

Since p ≥ 7 and r ≥ 5, the coefficients on h, h^{2}, h^{3} and h^{4} must all be divisible by p. The coefficient on h is au + a(a − 1), so u ≡ −(a − 1) (mod p). The coefficient on h^{2} is then

∑_{n=0}^{a−1}(v + un + n^{2}) + ∑_{0≤n<m≤a−1}(u + 2n)(u + 2m) = av − \frac{1}{6}a(a − 1)(2a − 1)

after substituting u ≡ −(a − 1). Hence v ≡ \frac{1}{6}(a − 1)(2a − 1). The coefficient on h^{3} is

∑_{\begin{matrix}0≤n,m≤a−1 \\ n≠m \end{matrix}}(v + un + n^{2})(u + 2m) + ∑_{0≤n<m<l≤a−1}(u + 2n)(u + 2m)(u + 2l).

Unfortunately, after substituting u ≡ −(a − 1), this expression is identically zero in v. Hence we are forced to consider the coefficient on h^{4}, which is calculated using

\begin{matrix}∑_{0≤n<m≤a−1}(v + un + n^{2})(v + um + m^{2}) + ∑_{\begin{matrix}0≤n,m,l≤a−1 \\ m≠n≠l \\ m<l \end{matrix}}(v + un + n^{2})(u + 2m)(u + 2l) \\ + ∑_{0≤n<m<l<k≤a−1}(u + 2n)(u + 2m)(u + 2l)(u + 2k). \end{matrix}

Substituting u ≡ −(a − 1) and v ≡ \frac{1}{6}(a − 1)(2a − 1), this yields

\frac{1}{360}a(a + 1)(a − 1)(2a + 1)(2a − 1),

which is congruent to 0 (mod p). Since 2 ≤ a ≤ p − 2, we must have 2a + 1 ≡ 0 or 2a − 1 ≡ 0 so that a = \frac{p−1}{2} or a = \frac{p+1}{2}. □

Corollary 3.22. If p ≥ 11 and r ≥ 5, then there does not exist a kE-module of stable constant Jordan type [3]^{2}.

3.8. Non-realisation of the stable Jordan type [3][2][1] in certain cases

The final result of this chapter uses the above techniques to place restrictions on the values of p and r for which there exists a kE-module of stable constant Jordan type [3][2][1].

Theorem 3.23. If p ≥ 11 and r ≥ 6 or p ≥ 13 and r ≥ 5, then there does not exist a kE-module of stable constant Jordan type [3][2][1].

Proof. Using the same technique as before, we assume that M is a module of stable constant Jordan type [3][2][1] and write

(1+αh)(1+βh)(1+(β +1)h)(1+γh)(1+(γ +1)h)(1+(γ +2)h) ≡ 1 (mod (p, h^{p−1}))

where α = c_{1}(F_{1}(M)), β = c_{1}(F_{2}(M)) and γ = c_{1}(F_{3}(M)). The coefficient on h is α + 2β + 3γ + 4, which in any case must be congruent to 0 (mod p). Hence α ≡ −2β − 3γ − 4 (mod p). Substituting this and calculating the coefficient on h^{3} yields a factorisation

−(β + γ + 1)(β + γ + 2)(2β + 8γ + 9) ≡ 0 (mod p),

hence β ≡ −γ − 1, −γ − 2 or −4γ − \frac{9}{2}. Substituting β ≡ −γ − 1 into the expression for the coefficient on h^{2} yields

−3γ^{2} − 6γ − 5 ≡ 0 (mod p)

so that

γ ≡ −1 ± \frac{1}{3}u (mod p),

where u ∈ ℤ is an integer such that u^{2} ≡ −6. Substituting this and β ≡ −γ − 1 into the expression for the coefficient on h^{4} gives us \frac{7}{3}, which must be congruent to 0 (mod p). But this contradicts our assumption that p ≥ 11. Similarly, substituting β ≡ −γ − 2 into the expression for the coefficient on h^{2} again yields

−3γ^{2} − 6γ − 5 ≡ 0 (mod p),

and substituting these results and calculating the coefficient on h^{4} again gives us \frac{7}{3}, a contradiction. Hence we are left with the case β ≡ −4γ − \frac{9}{2}. Substituting this and calculating the coefficient on h^{2} yields

−30γ^{2} − 60γ − \frac{125}{4} ≡ 0 (mod p)

so that

γ ≡ −1 ± \frac{1}{12}u (mod p).

Substituting this and β ≡ −4γ − \frac{9}{2} into the expression for the coefficient on h^{4} yields −\frac{77}{192}, which is never congruent to zero so long as p ≠ 7 or 11. If we demand that p ≥ 11 and r ≥ 6 however, then the coefficient on h^{5} must be congruent to 0 (mod p), and one readily calculates this to be ±\frac{35}{96}u, which is impossible. This completes the proof. □

3.9. Discussion

While the techniques of the last three sections do produce new bounds on p and r for which there exist kE-modules having certain small Jordan types, one may well ask whether or not these new bounds are sharp. We now give several reasons why this is likely not the case.

We first note that very few stable Jordan types having only two blocks are compatible with the conjectures given in Section 2.8. For example, Rickard’s Conjecture 2.25 would rule out the realisation of all stable Jordan types [a][b] with a ≠ b except [2][1], [p − 1][p − 2] and [p − 1][1]. The first two are actually realised in rank two by the modules kE/J^{2} and J^{2}, respectively, although Conjecture 2.28 would rule out their realisation in higher ranks. The stable Jordan type [p − 1][1] is realised in any rank by the module k ⊕ Ω(k) per the discussion following Theorem 2.18. Both Rickard and Suslin’s conjectures would rule out the realisation of all stable Jordan types [a]^{2} except [1]^{2} and [p − 1]^{2}. The latter are realised in any rank by the modules k ⊕ k and Ω(k) ⊕ Ω(k), respectively.

A more conceptual argument against the sharpness of our bounds is the observation that the theory of Chern classes for vector bundles applies equally well to arbitrary nilvarieties of constant Jordan type. Specifically, a nilvariety of constant Jordan type is a subspace of Mat_{n}(k) spanned by a set of nilpotent matrices A_{1}, …, A_{r} such that every non-zero linear combination of the A_{i} has the same Jordan canonical form. The point we wish to emphasise here is that the matrices A_{i} are not required to commute with one another. In other words, because the vector bundle techniques of Benson and Pevtsova do not exploit the commutative structure of kE, it is unlikely that any approach based solely upon them will yield sharp information about kE-modules of constant Jordan type.

Finally, perhaps the most convincing evidence against the sharpness of our bounds is Benson’s Theorem 6.1, which was proved after our results had already been obtained. One of its consequences is the following, which appeared as Corollary 1.4 of [6]. In particular, for r ≥ 2, it rules out the realisation of any stable Jordan type [a][b] with 2 ≤ a, b ≤ p − 2.

Corollary 3.24. If r ≥ 2 and a kE-module has constant Jordan type containing no blocks of length 1 or p − 1, then the number of Jordan blocks of length less than p is divisible by p.

Proof. Recall from the proof of Theorem 2.18 that we have

Ω(M)↓_{⟨g_{α}⟩} ⊕ P_{M↓_{⟨gα⟩}} ≅ Ω(M ↓_{⟨g_{α}⟩}) ⊕ P_{M} ↓_{⟨g_{α}⟩}

for all non-zero α ∈ 𝔸^{r}(k). The number of Jordan blocks of length p in P_{M↓_{⟨gα⟩}} is equal to the total number of Jordan blocks of M by the way one computes projective covers of k(ℤ/p)-modules. Moreover, Ω(M ↓_{⟨g_{α}⟩}) is projective free, and the number of Jordan blocks of length p in P_{M} ↓_{⟨g_{α}⟩} is divisible by p since r ≥ 2. This shows that the number of Jordan blocks of length p in Ω(M) is congruent modulo p to minus the total number of Jordan blocks of M.

Now observe by the remarks following Theorem 2.18 that if M contains no Jordan blocks of length p−1, then Ω^{−1}(M) contains no Jordan blocks of length one. Replacing M by Ω^{−1}(M) in the above argument and applying Theorem 6.1, it follows that the total number of Jordan blocks of length p in M is divisible by p. By the same theorem, the total number of Jordan blocks of M is divisible by p, hence the number of remaining Jordan blocks must also be divisible by p. □

In light of the preceding remarks, it is likely better to view the results of this chapter in the context of arbitrary nilvarieties of constant Jordan type, a class of objects for which it remains unknown whether or not the bounds we have provided are sharp.

CHAPTER 4 Constant image layers and vector bundles for W -modules

This chapter expands on the work of Carlson, Friedlander and Suslin [17] related to modules with the constant image property. Our goal is to study the unique submodule of a kE-module that is maximal with respect to having the constant image property. This submodule is known to be well behaved when E has rank two, as it coincides with the module’s generic kernel. (See Chapter 5.) The best we are able to do here is show that every kE-module has a natural filtration with respect to these submodules in which almost all of the filtered subquotients are semisimple.

We also prove an interesting lemma relating the functors F_{i} (see Section 3.3) to powers of the radical of a module having the constant image property. After introducing the class of W -modules in rank two, which were also defined in [17], we use this lemma to calculate the images of all W -modules under the F_{i}.

4.1. The constant image property

The constant image property was introduced by Carlson, Friedlander and Pevtsova [16] and was investigated further by Carlson, Friedlander and Suslin [17]. We maintain our notation as in Chapter 2.

Definition 4.1. A kE-module M has the constant image property if Im(X_{α}, M) is independent of the choice of non-zero α ∈ 𝔸^{r}(k).

Note that M has the constant image property if and only if X_{α}M = Rad(M) for all non-zero α ∈ 𝔸^{r}(k). To see this, suppose that

m = X_{1}m_{1} + ⋯ + X_{r}m_{r} ∈ Rad(M)

with m_{1}, …, m_{r} ∈ M. If M has the constant image property, then for each 2 ≤ i ≤ r there exists m′_{i} ∈ M such that X_{1}m′_{i} = X_{i}m_{i}, hence m = X_{1}(m_{1} + ∑_{i=2}^{r} m′_{i}). It follows by another application of the constant image property that m ∈ Im(X_{α}, M) for all non-zero α ∈ 𝔸^{r}(k).

As was the case with the constant Jordan type property (see Definition 2.14), it is not clear from the definition that the constant image property is independent of the choice of generators g_{1}, …, g_{r} of E. The following proposition, which appeared as Proposition 2.6 (3) of [17], shows that this is indeed the case.

Proposition 4.2. A kE-module M has the constant image property if and only if Im(x, M) is independent of x ∈ J \ J^{2}.

Proof. One direction is clear. So suppose x ∈ J \ J^{2} and write x = X_{α} + y with α ∈ 𝔸^{r}(k) non-zero and y ∈ J^{2}. We have

Rad(M) = X_{α}M ⊆ xM + yM ⊆ Rad(M)

by the constant image property, hence xM + yM = Rad(M). Since yM ⊆ Rad^{2}(M), it follows from Nakayama’s lemma that xM = Rad(M). □

A nice feature of the constant image property is that it also remains invariant under field extensions. The following appeared as Proposition 2.5 (2) of [17].

Proposition 4.3. A kE-module M has the constant image property if and only if for every field extension K/k, Im(X_{α}, M ⊗_{k} K) is independent of the choice of non-zero α ∈ 𝔸^{r}(K).

Our next goal is to show that every kE-module with the constant image property also has constant Jordan type, a fact recorded as Proposition 2.8 of [17].

Proposition 4.4. If M is a kE-module with the constant image property and α is a non-zero point in 𝔸^{r}(k), then X_{α}^{i}M = Rad^{i}(M) for all i ≥ 0.

Proof. We proceed by induction on i, the case i = 1 being clear by the constant image property. If i ≥ 1 and X_{α}^{i}M = Rad^{i}(M), then by the commutativity of kE and the constant image property we have

X_{α}^{i+1}M = X_{α}.X_{α}^{i}M = X_{α}.J^{i}M = J^{i}.X_{α}M = J^{i}.JM = Rad^{i+1}(M)

as required. □

Proposition 4.5. If M is a kE-module with the constant image property, then M has constant Jordan type.

Proof. By Proposition 4.4, the rank of X_{α}^{i} acting on M is independent of the choice of non-zero α ∈ 𝔸^{r}(k) for all i ≥ 0. The proof now follows after noting that the Jordan type of a nilpotent matrix is completely determined by the ranks of its powers. □

We end this section by recording some facts about the constant image property that follow directly from the definition and Proposition 4.4.

Proposition 4.6. The class of kE-modules with the constant image property is closed under taking finite direct sums, quotients and radicals.

4.2. Constant image layers

Proposition 4.7. If M is a kE-module and N, N′ are submodules of M having the constant image property, then N + N′ has the constant image property.

Proof. We have N + N′ ≅ (N ⊕ N′)/Ker(ϕ), where ϕ: N ⊕ N′ → N + N′ is the surjection (n, n′) ↦ n + n′. The proof now follows from Proposition 4.6. □

Corollary 4.8. Every (non-zero) kE-module contains a unique (non-zero) submodule that is maximal with respect to having the constant image property.

Proof. This follows from Proposition 4.7 and the fact that for any kE-module M, Soc(M) has the constant image property. □

Definition 4.9. In light of Corollary 4.8, we denote the maximal submodule of M with the constant image property by I(M).

Definition 4.10. Let M be a kE-module. We define the constant image series of M inductively by I^{0}(M) = 0 and I^{n}(M)/I^{n−1}(M) = I(M/I^{n−1}(M)). For n > 0, we call the subquotients I^{n}(M)/I^{n−1}(M) the constant image layers of M.

Our goal in this section is to show for every kE-module M that the constant image layers I^{n}(M)/I^{n−1}(M) are semisimple for all n ≥ 2. The key result is the following.

Lemma 4.11. If M is a kE-module, then I(M/I(M)) = Soc(M/I(M)).

Proof. Let N be the preimage of Soc(M/I(M)) under the canonical surjection M → M/I(M). If L is the preimage of I(M/I(M)) under this map, then we have N ⊆ L ⊆ M since Soc(M/I(M)) has the constant image property. This means that I(L) = I(M) and N/I(M) = Soc(L/I(M)). Replacing M by L, it therefore suffices to show that if M/I(M) has the constant image property, then M = N. The statement is clear if M has the constant image property, so assume that M ≠ I(M).

As a first step towards proving the statement, we claim there exists an element m ∈ I(M) and linearly independent α, β ∈ 𝔸^{r}(k) such that m ∈ Im(X_{α}, N) \ Im(X_{β}, N). Suppose this is not the case. Then if m ∈ I(M), we have either m ∈ Im(X_{α}, N) for all non-zero α ∈ 𝔸^{r}(k) or m ∉ Im(X_{α}, N) for any α ∈ 𝔸^{r}(k). In the latter situation we cannot have m ∈ Rad(N). To see this, suppose otherwise and write

m = X_{1}n_{1} + ⋯ + X_{r}n_{r} for some n_{1}, …, n_{r} ∈ N.

Then m − (X_{1}n_{1} + ⋯ + X_{r−1}n_{r−1}) ∈ Im(X_{r}, N), but m ∉ Im(X_{r}, N) by assumption, so we must have X_{1}n_{1} + ⋯ + X_{r−1}n_{r−1} ∉ Im(X_{r}, N). Our original assumption then allows us to conclude that X_{1}n_{1} + ⋯ + X_{r−1}n_{r−1} ∉ Im(X_{α}, N) for any α ∈ 𝔸^{r}(k). In particular, we have X_{1}n_{1} + ⋯ + X_{r−1}n_{r−1} ∉ Im(X_{r−1}, N). Proceeding by induction, we eventually see that X_{1}n_{1} ∉ Im(X_{2}, N). Thus X_{1}n_{1} ∈ Rad(N) ⊆ I(M) and

X_{1}n_{1} ∈ Im(X_{1}, N) \ Im(X_{2}, N),

contradicting our original assumption. In summary, this shows that if m ∈ Rad(N), then m ∈ Im(X_{α}, N) for all non-zero α ∈ 𝔸^{r}(k). This means that N has the constant image property, and it follows that N = I(M). On the other hand, we assumed that M does not have the constant image property, so N properly contains I(M). This is a contradiction, and the claim now follows.

Having established the claim, we now suppose that M ≠ N. Then because M/I(M) is not semisimple, its radical length d is greater than one. Since the constant image property is closed under taking radicals, one sees that Rad^{d−2}(M/I(M)) also has the constant image property. Replacing M by Rad^{d−2}(M) + I(M) and N by

(N ∩ Rad^{d−2}(M)) + I(M),

we may therefore assume that M/I(M) has radical length two. Note that M/I(M) will continue to have a direct summand that is not semisimple. Because the constant image property is also closed under taking direct summands, we may assume further that M/I(M) contains no simple direct summand. These conditions imply that

Rad(M/I(M)) = Soc(M/I(M)) = N/I(M).

Now let m, α, β be as described in the claim and choose n ∈ N such that m = X_{α}n. Then since M/I(M) has the constant image property and n + I(M) is in the radical of M/I(M), there exists n′ ∈ M such that X_{β}(n′ + I(M)) = n + I(M). We have

X_{β}X_{α}n′ = X_{α}X_{β}n′ = X_{α}(n + m_{0}) for some m_{0} ∈ I(M)
= m + X_{α}m_{0} = m + X_{β}m_{1} for some m_{1} ∈ I(M)

since I(M) has the constant image property, thus m = X_{β}(X_{α}n′ − m_{1}). But we also have X_{α}n′ + I(M) ∈ Rad(M/I(M)) = N/I(M), forcing X_{α}n′ − m_{1} ∈ N. This contradicts our assumption that m ∉ Im(X_{β}, N), completing the proof. □

Theorem 4.12. For n ≥ 2 we have I^{n}(M)/I^{n−1}(M) = Soc(M/I^{n−1}(M)), hence all such constant image layers are semisimple.

Proof. For convenience, we write I^{n} = I^{n}(M). By Lemma 4.11 we have

I^{n}/I^{n−1} = I(M/I^{n−1}) ≅ I((M/I^{n−2})/(I^{n−1}/I^{n−2}))
= I((M/I^{n−2})/I(M/I^{n−2})) = Soc((M/I^{n−2})/I(M/I^{n−2}))
= Soc((M/I^{n−2})/(I^{n−1}/I^{n−2})) ≅ Soc(M/I^{n−1}).

□

4.3. W -modules and the constant image property in rank two

As observed by Carlson, Friedlander and Suslin [17], the constant image property is particularly well behaved in the case where E has rank two. Specifically, the authors of [17] introduced the family of k(ℤ/p)^{2}-modules called W -modules. As we shall see in this section, every W -module has the constant image property, and what’s more, a k(ℤ/p)-module has the constant image property if and only if it is the homomorphic image of some W -module.

For the remainder of this thesis, unless otherwise specified, we shall assume that r = 2, that is, E = ⟨g_{1}, g_{2}⟩ ≅ (ℤ/p)^{2}, where g_{1}, g_{2} are fixed commuting generators having order p. Recall that the Jacobson radical J = J(kE) is generated by the elements X_{1} = g_{1} − 1 and X_{2} = g_{2} − 1.

Definition 4.13. Let n and d be positive integers such that 1 ≤ d ≤ n and d ≤ p. If V is the free kE-module of rank n with generators v_{1}, …, v_{n}, we define W_{n,d} to be the quotient V/U, where U is the kE-submodule of V generated by the elements

X_{1}v_{1}, X_{2}v_{n}, X_{1}^{d}v_{i} for 1 ≤ i ≤ n, X_{2}v_{i} − X_{1}v_{i+1} for 1 ≤ i ≤ n − 1.

Any kE-module of the form W_{n,d} is called a W -module.

W -modules can be visualised using module diagrams as introduced in Section 2.4. For example, if p is any prime number and n ≥ 2, then the module W_{n,2} is represented by the diagram

[figure: v_{1}; v_{2}; v_{3}; ⋯; v_{n−1}; v_{n}; •; •; •]

Similarly, for p ≥ 3, the module W_{4,3} has diagram

[figure: v_{1}; v_{2}; v_{3}; v_{4}; •; •; •; •; •]

Observe that both of the above diagrams have what can be roughly described as a ‘W’ shape, hence the terminology ‘W -module’.

It is easy to verify that every W -module has the constant image property, thus has constant Jordan type. The following appeared as Proposition 3.3 of [17].

Proposition 4.14. If 1 ≤ d ≤ n and d ≤ p, then the kE-module W_{n,d} has the constant image property.

Proof. It suffices to show that X_{α}W_{n,d} = Rad(W_{n,d}) for all non-zero α ∈ 𝔸^{2}(k). Without loss of generality, we may assume that α = (1, λ) for some λ ∈ k. Observe that Rad(W_{n,d}) is generated by the elements X_{1}v_{2}, …, X_{1}v_{n}. We have

X_{1}v_{i} = X_{α}v_{i} − λX_{2}v_{i} = X_{α}v_{i} − λX_{1}v_{i+1}.

Continuing in this way we obtain X_{1}v_{i} = ∑_{j=i}^{n}(−λ)^{j−i}X_{α}v_{j} so that

X_{α}W_{n,d} = X_{1}W_{n,d} = Rad(W_{n,d}).  □

Corollary 4.15. If 1 ≤ d ≤ n and d ≤ p, then W_{n,d} has constant Jordan type [d]^{n−d+1}[d − 1] … [1].

Proof. The fact that W_{n,d} has constant Jordan type follows from Proposition 4.5. Calculating the Jordan type is accomplished by calculating the Jordan type of X_{1} on W_{n,d}, using the corresponding module diagram as illustrated above. □

Not only do W -modules have the constant image property, but it can also be shown that every kE-module with the constant image property is realised as the homomorphic image of some W_{n,d}. In particular, we record the following theorem, which appeared as Theorem 5.4 of [17].

Theorem 4.16. If M is a kE-module with the constant image property having radical length d, then there exists an integer n ≥ d and a surjective kE-module homomorphism W_{n,d} → M.

4.4. Vector bundles for W -modules

Motivated by the central role W -modules play in the theory of kE-modules with the constant image property, we now calculate the vector bundles F_{i}(W_{n,d}) for 1 ≤ i ≤ d. In [17, Proposition 6.4], the authors calculated the kernel bundle Ker θ_{W_{n,d}}. Referring to the proof of Proposition 3.7, our calculation is in some sense a refinement of their result. We begin with a general lemma, which may be thought of as a vector bundle analog of Proposition 4.4.

Lemma 4.17. If M is a kE-module with the constant image property in any rank r, then for all 0 ≤ j < i ≤ p we have F_{i}(M) ≅ F_{i−j}(Rad^{j}(M)).

Proof. By the definition of the operator θ_{M}, the map θ_{M}^{j}: M̃(−i) → M̃(−i + j) induces a natural map of vector bundles

M̃(−i) \underrightarrow{θ̂_{M}^{j}} { \atop Rad^{j}(M)}(−i + j).

We claim that this map is surjective. Note by Proposition 3.5 that its fibre at a point x ∈ ℙ^{r−1} is

M ⊗_{k} k(x) ⟶^{X_{α}^{j}} Rad^{j}(M) ⊗_{k} k(x),

where ᾱ ∈ ℙ^{r−1}(k(x)) corresponds to the generic point of x. Since M has the constant image property, so does M ⊗_{k} k(x) by Proposition 4.3, hence the map X_{α}^{j} is surjective by Proposition 4.4. It now follows from Proposition 3.4 that θ̂_{M}^{j} is surjective.

Having established the claim, consider the composition of maps

M̃(−i) \underrightarrow{θ̂_{M}^{j}} { \atop Rad^{j}(M)}(−i + j) ⟶^{θ_{Rad^{j} (M)}^{i−j}} { \atop Rad^{j}(M)}.

Since the map on the left is surjective, the image of the composition is equal to that of the map on the right. But the image of the composition is also isomorphic to Im{θ_{M}^{j}: M̃(−i) → M̃}. It follows that

F_{i}(M) = \frac{Ker θ_{M} ∩ Im θ_{M}^{i−1}}{Ker θ_{M} ∩ Im θ_{M}^{i}} ≅ \frac{Ker θ_{M} ∩ Im θ_{Rad^{j}(M)}^{i−j−1}}{Ker θ_{M} ∩ Im θ_{Rad^{j}(M)}^{i−j}} ≅ \frac{Ker θ_{Rad^{j}(M)} ∩ Im θ_{Rad^{j}(M)}^{i−j−1}}{Ker θ_{Rad^{j}(M)} ∩ Im θ_{Rad^{j}(M)}^{i−j}}
= F_{i−j}(Rad^{j}(M))

as required. □

Returning to the rank two case, we recall that Grothendieck [28] has classified the vector bundles on ℙ^{1}. In particular, every such bundle is a direct sum of line bundles

O(n_{1}) ⊕ ⋯ ⊕ O(n_{t}),

where the integers n_{1}, …, n_{t} are uniquely determined up to reordering. Given this classification, we now present the main theorem of this section.

Theorem 4.18. If 1 ≤ d ≤ n and d ≤ p, then F_{i}(W_{n,d}) ≅ O(−n+i) for 1 ≤ i ≤ d−1, and F_{d}(W_{n,d}) ≅ O^{⊕(n−d+1)}.

Proof. We proceed by induction on d, the case d = 1 being trivial. So suppose d > 1. Since the trivial bundle W̃_{n,d} has a filtration with filtered quotients F_{i}(M)(j) for 0 ≤ j < i ≤ d by Proposition 3.7, we have

1 = c(W̃_{n,d}) = ∏_{0≤j<i≤d} F_{i}(W_{n,d})(j).

Comparing the first Chern numbers using Lemma 3.10 gives us

0 = ∑_{i=1}^{d−1} (i · c_{1}(F_{i}(W_{n,d})) + \frac{1}{2}i(i − 1)) + ∑_{j=0}^{d−1} j · c_{1}(F_{d}(W_{n,d})).  (4.1)

Note that Rad(W_{n,d}) is also a W -module, being isomorphic to W_{n−1,d−1}. Hence by induction and Lemma 4.17 we have

F_{i}(W_{n,d}) ≅ F_{i−1}(Rad(W_{n,d})) ≅ F_{i−1}(W_{n−1,d−1}) ≅ O(−n + i)

for 2 ≤ i ≤ d − 1 and

F_{d}(W_{n,d}) ≅ F_{d−1}(Rad(W_{n,d})) ≅ F_{d−1}(W_{n−1,d−1}) ≅ O^{⊕(n−d+1)}.

Substituting this into (4.1) and simplifying yields

0 = c_{1}(F_{1}(W_{n,d})) + n − 1

so that F_{1}(W_{n,d}) ≅ O(−n + 1). This completes the proof. □

The following corollary shows that the functor F_{1} : cJt(kE) → Vec(ℙ^{1}) is essentially surjective.

Corollary 4.19. If F is a vector bundle on ℙ^{1}, then there exists a finitely generated kE-module M of constant Jordan type such that F_{1}(M) ≅ F.

Proof. By the above remarks, it suffices to realise all line bundles on ℙ^{1} since F_{1} distributes over direct sums. By Theorem 4.18, we have F_{1}(W_{n,2}) ≅ O(−n + 1) for all n ≥ 2 and F_{1}(k) ≅ F_{1}(W_{1,1}) ≅ O. To realise the line bundles with positive first Chern numbers, we use Proposition 3.9 to obtain

F_{1}(W_{n,2}^{∗}) ≅ F_{1}(W_{n,2})^{∨} ≅ O(−n + 1)^{∨} ≅ O(n − 1) for all n ≥ 2.  □

CHAPTER 5 Generic kernels and a question of Carlson, Friedlander and Suslin

This chapter provides a thorough treatment of Question 8.11 of Carlson, Friedlander and Suslin [17] regarding generic kernels for modules of constant Jordan type in rank two. Aside from giving a detailed answer to this question, we derive a particularly nice duality formula for a module of constant rank that relates its generic kernel to that of its dual. We make a special point of showing that this formula is independent of the Hopf algebra structure on kE. The exposition of background material closely follows that of [17], where all relevant concepts were originally introduced.

5.1. Generic kernels

Throughout this chapter we continue to require that E has rank two, unless otherwise specified. We now introduce the primary structure of interest throughout the rest of the thesis, namely the generic kernel of a kE-module, which was defined by Carlson, Friedlander and Suslin [17]. There, the authors showed that the generic kernel 𝔎(M) of a kE-module M coincides with the maximal submodule of M having the constant image property. (See Section 4.2.)

Let M be a finite dimensional kE-module. Note for any non-zero point

α = (λ_{1}, λ_{2}) ∈ 𝔸^{2}(k)

that Ker(X_{α}, M) is uniquely determined by the class ᾱ = [λ_{1} : λ_{2}] ∈ ℙ^{1}(k). If S is a subset of ℙ^{1}(k), we denote by _{S}M the vector space ∑_{ᾱ∈S} Ker(X_{α}, M). Observe that SM is in fact a kE-submodule of M, because kE is a commutative ring. Recall that a subset S ⊆ ℙ^{1}(k) is cofinite if its complement in ℙ^{1}(k) is a finite set.

Definition 5.1 (Carlson, Friedlander and Suslin [17]). If M is a finite dimensional kE-module, we define the generic kernel of M to be the submodule

𝔎(M) = ⋂_{S ⊆ ℙ^{1}(k) cofinite}_{S}M

of M.

Note that there always exists a cofinite S ⊆ ℙ^{1}(k) such that 𝔎(M) = _{S}M. To see this, suppose otherwise and observe from the definition that 𝔎(M) ⊆ _{ℙ^{1}(k)}M. Since 𝔎(M) ≠ _{ℙ^{1}(k)}M, there exists a cofinite S_{1} ⊆ ℙ^{1}(k) such that _{ℙ^{1}(k)}M ⊈ _{S_{1}}M. Similarly, because 𝔎(M) ≠ _{S_{1}}M, there exists a cofinite S_{2} ⊆ ℙ^{1}(k) such that _{S_{1}}M ⊈ _{S_{2}}M. Writing S_{1} = ℙ^{1}(k) \ T_{1} and S_{2} = ℙ^{1}(k) \ T_{2} with T_{1}, T_{2} ⊆ ℙ^{1}(k) finite, we then have

ℙ^{1}(k)M ⊋ _{ℙ^{1}(k)\T_{1}}M ⊋ _{ℙ^{1}(k)\(T_{1}∪T_{2})}M ⊋ 𝔎(M).

Continuing in this way, we obtain a strictly descending chain of submodules

ℙ^{1}(k)M ⊋ _{ℙ^{1}(k)\T_{1}}M ⊋ _{ℙ^{1}(k)\(T_{1}∪T_{2})}M ⊋ ⋯ ⊋ _{ℙ^{1}(k)\(T_{1}∪···∪T_{n})}M ⊋ ⋯,

contradicting the fact that M is finite dimensional.

Using the identity _{T} (_{S}M) = _{S ∩ T} M for all cofinite S, T ⊆ ℙ^{1}(k), one also sees from the definition that 𝔎(𝔎(M)) = 𝔎(M).

The following theorem appeared as Theorem 7.10 of [17]. It provides an essential link between the generic kernel and the constant image property.

Theorem 5.2. If M is a kE-module, then the generic kernel 𝔎(M) has the constant image property. Moreover, if N is any submodule of M having the constant image property, then N ⊆ 𝔎(M).

Proof. We shall prove the first statement to demonstrate the geometric nature of the argument. The proof of the second statement is more difficult, and we leave the reader to study its proof in [17].

It suffices to show that Im(X_{α}, 𝔎(M)) = Im(X_{1}, 𝔎(M)) for each non-zero α ∈ 𝔸^{2}(k). Let S ⊆ ℙ^{1}(k) be cofinite such that 𝔎(M) = _{S}M = ∑_{β̄∈S} Ker(X_{β}, M). We may assume that S contains neither ᾱ nor [0 : 1], the latter corresponding to the element X_{1} ∈ J. Let m be a generator of 𝔎(M) lying in Ker(X_{β}, M) for some β̄ ∈ S, and let β ∈ 𝔸^{2}(k) be any point lying above β̄. Since α and β are linearly independent, X_{α} and X_{β} generate J. It follows that

\begin{aligned}Rad(kE.m) & = J.m = (X_{α}, X_{β}).m = X_{α}.kE.m + X_{β}.kE.m \\ & = X_{α}.kE.m + kE.X_{β}.m = X_{α}.kE.m. \end{aligned}

By a similar argument we have Rad(kE.m) = X_{1}.kE.m. Summing over all generators of the above form completes the proof. □

5.2. Generic kernels for modules of constant rank

For the moment, we temporarily relax our condition that E has rank two in order to introduce modules of constant rank in a more general context.

Definition 5.3. A kE-module M has constant rank if rank(X_{α}, M) is independent of the choice of non-zero α ∈ 𝔸^{r}(k). In particular, if M has constant Jordan type then M has constant rank.

Much of our future discussion will involve showing that certain duality results are independent of the Hopf algebra structure on kE. With this goal in mind we record the following lemma, which may be found in Jantzen [31, I.2.4].

Lemma 5.4. Let τ: kE → kE be the antipode for any Hopf algebra structure on kE. If x ∈ J and α is the point in 𝔸^{r}(k) such that x ≡ X_{α} (mod J^{2}), then

τ(x) ≡ −X_{α} (mod J^{2}).

We now give a proposition that implies, in particular, that M has constant rank if and only if M^{∗} has constant rank. The proof is similar to that of Proposition 2.11. The main feature here is that we avoid fixing the Hopf algebra structure on kE.

Proposition 5.5. Let τ: kE → kE be the antipode for any Hopf algebra structure on kE. Let M be a kE-module and let M_{τ}^{∗} denote the dual kE-module structure on Hom_{k}(M, k) induced by τ. If α is a point in 𝔸^{r}(k) such that the rank of X_{α} on M is maximal amongst all points in 𝔸^{r}(k), then the rank of X_{α} on M_{τ}^{∗} is equal to the rank of X_{α} on M.

Proof. Observe that X_{α} + τ(X_{α}) ∈ J^{2} for all α ∈ 𝔸^{r}(k) by Lemma 5.4. If X_{α} achieves the maximal rank on M, then it follows from the second part of Theorem 2.5 that rank(X_{α}, M) = rank(τ(X_{α}), M). The proof is complete after noting that the representation affording M_{τ}^{∗} is obtained from the representation affording M by precomposing with τ and then composing with the transpose map. □

Returning to the case in which E has rank two, our next goal is to show that 𝔎(M) is especially well behaved whenever M has constant rank. The following appeared as Proposition 7.6 of [17].

Lemma 5.6. Let M be a finite dimensional kE-module.

(1) For any ᾱ ∈ ℙ^{1}(k), we have Ker(X_{α}, M) ⊆ 𝔎(M) if and only if the rank of X_{α} is maximal amongst all points in 𝔸^{2}(k).

(2) The set of points ᾱ ∈ ℙ^{1}(k) such that X_{α} achieves the maximal rank on M is the largest subset S ⊆ ℙ^{1}(k) such that 𝔎(M) = _{S}M.

Proof. Write 𝔎(M) = _{S}M for some cofinite S ⊆ ℙ^{1}(k). Since S is an infinite set, it is dense in ℙ^{1}(k), hence intersects every open set of ℙ^{1}(k) non-trivially. Now the set of points ᾱ ∈ ℙ^{1}(k) such that rank(X_{α}, M) is maximal forms an open subset of ℙ^{1}(k). Thus there exists a point ᾱ ∈ ℙ^{1}(k) such that Ker(X_{α}, M) ⊆ 𝔎(M) and such that X_{α} has maximal rank on M. For any β̄ ∈ ℙ^{1}(k) we then have

dim_{k} Ker(X_{β}, M) ≥ dim_{k} Ker(X_{α}, M)
= dim_{k} Ker(X_{α}, 𝔎(M))
= dim_{k} Ker(X_{β}, 𝔎(M)).

The inequality follows from the fact that rank(X_{α}, M) is maximal, the first equality follows from the fact that Ker(X_{α}, M) ⊆ 𝔎(M), and the second equality follows from the fact that 𝔎(M) has the constant image property. Note that X_{β} has maximal rank on M if and only if the inequality is an equality, which is true if and only if Ker(X_{β}, M) ⊆ 𝔎(M). This proves part (i). Part (ii) follows by the same argument.

□

As an immediate consequence, we obtain the following characterisation of the constant rank property in terms of the generic kernel.

Theorem 5.7. A kE-module M has constant rank if and only if 𝔎(M) = _{ℙ^{1}(k)}M. In this case we have Ker(X_{α}, M) ⊆ 𝔎(M) for all non-zero α ∈ 𝔸^{2}(k).

5.3. The generic kernel filtration and the main question

The goal of this section is to introduce the main question concerning generic kernels and modules of constant Jordan type that we wish to address in this chapter.

Let M be a kE-module and N a submodule of M. If I is an ideal of kE and i ∈ ℕ, then we denote by I^{−i}N the set of elements m ∈ M satisfying I^{i}m ⊆ N. Similarly, for x ∈ kE, we denote by x^{−i}N the set of elements m ∈ M for which x^{i}m ∈ N. It is immediately verified that both of these constructions are submodules of M, because kE is a commutative ring.

Armed with Theorem 5.7, we now introduce the main proposition of the section. It originally appeared as part of [17, Theorem 8.10]. We provide an alternative proof here.

Proposition 5.8. If M is a kE-module of constant rank and α is a non-zero point in 𝔸^{2}(k), then X_{α}^{−i}𝔎(M) = J^{−i}𝔎(M) for all i ≥ 0.

Proof. The leftwards containment is clear for each i ≥ 0, for if m ∈ M satisfies J^{i}m ⊆ 𝔎(M), then we obviously have X_{α}^{i}m ∈ 𝔎(M). We are therefore left to prove the reverse containment, proceeding by induction on i. For i = 1, let m ∈ M such that X_{α}m ∈ 𝔎(M). Given β ∈ 𝔸^{2}(k), we then have X_{β}X_{α}m ∈ J𝔎(M). By Theorem 5.2 and the constant image property, there exists m′ ∈ 𝔎(M) such that

X_{α}m′ = X_{β}X_{α}m = X_{α}X_{β}m.

It follows by Theorem 5.7 that m′ − X_{β}m ∈ Ker(X_{α}, M) ⊆ 𝔎(M), hence X_{β}m is also in 𝔎(M). Since the elements X_{β} generate J, this shows that Jm ⊆ 𝔎(M).

Now let i > 1 and assume by induction that X_{α}^{−i}𝔎(M) = J^{−i}𝔎(M). Letting m ∈ M satisfy X_{α}^{i}m ∈ 𝔎(M), we then have X_{α}^{i}X_{α}m ∈ 𝔎(M), hence

X_{α}m ∈ X_{α}^{−i}𝔎(M) = J^{−i}𝔎(M).

It follows that X_{α}J^{i}m ⊆ 𝔎(M) so that J^{i}m ∈ X_{α}^{−1}𝔎(M) = J^{−1}𝔎(M) by the opening step of the induction. This shows that J^{i+1}m = JJ^{i}m ⊆ 𝔎(M) as required. □

The following corollary will be used implicitly in Section 5.8 whenever we invoke the duality provided by Theorem 5.23.

Corollary 5.9. If M is a kE-module of constant rank, then M = J^{−p+1}𝔎(M) and J^{p}𝔎(M) = 0.

Proof. For any non-zero α ∈ 𝔸^{2}(k) we have X_{α}^{p}M = 0, hence

X_{α}^{p−1}M ⊆ Ker(X_{α}, M) ⊆ 𝔎(M)

by Theorem 5.7. It follows that M ⊆ X_{α}^{−p+1}𝔎(M) = J^{−p+1}𝔎(M) by Proposition 5.8. Finally, we have J^{p}𝔎(M) = 0 by Proposition 4.4. □

If M is a kE-module of constant rank, then Corollary 5.9, Theorem 5.2 and Proposition 4.6 show that M has a filtration

0 = J^{p}𝔎(M) ⊆ J^{p−1}𝔎(M) ⊆ ⋯ ⊆ J^{−p+2}𝔎(M) ⊆ J^{−p+1}𝔎(M) = M  (5.1)

in which J^{i}𝔎(M) has constant Jordan type for all i ≥ 0. Moreover, if M itself has constant Jordan type, then the previous statement is also true for i = −p + 1. We call (5.1) the generic kernel filtration of M. The above observations lead naturally to our main question, which appeared as Question 8.11 of [17].

Question 5.10. If M is a finite dimensional kE-module of constant Jordan type, is it true that J^{−i}𝔎(M) also has constant Jordan type for all i ≥ 0?

As we shall see later in the chapter, Question 5.10 has an affirmative answer whenever p = 3 or J^{2}𝔎(M) = 0. We shall also see that these restrictions are sharp, that is, we will construct a counterexample to Question 5.10 in the first interesting case, namely that in which p ≥ 5, J^{2}𝔎(M) ≠ 0 and J^{3}𝔎(M) = 0.

5.4. Generic images

In order to prove statements about duality and generic kernels in the sequel, we now introduce the concept of the generic image of a kE-module. As with the preceding material in this chapter, the generic image was defined and investigated by Carlson, Friedlander and Suslin [17].

Let M be a finite dimensional kE-module. Observe for any non-zero point α ∈ 𝔸^{2}(k) that Im(X_{α}, M) is uniquely determined by the class ᾱ ∈ ℙ^{1}(k). If S is a subset of ℙ^{1}(k), we denote by M_{S} the vector space ⋂_{ᾱ∈S} Im(X_{α}, M). As before, the fact that kE is a commutative ring implies that M_{S} is a kE-submodule of M.

Definition 5.11 (Carlson, Friedlander and Suslin [17]). If M is a finite dimensional kE-module, then the generic image of M is defined to be the submodule

ℑ(M) = ∑_{S ⊆ ℙ^{1}(k) cofinite} M_{S}

of M.

As was the case with generic kernels, there always exists a cofinite S ⊆ ℙ^{1}(k) such that ℑ(M) = M_{S}. To see this, suppose otherwise and observe from the definition that M_{ℙ^{1}(k)} ⊆ ℑ(M). Since ℑ(M) ≠ M_{ℙ^{1}(k)}, there exists a cofinite S_{1} ⊆ ℙ^{1}(k) such that M_{S_{1}} ⊈ M_{ℙ^{1}(k)}. Similarly, because ℑ(M) ≠ M_{S_{1}}, there exists a cofinite S_{2} ⊆ ℙ^{1}(k) such that M_{S_{2}} ⊈ M_{S_{1}}. Writing S_{1} = ℙ^{1}(k)\T_{1} and S_{2} = ℙ^{1}(k)\T_{2} with T_{1}, T_{2} ⊆ ℙ^{1}(k) finite, we then have

M_{ℙ^{1}(k)} ⊊ M_{ℙ^{1}(k)\T_{1}} ⊊ M_{ℙ^{1}(k)\(T_{1}∪T_{2})} ⊊ ℑ(M).

Continuing in this way, we obtain a strictly ascending chain of submodules

M_{ℙ^{1}(k)} ⊊ M_{ℙ^{1}(k)\T_{1}} ⊊ M_{ℙ^{1}(k)\(T_{1}∪T_{2})} ⊊ ⋯ ⊊ M_{ℙ^{1}(k)\(T_{1}∪···∪T_{n})} ⊊ ⋯,

contradicting the fact that M is finite dimensional.

Given any Hopf algebra structure on kE, recall that if M is a kE-module and N is a submodule of M, then the orthogonal complement

N^{⊥} = {f ∈ Hom_{k}(M, k) | f(n) = 0 for all n ∈ N}

is naturally a kE-submodule of Hom_{k}(M, k), and we have (N^{⊥})^{⊥} ≅ N. The following appeared as Proposition 8.4 of [17]. It relates the generic image to the generic kernel through duality and orthogonality.

Proposition 5.12. If M is a kE-module and M ˜∗ is the Lie theoretic dual of M (see Section 2.2), then

ℑ(M) ≅ 𝔎(M ˜∗)^{⊥}  and  𝔎(M) ≅ ℑ(M ˜∗)^{⊥}.

Proof. If ϕ: M → M is any k-linear map, we denote by ϕ^{∗} the dual linear map Hom_{k}(M, k) → Hom_{k}(M, k) given by ϕ^{∗}(f) = f ∘ ϕ. In other words, ϕ^{∗} is the image of ϕ under the transpose operation. We have

Im(ϕ)^{⊥} = {f: M → k | f ∘ ϕ = 0} = Ker(ϕ^{∗}).

In the case where ϕ is given by the action of X_{α} on M, we have X_{α}^{∗} = −σ̃(X_{α}), where σ̃ is the Lie theoretic antipode of kE. It follows that

Im(X_{α}, M)^{⊥} = Ker(X_{α}, M ˜∗).

A standard result in linear algebra states that if {V_{i}} is any family of subspaces of a finite dimensional vector space V, then

( ⋂_{i} V_{i})^{⊥} = ∑_{i} V_{i}^{⊥}

as subspaces of V^{∗}. For any cofinite S ⊆ ℙ^{1}(k), we thus have

( ⋂_{ᾱ∈S} Im(X_{α}, M))^{⊥} = ∑_{ᾱ∈S} Ker(X_{α}, M ˜∗).

Choosing S such that 𝔎(M) = _{S}M and ℑ(M) = M_{S}, we then have ℑ(M)^{⊥} ≅ 𝔎(M ˜∗), establishing the first isomorphism. The second follows by replacing M with M ˜∗. □

Remark 5.13. In [17], the authors state that Proposition 5.12 applies to the group theoretic dual M^{∗}, whereas their proof (given above) only applies to the Lie theoretic dual M ˜∗. (See Section 2.2.) Through a personal conversation, Jon Carlson provided the following argument, which shows that Proposition 5.12 is actually independent of the choice of Hopf algebra structure on kE.

Let S be any subset of J \ J^{2} and define \overline{_{S}M} to be the submodule ∑_{x∈S} Ker(x, M) of M. We then define 𝔎̄(M) to be the intersection of all submodules \overline{_{S}M} for which the image of S + J^{2} in ℙ(J/J^{2}) ≅ ℙ^{1}(k) is cofinite. It follows directly from the definitions that 𝔎(M) ⊆ 𝔎̄(M). Moreover, an argument similar to that given in the proof of Theorem 5.2 shows that 𝔎̄(M) has the constant image property. By Theorem 5.2 we then have 𝔎̄(M) ⊆ 𝔎(M), hence 𝔎̄(M) = 𝔎(M).

Proposition 5.14 (Carlson). Let τ, θ: kE → kE be antipodes for any two Hopf algebra structures on kE. If M is a finite dimensional kE-module and M_{τ}^{∗} and M_{θ}^{∗} denote the kE-modules structures on Hom_{k}(M, k) induced by τ and θ, respectively, then 𝔎(M_{τ}^{∗}) = 𝔎(M_{θ}^{∗}).

Proof. Let x ∈ J and let α be the point in 𝔸^{2}(k) such that x ≡ X_{α} (mod J^{2}). By Lemma 5.4, we have

τ(x) ≡ θ(x) ≡ −X_{α} (mod J^{2}).  (5.2)

We may therefore write θ(x) = τ(x) + y(x) with y(x) ∈ J^{2}.

Now choose S ⊆ J \ J^{2} such that the image of S + J^{2} in ℙ(J/J^{2}) is cofinite, and such that 𝔎(M_{θ}^{∗}) = \overline{_{S}M_{θ}^{∗}}. It follows from the definitions of M_{τ}^{∗} and M_{θ}^{∗} that

Ker(x, M_{θ}^{∗}) = Ker(x + τ(y(x)), M_{τ}^{∗}).  (5.3)

Letting S′ = {x+τ(y(x)) | x ∈ S}, note by (5.2) that the image of S′ +J^{2} in ℙ(J/J^{2}) is equal to that of S + J^{2}, hence is cofinite. It now follows from (5.3) that

𝔎(M_{θ}^{∗}) = \overline{_{S}M_{θ}^{∗}} = \overline{_{S′}M_{τ}^{∗}} ⊇ 𝔎(M_{τ}^{∗}).

A symmetric argument shows that we also have 𝔎(M_{θ}^{∗}) ⊆ 𝔎(M_{τ}^{∗}). □

Corollary 5.15. If τ is the antipode for any Hopf algebra structure on kE and M is a finite dimensional kE-module, then

ℑ(M) ≅ 𝔎(M_{τ}^{∗})^{⊥}  and  𝔎(M) ≅ ℑ(M_{τ}^{∗})^{⊥}.

Proof. This follows from Propositions 5.12 and 5.14 by setting θ = σ̃. □

5.5. The constant kernel property

The following definition can be thought of as being dual to that of the constant image property.

Definition 5.16 (Carlson, Friedlander and Suslin [17]). Given any rank r, a finite dimensional kE-module M has the constant kernel property if Ker(X_{α}, M) is independent of the choice of non-zero α ∈ 𝔸^{r}(k).

Proposition 5.17. A kE-module M has the constant kernel property if and only if M ˜∗ has the constant image property.

Proof. As in the proof of Proposition 5.12, we have Ker(X_{α}, M) = Im(X_{α}, M ˜∗)^{⊥} for any non-zero α ∈ 𝔸^{r}(k). Taking orthogonal complements shows that Ker(X_{α}, M) is independent of α if and only if Im(X_{α}, M ˜∗) is. □

Corollary 5.18. Let τ be the antipode for any Hopf algebra structure on kE. If M is a finite dimensional kE-module and M_{τ}^{∗} denotes the dual kE-module structure induced from τ, then M has the constant kernel property if and only if M_{τ}^{∗} has the constant image property.

Proof. By Propositions 5.17 and 4.2, we know that M has the constant kernel property if and only if Im(x, M ˜∗) is independent of x ∈ J \ J^{2}. Choosing a non-zero α ∈ 𝔸^{r}(k) and applying Lemma 5.4 to σ̃, we have

τ(X_{α}) ≡ σ̃(X_{α}) = −X_{α} (mod J^{2}).

Writing τ(X_{α}) = −X_{α} + y with y ∈ J^{2}, we then have

Im(X_{α}, M_{τ}^{∗}) = Im(X_{α} + σ̃(y), M ˜∗).

Because X_{α} + σ̃(y) ∈ J \ J^{2}, this shows that M has the constant kernel property if and only if Im(X_{α}, M_{τ}^{∗}) is independent of α, i.e., if and only if M_{τ}^{∗} has the constant image property. □

The following two results appear as Lemma 8.5 and Theorem 8.6 of [17], respectively. Given our previous remarks, one readily verifies that their proofs have no dependence on the choice of Hopf algebra structure on kE. We therefore use the group theoretic antipode σ without any loss of generality.

Lemma 5.19. If M is a kE-module of constant rank, then

Ker(X_{α}, M/J𝔎(M)) = 𝔎(M)/J𝔎(M).

In particular, M/J𝔎(M) has the constant kernel property, that is, (𝔎(M)/J𝔎(M))^{∗} has the constant image property.

Proof. If m ∈ 𝔎(M), then we clearly have X_{α}m ∈ J𝔎(M), hence

𝔎(M)/J𝔎(M) ⊆ Ker(X_{α}, M/J𝔎(M)).

Conversely, if m ∈ M and X_{α}m ∈ J𝔎(M), then it follows by Theorem 5.2 and the constant image property that there exists m′ ∈ 𝔎(M) such that X_{α}m′ = X_{α}m. We then have m − m′ ∈ Ker(X_{α}, M) ⊆ 𝔎(M) by Theorem 5.7, forcing m ∈ 𝔎(M). This shows that Ker(X_{α}, M/J𝔎(M)) ⊆ 𝔎(M)/J𝔎(M), proving the first statement. The second follows from Corollary 5.18. □

Theorem 5.20. If M is a finite dimensional kE-module of constant rank, then ℑ(M) = J𝔎(M).

Proof. Since 𝔎(M) has the constant image property, Im(X_{α}, 𝔎(M)) = J𝔎(M) for all non-zero α ∈ 𝔸^{2}(k). Letting S ⊆ ℙ^{1}(k) be cofinite such that M_{S} = ℑ(M), we thus have ℑ(M) = ∑_{ᾱ∈S} Im(X_{α}, M) ⊇ ∑_{ᾱ∈S} Im(X_{α}, 𝔎(M)) = J𝔎(M). To prove the reverse containment, note that (M/J𝔎(M))^{∗} ⊆ 𝔎(M^{∗}) by Lemma 5.19 and Theorem 5.2. Applying Corollary 5.15 now yields

ℑ(M) = (𝔎(M^{∗}))^{⊥} ⊆ ((M/J𝔎(M))^{∗})^{⊥} = J𝔎(M).  □

5.6. Duality for subquotients in the generic kernel filtration

In this section we introduce a duality formula that will be used to study the generic kernel filtration (5.1) and Question 5.10 in terms of the generic kernels of both M and M^{∗}. As before, our approach will illustrate that the following results are independent of the choice of Hopf algebra structure on kE.

Lemma 5.21. Let τ be the antipode for any Hopf algebra structure on kE. If M is a kE-module of constant rank and M_{τ}^{∗} denotes the dual kE-module structure induced by τ, then 𝔎(M_{τ}^{∗}) ≅ (M/J𝔎(M))_{τ}^{∗}.

Proof. By Corollary 5.15 and Theorem 5.20 we have

𝔎(M_{τ}^{∗}) = (ℑ(M))^{⊥} = (J𝔎(M))^{⊥} ≅ (M/J𝔎(M))_{τ}^{∗}.  □

Lemma 5.22. Let τ be the antipode for any Hopf algebra structure on kE. If M is a kE-module in any rank r and N is a submodule of M, then

(i) J^{−1}(N^{⊥}) = (JN)^{⊥} and
(ii) J(N^{⊥}) = (J^{−1}N)^{⊥},

where the action of kE on Hom_{k}(M, k) is that induced by τ.

Proof. To prove (i), suppose f ∈ J^{−1}(N^{⊥}). For any n ∈ JN, we may write n = X_{1}n_{1} + ⋯ + X_{r}n_{r} with n_{i} ∈ N. We then have

f(n) = f(X_{1}n_{1}) + ⋯ + f(X_{r}n_{r}) = (τ(X_{1}).f)(n_{1}) + ⋯ + (τ(X_{r}).f)(n_{r}) = 0

since each τ(X_{i}) lies in J. This shows that f ∈ (JN)^{⊥}. Conversely, if f ∈ (JN)^{⊥} and n ∈ N, then

(X_{i}.f)(n) = f(τ(X_{i})n) = 0 for all 1 ≤ i ≤ r

since τ(X_{i})n ∈ JN. It follows that X_{i}.f ∈ N^{⊥} for all i, that is, J.f ⊆ N^{⊥}.

To prove (ii), note that

J^{−1}(N) = J^{−1}((N^{⊥})^{⊥}) = (J(N^{⊥}))^{⊥}

by (i) so that (J^{−1}N)^{⊥} = J(N^{⊥}). □

We are now in a position to prove the main theorem of this section.

Theorem 5.23. Let τ be the antipode for any Hopf algebra structure on kE. If M is a kE-module of constant rank, then for all a, b ∈ ℤ with a ≤ b we have

J^{a}𝔎(M_{τ}^{∗})/J^{b}𝔎(M_{τ}^{∗}) ≅ (J^{−b+1}𝔎(M)/J^{−a+1}𝔎(M))_{τ}^{∗}.

Proof. We first need to prove that J^{−a}J𝔎(M) = J^{−a+1}𝔎(M) for all a > 0. For the leftwards containment, note that if m ∈ M and J^{a−1}m ∈ 𝔎(M), then we clearly have J^{a}m ⊆ J𝔎(M). Conversely, suppose that m ∈ M satisfies J^{a}m ⊆ J𝔎(M). For any y ∈ J^{a−1}, we then have X_{1}ym ∈ J𝔎(M). Because 𝔎(M) has the constant image property, there exists m′ ∈ 𝔎(M) such that X_{1}m′ = X_{1}ym. It follows by Theorem 5.7 that m′ − ym ∈ 𝔎(M), hence ym ∈ 𝔎(M). This shows that J^{a−1}m ⊆ 𝔎(M).

Having established the above equality, it follows by Lemmas 5.21 and 5.22 that

\frac{J^{a}𝔎(M_{τ}^{∗})}{J^{b}𝔎(M_{τ}^{∗})} ≅ \frac{J^{a}(J𝔎(M))^{⊥}}{J^{b}(J𝔎(M))^{⊥}} = \frac{(J^{−a+1}𝔎(M))^{⊥}}{(J^{−b+1}𝔎(M))^{⊥}} ≅ (\frac{J^{−b+1}𝔎(M)}{J^{−a+1}𝔎(M)})_{τ}^{∗} .  □

5.7. Blocks of small length in certain subquotients

When proving that Question 5.10 has an affirmative answer in the case p = 3, it will be important to know that if M is a kE-module of constant Jordan type, then the number of blocks of length one in the action of X_{α} on J^{−1}𝔎(M) is the same for all non-zero α ∈ 𝔸^{2}(k). In this section we prove a more general statement about blocks having small length in the action of X_{α} on the subquotients J^{−i}𝔎(M)/J^{j}𝔎(M) for i ≥ 1, j ≥ 2. We begin with a key lemma.

Lemma 5.24. If M is a kE-module of constant rank, then for all non-zero α ∈ 𝔸^{2}(k) and all 0 ≤ n ≤ i we have

Ker(X_{α}, M) ∩ Im(X_{α}^{n}, M) = Ker(X_{α}, J^{−i}𝔎(M)) ∩ Im(X_{α}^{n}, J^{−i}𝔎(M)).

Proof. The leftwards containment is clear, so let m ∈ Ker(X_{α}, M) ∩ Im(X_{α}^{n}, M). Note that Ker(X_{α}, M) ⊆ 𝔎(M) by Theorem 5.7. It follows that

Ker(X_{α}, J^{−i}𝔎(M)) ⊆ Ker(X_{α}, M) ⊆ Ker(X_{α}, 𝔎(M)) ⊆ Ker(X_{α}, J^{−i}𝔎(M)),

whence equality holds throughout. In particular, we have m ∈ Ker(X_{α}, J^{−i}𝔎(M)). Also, there exists m′ ∈ M such that X_{α}^{n}m′ = m. Since m ∈ 𝔎(M), Proposition 5.8 implies m′ ∈ X_{α}^{−n}𝔎(M) = J^{−n}𝔎(M) ⊆ J^{−i}𝔎(M), hence m ∈ Im(X_{α}^{n}, J^{−i}𝔎(M)). □

Theorem 5.25. Let M be a kE-module of constant rank and let i ≥ 1, j ≥ 2. If n ≤ min{i, j −1}, then the number of Jordan blocks of length n in the action of X_{α} on J^{−i}𝔎(M)/J^{j}𝔎(M) is equal to the number of Jordan blocks of length n in the action of X_{α} on M.

Proof. It suffices to show that

\frac{Ker(X_{α}, M) ∩ X_{α}^{n−1}M}{Ker(X_{α}, M) ∩ X_{α}^{n}M} ≅ \frac{Ker (X_{α}, \frac{J^{−i}𝔎(M)}{J^{j}𝔎(M)}) ∩ X_{α}^{n−1} \frac{J^{−i}𝔎(M)}{J^{j}𝔎(M)}}{Ker (X_{α}, \frac{J^{−i}𝔎(M)}{J^{j}𝔎(M)}) ∩ X_{α}^{n} \frac{J^{−i}𝔎(M)}{J^{j}𝔎(M)}}  (5.4)

since the dimension of the left hand side is equal to the number of Jordan blocks of length n in the action of X_{α} on M, and the dimension of the right hand side is equal to the number of Jordan blocks of length n in the action of X_{α} on J^{−i}𝔎(M)/J^{j}𝔎(M).

Note by Lemma 5.24 that we have

\frac{Ker(X_{α}, M) ∩ X_{α}^{n−1}M}{Ker(X_{α}, M) ∩ X_{α}^{n}M} = \frac{Ker(X_{α}, J^{−i}𝔎(M)) ∩ X_{α}^{n−1}J^{−i}𝔎(M)}{Ker(X_{α}, J^{−i}𝔎(M)) ∩ X_{α}^{n}J^{−i}𝔎(M)}.

By the second isomorphism theorem we also have

\frac{Ker(X_{α}, J^{−i}𝔎(M)) ∩ X_{α}^{n−1}J^{−i}𝔎(M)}{Ker(X_{α}, J^{−i}𝔎(M)) ∩ X_{α}^{n}J^{−i}𝔎(M)} ≅

\frac{(Ker(X_{α}, J^{−i}𝔎(M)) ∩ X_{α}^{n−1}J^{−i}𝔎(M)) + X_{α}^{n}J^{−i}𝔎(M)}{X_{α}^{n}J^{−i}𝔎(M)}.

Consider the natural injective k-linear map

\begin{matrix}\frac{(Ker(X_{α}, J^{−i}𝔎(M)) ∩ X_{α}^{n−1}J^{−i}𝔎(M)) + X_{α}^{n}J^{−i}𝔎(M)}{X_{α}^{n}J^{−i}𝔎(M)} & ⟶ \\ \frac{(Ker (X_{α}, \frac{J^{−i}𝔎(M)}{J^{j}𝔎(M)}) ∩ X_{α}^{n−1} \frac{J^{−i}𝔎(M)}{J^{j}𝔎(M)}) + X_{α}^{n} \frac{J^{−i}𝔎(M)}{J^{j}𝔎(M)}}{X_{α}^{n} \frac{J^{−i}𝔎(M)}{J^{j}𝔎(M)}} , \end{matrix}

where the latter term is isomorphic to the right hand side of (5.4). In order to prove the theorem, we need to show that this map is also surjective.

Let m ∈ J^{−i}𝔎(M) such that

m + J^{j}𝔎(M) ∈ Ker (X_{α}, \frac{J^{−i}𝔎(M)}{J^{j}𝔎(M)}) ∩ X_{α}^{n−1} \frac{J^{−i}𝔎(M)}{J^{j}𝔎(M)}.

Because X_{α}m ∈ J^{j}𝔎(M), it follows from Theorem 5.2 and Proposition 4.4 that there exists m′ ∈ 𝔎(M) such that X_{α}m = X_{α}^{j}m′. We then have X_{α}(m − X_{α}^{j−1}m′) = 0. Note by our choice of m that there also exists m′′ ∈ J^{−i}𝔎(M) such that

m − X_{α}^{n−1}m′′ ∈ J^{j}𝔎(M) = X_{α}^{j}𝔎(M),

hence m ∈ X_{α}^{n−1}J^{−i}𝔎(M). Finally, we have X_{α}^{j−1}m′ ∈ X_{α}^{n}𝔎(M) ⊆ X_{α}^{n}J^{−i}𝔎(M) since n ≤ j − 1. Combining the above information now yields

\begin{matrix}m = (m − X_{α}^{j−1}m′) + X_{α}^{j−1}m′ & ∈ \\ (Ker(X_{α}, J^{−i}𝔎(M)) ∩ X_{α}^{n−1}J^{−i}𝔎(M)) + X_{α}^{n}J^{−i}𝔎(M) \end{matrix}

as required. □

We now show that Theorem 5.25 allows us to generalise Lemma 3.3 of Benson [6]. The main distinction here is that the latter result only applies to kE-modules of constant Jordan type containing no blocks of length one.

Theorem 5.26. If M is any kE-module of constant Jordan type, then the subquotient J^{−1}𝔎(M)/J^{2}𝔎(M) also has constant Jordan type.

Proof. Fix a non-zero α ∈ 𝔸^{2}(k) and let [3]^{a}[2]^{b}[1]^{c} be the Jordan type of X_{α} on J^{−1}𝔎(M)/J^{2}𝔎(M). Note that the total number of Jordan blocks in the action of X_{α} on M is independent of α and equal to dim_{k} Ker(X_{α}, M), which in turn is equal to dim_{k} Ker(X_{α}, J^{−1}𝔎(M)) by Theorem 5.7. We also have

dim_{k} Ker(X_{α}, J^{−1}𝔎(M)) = dim_{k} J^{−1}𝔎(M)/X_{α}J^{−1}𝔎(M)

= dim_{k} \frac{J^{−1}𝔎(M)/J^{2}𝔎(M)}{X_{α}J^{−1}𝔎(M)/J^{2}𝔎(M)}.

The latter is the total number of Jordan blocks of X_{α} acting on J^{−1}𝔎(M)/J^{2}𝔎(M). This shows that a + b + c is independent of α. We also have

3a + 2b + c = dim_{k} J^{−1}𝔎(M)/J^{2}𝔎(M),

which is clearly independent of α. Finally, Theorem 5.25 implies that c is equal to the number of Jordan blocks of length one in the action of X_{α} on M, which is also independent of α. The fact that J^{−1}𝔎(M)/J^{2}𝔎(M) has constant Jordan type now follows from the non-singularity of the matrix

(\begin{matrix}1 & 1 & 1 \\ 3 & 2 & 1 \\ 0 & 0 & 1 \end{matrix}) ∈ GL(3, ℤ).  □

5.8. Question 5.10 in the cases p = 3 and J^{2}𝔎(M) = 0

In this section we identify two special cases in which Question 5.10 has an affirmative answer.

Theorem 5.27. If p = 3 and M is a kE-module of constant Jordan type, then J^{−i}𝔎(M) has constant Jordan type for all i ≥ 0.

Proof. If M is a k(ℤ/3)^{2}-module of constant Jordan type, then M = J^{−2}𝔎(M) by Corollary 5.9. Hence we need only check that J^{−1}𝔎(M) has constant Jordan type. This involves essentially the same argument as given in the proof of Theorem 5.26. We again fix a non-zero α ∈ 𝔸^{2}(k) and let [3]^{a}[2]^{b}[1]^{c} be the Jordan type of X_{α} on J^{−1}𝔎(M). Theorem 5.7 implies that the total number of Jordan blocks of X_{α} acting on J^{−1}𝔎(M) is equal to that of X_{α} on M, hence a + b + c is independent of α. We have 3a + 2b + c = dim_{k} J^{−1}𝔎(M) independent of α, and Theorem 5.25 tells us that c is equal to the number of Jordan blocks of length one in the action of X_{α} on M, hence is independent of α. The fact that J^{−1}𝔎(M) has constant Jordan type again follows from the non-singularity of the matrix

(\begin{matrix}1 & 1 & 1 \\ 3 & 2 & 1 \\ 0 & 0 & 1 \end{matrix}) ∈ GL(3, ℤ).  □

In order to address the case in which J^{2}𝔎(M) = 0, we make the following observation. Let M be any kE-module and let n ≥ 0. For each non-zero α ∈ 𝔸^{2}(k) there is a natural short exact sequence

0 ⟶ L_{n,α} ⟶ Im(X_{α}^{n}, M/J^{j+1}𝔎(M)) ⟶ Im(X_{α}^{n}, M/J^{j}𝔎(M)) ⟶ 0  (5.5)

where

L_{n,α} = Im(X_{α}^{n}, M/J^{j+1}𝔎(M)) ∩ J^{j}𝔎(M)/J^{j+1}𝔎(M).

If M/J^{j+1}𝔎(M) has constant Jordan type, then the dimension of the middle term is independent of α. We now introduce a criterion under which the dimensions of the outer two terms would also be independent of α.

Lemma 5.28. Suppose M/J^{j+1}𝔎(M) has constant Jordan type and fix n ≥ 0. If there exists a submodule N of M/J^{j+1}𝔎(M) such that L_{n,α} = Im(X_{α}^{n}, N) for all non-zero α ∈ 𝔸^{2}(k), then rank(X_{α}^{n}, M/J^{j}𝔎(M)) is independent of α.

Proof. Since M/J^{j+1}𝔎(M) has constant Jordan type, we see from the sequence (5.5) that

rank(X_{α}^{n}, N) + rank(X_{α}^{n}, M/J^{j}𝔎(M))

is constant as a function of α. Following the proof of Proposition 2.2, there exists a dense open subset U ⊆ 𝔸^{2}(k) such that α ∈ U if and only if rank(X_{α}^{n}, M/J^{j}𝔎(M)) is maximal. Similarly, there exists a dense open subset V ⊆ 𝔸^{2}(k) such that α ∈ V if and only if rank(X_{α}^{n}, N) is maximal. Since 𝔸^{2}(k) is irreducible, U ∩ V is non-empty. Choosing α ∈ U ∩ V, we simultaneously have rank(X_{α}^{n}, M/J^{j}𝔎(M)) maximal and rank(X_{α}^{n}, N) maximal. This forces both summands to be constant in α. □

We now use the above theory to show that Question 5.10 is true in the special case where J^{2}𝔎(M) = 0. We make use of the group theoretic dual M^{∗} without any loss of generality, because the proof is independent of the choice of Hopf algebra structure on kE.

Theorem 5.29. If M is a kE-module of constant Jordan type such that J^{2}𝔎(M) = 0, then J^{−i}𝔎(M) has constant Jordan type for all i ≥ 0.

Proof. Since J^{2}𝔎(M) = 0, we have M^{∗} = J^{−1}𝔎(M^{∗}) by Theorem 5.23. We first show that M^{∗}/J^{j}𝔎(M^{∗}) has constant Jordan type for all j ≥ 0 using downwards induction on j, beginning at M^{∗} = M^{∗}/J^{p}𝔎(M^{∗}). So assume that M^{∗}/J^{j+1}𝔎(M^{∗}) has constant Jordan type and note for a non-zero α ∈ 𝔸^{2}(k) that any Jordan block in the action of X_{α} on M^{∗}/J^{j}𝔎(M^{∗}) has length at most j + 1. If n ≤ j, let m ∈ M^{∗}/J^{j+1}𝔎(M^{∗}) such that X_{α}^{n}m ∈ J^{j}𝔎(M^{∗})/J^{j+1}𝔎(M^{∗}). Since 𝔎(M^{∗})/J^{j+1}𝔎(M^{∗}) has the constant image property, there exists m′ ∈ 𝔎(M^{∗})/J^{j+1}𝔎(M^{∗}) such that X_{α}^{n}m′ = X_{α}^{n}m. Applying the sequence (5.5) to M^{∗}, this yields

L_{n,α} = Im(X_{α}^{n}, 𝔎(M^{∗})/J^{j+1}𝔎(M^{∗})) for all non-zero α ∈ 𝔸^{2}(k).

It follows from Lemma 5.28 that rank(X_{α}^{n}, M^{∗}/J^{j}𝔎(M^{∗})) is independent of α for all n ≤ j. If [1]^{a_{1}} … [j + 1]^{a_{j+1}} is the Jordan type of X_{α} on 𝔎(M^{∗})/J^{j+1}𝔎(M^{∗}), then we have

a_{1} + 2a_{2} + 3a_{3} + ⋯ + ja_{j} + (j + 1)a_{j+1} = dim_{k} M^{∗}/J^{j}𝔎(M^{∗})
a_{2} + 2a_{3} + ⋯ + (j − 1)a_{j} + ja_{j+1} = rank(X_{α}, M^{∗}/J^{j}𝔎(M^{∗}))
⋮
a_{j} + 2a_{j+1} = rank(X_{α}^{j−1}, M^{∗}/J^{j}𝔎(M^{∗}))
a_{j+1} = rank(X_{α}^{j}, M^{∗}/J^{j}𝔎(M^{∗})).

Because the terms on the right are independent of α, these equations have a solution for (a_{1}, …, a_{j+1}) that is also independent of α. This shows that M^{∗}/J^{j}𝔎(M^{∗}) has constant Jordan type.

Having proved the claim, setting j = −i + 1, it now follows by another application of Theorem 5.23 that J^{−i}𝔎(M) has constant Jordan type for all i ≥ 0 as required. □

5.9. Some counterexamples to Question 5.10

We now present an example of a kE-module M of constant Jordan type such that J^{−1}𝔎(M) does not have constant Jordan type.

Example 5.30. Let M be the 26 dimensional kE-module having basis

a, b, c, …, x, y, z

Chapter 5.9 Page 69

on which g_{1} acts via the matrix

( \begin{matrix}a & 1 \\ b & & 1 \\ c & 1 & & 1 \\ d & & & & 1 \\ e & & & & & 1 \\ f & & & & & & 1 \\ g & & 1 & & & & & 1 \\ h & & & & & & & & 1 \\ i & & & & & & & & & 1 \\ j & 1 & & & & & & & & & 1 \\ k & & & 1 & & & & & & & & 1 \\ l & & & & 1 & & & & & & & & 1 \\ m & & & & & & & & & & & & & 1 \\ n & & & & & & & & & & & & & & 1 \\ o & & & & & & & & & & & & & & & 1 \\ p & & & & & 1 & & & & & & & & & & & 1 \\ q & & & & & & & & & & & & & & & & & 1 \\ r & & & & & & & & & & & & & & & & & & 1 \\ s & & & & & & & 1 & & & & & & & & & & & & 1 \\ t & & & & & & & & & 1 & & & & & & & & & & & 1 \\ u & & & & & & & & & & 1 & & & & & & & & & & & 1 \\ v & & & & & & & & & & & & & & & 1 & & & & & & & 1 \\ w & & & & & & & & & & & & & & & & 1 & & & & & & & 1 \\ x & & & & & & & & & & & & & & & & & & 1 & & & & & & 1 \\ y & & & & & & & & & & & & & & & & & & & & & 1 & & & & 1 \\ z & & & & & & & & & & & & & & & & & & & & & & & 1 & & & 1 \\ & a & b & c & d & e & f & g & h & i & j & k & l & m & n & o & p & q & r & s & t & u & v & w & x & y & z \end{matrix} )

and g_{2} acts via the matrix

( \begin{matrix}a & 1 \\ b & & 1 \\ c & & & 1 \\ d & 1 & & & 1 \\ e & & & & & 1 \\ f & & & & & & 1 \\ g & & & & & & & 1 \\ h & & 1 & & & & & & 1 \\ i & & & & & & & & & 1 \\ j & & & & & & & & & & 1 \\ k & & & & & & & & & & & 1 \\ l & & & 1 & & & & & & & & & 1 \\ m & & & & 1 & & & & & & & & & 1 \\ n & 1 & & & & & & & & & & & & & 1 \\ o & & & & & & & & & & & & & & & 1 \\ p & & & & & & & & & & & & & & & & 1 \\ q & & & & & 1 & & & & & & & & & & & & 1 \\ r & & & & & & & & & & & & & & & & & & 1 \\ s & & & & & & 1 & & & & & & & & & & & & & 1 \\ t & & & & & & & & 1 & & & & & & & & & & & & 1 \\ u & & & & & & & & & 1 & & & & & & & & & & & & 1 \\ v & & & & & & & & & & & & & & 1 & & & & & & & & 1 \\ w & & & & & & & & & & & & & & & 1 & & & & & & & & 1 \\ x & & & & & & & & & & & & & & & & & 1 & & & & & & & 1 \\ y & & & & & & & & & & & & & & & & & & & & 1 & & & & & 1 \\ z & & & & & & & & & & & & & & & & & & & & & & 1 & & & & 1 \\ & a & b & c & d & e & f & g & h & i & j & k & l & m & n & o & p & q & r & s & t & u & v & w & x & y & z \end{matrix} ) .

Here we have omitted entries equal to zero. This module structure is illustrated in the following diagram, where X_{1} = g_{1} − 1 acts via single edges downwards to the left, X_{2} = g_{2} − 1 acts via double edges downwards to the right, and the dotted edges indicate that X_{1}.a = c + j and X_{2}.a = d + n.

[figure: a; b; c; d; e; f; g; h; i; j; k; l; m; n; o; p; q; r; s; t; u; v; w; x; y; z]

We first prove that M has constant Jordan type by verifying that rank(X_{α}^{n}, M) is independent of the choice of non-zero α ∈ 𝔸^{2}(k) for n = 1, 2, 3. By inspection of the diagram we see that rank(X_{2}, M) = 13. For any λ ∈ k, the image of X_{1} + λX_{2} on M is spanned by the elements

c + j + λ(d + n), g + λh, k + λl, l + λm, p + λq, s, t, u, v, w, x, y, z,

which are readily confirmed to be linearly independent. Hence rank(X_{1} + λX_{2}, M) = 13 as well. Next observe that rank(X_{2}^{2}, M) = 5, and for any λ ∈ k, the image of (X_{1} + λX_{2})^{2} on M is spanned by the linearly independent elements

k + u + 2λl + λ^{2}(m + v), s + λ^{2}t, w + λ^{2}x, y, z

so that rank((X_{1} + λX_{2})^{2}, M) = 5. Finally, we have rank(X_{2}^{3}, M) = 2, and for any λ ∈ k, the image of (X_{1} + λX_{2})^{3} on M is the submodule spanned by

λ^{3}y, y + λ^{3}z, z,

which in any case has dimension two. Thus rank((X_{1} + λX_{2})^{3}, M) = 2 as required. This shows that M has constant Jordan type [4]^{2}[3][2]^{5}[1]^{5}.

To see that J^{−1}𝔎(M) does not have constant Jordan type, note that 𝔎(M) is the submodule of M generated by

f, g, h, i, j, k, l, m, n, o, p, q, r

since this is the largest submodule of M having the constant image property. It follows that J^{−1}𝔎(M) is the submodule of M generated by

f, b, i, j, c, d, n, o, e, r.

Observe that J^{−1}𝔎(M) has a direct summand N generated by the elements f, b, i, j. Clearly N cannot have constant Jordan type because, for example, X_{2}^{3}N = span_{k}(y), whereas X_{1}^{3}N = 0. By Theorem 2.18 (ii), this shows that J^{−1}𝔎(M) cannot have constant Jordan type.

While Example 5.30 does answer Question 5.10 in the negative, one might be tempted to ask whether or not there exists a ‘magic number’ i > 1 for which J^{−i}𝔎(M) has constant Jordan type whenever M does. We now present a family of counterexamples to this question. The first diagram represents a kE-module of constant Jordan type satisfying J^{3}𝔎(M) = 0 for which J^{−2}𝔎(M) does not have constant Jordan type. The second represents such a module for which J^{−3}𝔎(M) does not have constant Jordan type. Higher counterexamples may be constructed similarly by following the inherent pattern.

[figure: •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •]

[figure: •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •]

CHAPTER 6 On the structure of the subquotient J^{−1}𝔎(M)/J^{2}𝔎(M)

In this chapter we examine a theorem of Benson related to Rickard’s Conjecture 2.25. The proof of that theorem involves studying the subquotient J^{−1}𝔎(M)/J^{2}𝔎(M) of a kE-module M of constant Jordan type containing no blocks of length one. Although we come nowhere near obtaining a full structure theorem for these subquotients, we do prove that they contain certain ‘diamond shaped’ submodules isomorphic to those of type D_{p}V D_{p}V … V D_{p} described in Section 2.7.

6.1. Benson’s theorem on Rickard’s conjecture

The following theorem is the main result of Benson [6]. It proves the special case of Rickard’s Conjecture 2.25 in which a_{1} = 0.

Theorem 6.1. If r ≥ 2 and M is a kE-module of constant Jordan type that contains no blocks of length one, then the total number of Jordan blocks of M is divisible by p.

Proof. It suffices to restrict to the rank two case since the restriction of M to any rank two shifted subgroup of kE^{×} will be a module for that subgroup having the same constant Jordan type.

Now consider the subquotient J^{−1}𝔎(M)/J^{2}𝔎(M). It follows from Theorem 5.26 and its proof that this is also a module of constant Jordan type containing no blocks of length one. Furthermore, the same proof shows that the number of Jordan blocks of J^{−1}𝔎(M)/J^{2}𝔎(M) is equal to the number of Jordan blocks of M, which in turn is equal to the number of Jordan blocks of 𝔎(M)/J^{2}𝔎(M). The latter quantity is equal to dim_{k} 𝔎(M)/J𝔎(M). It therefore suffices to assume that M = J^{−1}𝔎(M)/J^{2}𝔎(M) and to prove for every direct summand N of 𝔎(M) that the dimension of N/JN is divisible by p.

So let N be an indecomposable direct summand of 𝔎(M). Note that N is also an indecomposable module for kE/J^{2}, and all such modules correspond to representations of the Kronecker quiver. Since 𝔎(M) has the constant image property, the map

X_{α}: 𝔎(M)/J𝔎(M) ⟶ J𝔎(M)

is surjective for any non-zero α ∈ 𝔸^{2}(k). By the Kronecker classification, this means that N is isomorphic to a W -module of radical length two, hence the structure of N is given by the module diagram

[figure: v_{0}; v_{1}; v_{3}; ⋯; v_{n−2}; v_{n−1}; w_{1}; w_{2}; w_{n−1}]

as described in Section 4.3. Note that we have altered the labeling of the top vertices for the sake of this proof.

Since M contains no Jordan blocks of length one, we have Ker(X_{α}, M) ⊆ Im(X_{α}, M) for all non-zero α ∈ 𝔸^{2}(k). It follows from Proposition 2.10 that this remains true after extending k to the field K = k[[t]][t^{−1}] of Laurent power series. The kernel of the element X_{1} + tX_{2} ∈ kE ⊗_{k} K on N ⊗_{k} K is one dimensional, spanned by

v_{0} − tv_{1} + t^{2}v^{2} − ⋯ ± t^{n−1}v_{n−1}.

Hence there exists an element ∑_{i}(−1)^{i}t^{i}u_{i} ∈ M ⊗_{k} K such that

(X_{1} + tX_{2}) ∑_{i}(−1)^{i}t^{i}u_{i} = ∑_{j=0}^{n−1}(−1)^{j}t^{j}v_{j}.  (6.1)

Observe for any non-zero α ∈ 𝔸^{2}(k) that the linear map

X_{α}: M/𝔎(M) ⟶ 𝔎(M)/J𝔎(M)

is injective by Theorem 5.7. In particular, because X_{1} and X_{2} are injective, we have u_{i} = 0 for all i < 0.

We also claim that u_{i} = 0 for all i ≥ n−1. Suppose first that u_{n−1} ≠ 0. Let m > n−1 be the smallest integer such that there exists a linear relation ∑_{i=n−1}^{m} λ_{i}u_{i} = 0. Such an integer exists because M is finite dimensional. Then u_{n−1}, …, u_{m−1} are linearly independent and u_{m} is a linear combination of them. It follows from (6.1) that

X_{2}u_{m} ∈ span_{k}(X_{2}u_{n−1}, …, X_{2}u_{m−1}) = span_{k}(X_{1}u_{n}, …, X_{1}u_{m})
⊆ span_{k}(X_{1}u_{n−1}, …, X_{1}u_{m−1}).

This shows for any non-zero α ∈ 𝔸^{2}(k) that we have an injective map

X_{α}: span_{k}(u_{n−1}, …, u_{m−1}) ⟶ span_{k}(X_{1}u_{n−1}, …, X_{1}u_{m−1}),

which is necessarily an isomorphism. Letting X′_{1} denote the linear map inverse to X_{1}, the composition

X′_{1} ∘ (λX_{1} − X_{2}) = λ.Id − (X′_{1} ∘ X_{2})

is also an isomorphism for all λ ∈ k. In particular, its injectivity implies that X′_{1} ∘ X_{2} has no non-zero eigenvector, contradicting the fact that k is algebraically closed. This shows that u_{n−1} = 0. The relations X_{1}u_{i+1} + X_{2}u_{i} = 0 for all i ≥ n − 1 then show that u_{i} = 0 for all i ≥ n − 1.

The above remarks applied to (6.1) now yield the equations

X_{1}u_{0} = v_{0} ; X_{1}u_{1} − X_{2}u_{0} = v_{1} ; ⋮ ; X_{1}u_{n−2} − X_{2}u_{n−3} = v_{n−2} ; −X_{2}u_{n−2} = v_{n−1}.  (6.2)

By the relations defining the W -module structure of N, it follows that

X_{1}(X_{1}u_{1} − 2v_{1}) = X_{1}(X_{2}u_{0} − v_{1}) = X_{2}v_{0} − X_{1}v_{1} = w_{1} − w_{1} = 0,

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}) 2 ≤ i ≤ n − 2,

X_{2}(X_{1}u_{n−2} − (n − 1)v_{n−2}) = −nw_{n−1}.

This shows that there exists a submodule of 𝔎(M) whose structure is indicated in the following diagram.

[figure: X_{1}u_{1} − 2v_{1}; X_{1}u_{2} − 3v_{2}; ⋯; X_{1}u_{n−2} − (n − 1)v_{n−2}; X_{1}^{2}u_{2} − 3w_{2}; X_{1}^{2}u_{3} − 4v_{3}; ⋯; −nw_{n−1}]

Note that the elements represented by vertices in the above graph are not necessarily linearly independent, or even non-zero. Nonetheless, we have

−nw_{n−1} ∈ (X_{2}X_{1}^{−1})^{n−2}{0}

where, for a subspace V of 𝔎(M), X_{1}^{−1}V denotes the set of m ∈ 𝔎(M) for which X_{1}m ∈ V. Because N is indecomposable, it follows from the W -module structure of N that w_{n−1} ∉ (X_{2}X_{1}^{−1})^{n−2}{0}, hence we must have −n = 0 in k. This gives us

dim_{k} N/JN = n ≡ 0 (mod p)

as required. □

6.2. Diamond shaped submodules of J^{−1}𝔎(M)/J^{2}𝔎(M)

Returning to the rank two case, we are now in a position to prove the main theorem of the chapter.

Theorem 6.2. Let M be a kE-module of constant Jordan type containing no blocks of length one and let N ≅ W_{n,2} be a direct summand of 𝔎(M)/J^{2}𝔎(M) with n minimal. Label the top and bottom vertices of N by v_{0}, …, v_{n−1} and w_{1}, …, w_{n−1}, respectively, as in the proof of Theorem 6.1. Then there exist elements v′_{i} ∈ 𝔎(M)/J^{2}𝔎(M) for 0 ≤ i ≤ n − 1, and u′_{i} ∈ J^{−1}𝔎(M)/J^{2}𝔎(M) for 1 ≤ i ≤ n − 1 with p ∤ i such that

(i) v′_{i} ≡ v_{i} (mod J𝔎(M)/J^{2}𝔎(M)) for all 0 ≤ i ≤ n − 1, and
(ii) X_{1}u′_{i} = v′_{i−1} and X_{2}u′_{i} = v′_{i} for all 1 ≤ i ≤ n − 1 with p ∤ i.

In other words, we obtain a submodule of J^{−1}𝔎(M)/J^{2}𝔎(M) having diagram

[figure: u′_{1}; u′_{2}; u′_{n−1}; v′_{0}; v′_{1}; v′_{2}; ⋯; v′_{n−2}; v′_{n−1}; w_{1}; w_{2}; w_{n−1}]

where u′_{i} exists if p ∤ i.

Proof. Consider the submodule L of N constructed in the proof of Theorem 6.1 whose structure is illustrated in the following diagram.

[figure: X_{1}u_{1} − 2v_{1}; X_{1}u_{2} − 3v_{2}; X_{1}u_{3} − 4v_{3}; ⋯; X_{1}^{2}u_{2} − 3w_{2}; X_{1}^{2}u_{3} − 4w_{3}; ⋯]

We claim that the top vertices of L lie in J𝔎(M)/J^{2}𝔎(M). We first write

𝔎(M)/J^{2}𝔎(M) ≅ ⨁_{j} W_{n_{j},2}

by the Kronecker classification and let ϕ_{j}: L → W_{n_{j},2} be the composition of the inclusion of L in 𝔎(M)/J^{2}𝔎(M) with the projection of 𝔎(M)/J^{2}𝔎(M) onto W_{n_{j},2}. Fixing j, we label the top and bottom vertices of W_{n_{j},2} by a_{0}, …, a_{n_{j}−1} and b_{1}, …, b_{n_{j}−1}, respectively. Because X_{1}(X_{1}u_{1} − 2v_{1}) = 0 in L, we have

ϕ_{j}(X_{1}u_{1} − 2v_{1}) ≡ λ_{0}a_{0} (mod Soc(W_{n_{j},2}))

for some λ_{0} ∈ k. We then have X_{1}ϕ_{j}(X_{1}u_{2} − 3v_{2}) = X_{2}ϕ_{j}(X_{1}u_{1} − 2v_{1}) = λ_{0}b_{1} so that ϕ_{j}(X_{1}u_{2} − 3v_{2}) ≡ λ_{1}a_{0} + λ_{0}a_{1} (mod Soc(W_{n_{j},2})) for some λ_{1} ∈ k. Continuing in this way, we have

ϕ_{j}(X_{1}u_{i} − (i + 1)v_{i}) ≡ ∑_{l=0}^{i−1} λ_{i−l−1}a_{l} (mod Soc(W_{n_{j},2})) for 1 ≤ i ≤ n − 2.

We also have X_{2}ϕ_{j}(X_{1}u_{n−2} − (n − 1)v_{n−2}) = 0, hence

0 = ∑_{l=0}^{n−3} λ_{i−l−1}X_{2}a_{l} = ∑_{l=0}^{n−3} λ_{i−l−1}b_{l+1}.

Because n was assumed to be minimal, b_{n−2} ≠ 0. It follows that each λ_{i} = 0 so that ϕ_{j}(X_{1}u_{i} − (i + 1)v_{i}) ∈ Soc(W_{n_{j},2}). But if m ∈ L, then m = ∑_{j} ϕ_{j}(m), thus

X_{1}u_{i} − (i + 1)v_{i} ∈ Soc(𝔎(M)/J^{2}𝔎(M)) = J𝔎(M)/J^{2}𝔎(M)

for all i as required.

Now since X_{1}u_{i} − (i + 1)v_{i} ∈ J𝔎(M)/J^{2}𝔎(M) for 1 ≤ i ≤ n − 2 and 𝔎(M)/J^{2}𝔎(M) has the constant image property, there exist s_{i} ∈ 𝔎(M)/J^{2}𝔎(M) such that

X_{1}u_{i} − (i + 1)v_{i} = X_{1}s_{i}.

Thus if p ∤ (i + 1), then X_{1} (\frac{1}{i+1}(u_{i} − s_{i})) = v_{i} so that

X_{1}X_{2} (\frac{1}{i+1}(u_{i} − s_{i})) = X_{2}X_{1} (\frac{1}{i+1}(u_{i} − s_{i})) = X_{2}v_{i} = w_{i+1}

and

X_{2} (\frac{1}{i+1}(u_{i} − s_{i})) ≡ v_{i+1} (mod Ker(X_{1}, 𝔎(M)/J^{2}𝔎(M))).

Write X_{2} (\frac{1}{i+1}(u_{i} − s_{i})) = v_{i+1} + t_{i+1} with t_{i+1} ∈ Ker(X_{1}, 𝔎(M)/J^{2}𝔎(M)). We claim that each t_{i+1} ∈ J𝔎(M)/J^{2}𝔎(M). If p ∤ (i + 1) and p ∤ (i + 2), then it follows from the equations (6.2) that

v_{i+1} + t_{i+1} = X_{2} (\frac{1}{i+1}(u_{i} − s_{i})) = \frac{1}{i+1}X_{2}u_{i} − \frac{1}{i+1}X_{2}s_{i}
= \frac{1}{i+1}(X_{1}u_{i+1} − v_{i+1}) − \frac{1}{i+1}X_{2}s_{i}
= \frac{1}{i+1} ((i + 2)X_{1} \frac{1}{i+2}(u_{i+1} − s_{i+1}) + X_{1}s_{i+1} − v_{i+1}) − \frac{1}{i+1}X_{2}s_{i}
= \frac{1}{i+1} ((i + 2)v_{i+1} − v_{i+1} + X_{1}s_{i+1}) − \frac{1}{i+1}X_{2}s_{i}
= v_{i+1} + \frac{1}{i+1}X_{1}s_{i+1} − \frac{1}{i+1}X_{2}s_{i},

hence t_{i+1} ∈ J𝔎(M)/J^{2}𝔎(M). On the other hand, if p | (i+2), then X_{1}u_{i+1} = X_{1}s_{i+1} and we have

v_{i+1} + t_{i+1} = \frac{1}{i+1}(X_{1}u_{i+1} − v_{i+1}) − \frac{1}{i+1}X_{2}s_{i} = \frac{1}{i+1}X_{1}s_{i+1} − \frac{1}{i+1}v_{i+1} − \frac{1}{i+1}X_{2}s_{i}.

The terms involving v_{i+1} then cancel so that t_{i+1} ∈ J𝔎(M)/J^{2}𝔎(M).

The elements u′_{i} and v′_{i} may now be constructed inductively. For p | i, we set v′_{i} = v_{i}. Now assume by induction that v′_{i} has been constructed and that v′_{i} = v_{i} + t′_{i} with t′_{i} ∈ J𝔎(M)/J^{2}𝔎(M). Because 𝔎(M)/J^{2}𝔎(M) has the constant image property, there exists s′_{i} ∈ 𝔎(M)/J^{2}𝔎(M) such that X_{1}s′_{i} = t′_{i}. We then have

X_{1} (\frac{1}{i+1}(u_{i} − s_{i}) + s′_{i}) = v_{i} + t′_{i} = v′_{i}.

So let u′_{i+1} = \frac{1}{i+1}(u_{i} − s_{i}) + s′_{i}. Then X_{2}u′_{i+1} = v_{i+1} + t_{i+1} + X_{2}s′_{i}, and the right two terms lie in J𝔎(M)/J^{2}𝔎(M) by the previous remarks. Setting v′_{i+1} = v_{i+1}+t_{i+1}+X_{2}s′_{i} now completes the induction. □

APPENDIX A Generic points in algebraic geometry

A.1. Generic points for affine varieties

In this section we recall some basic concepts from the Zariski-Weil theory of generic points for irreducible affine varieties. Let k be an algebraically closed field and let U = 𝔸^{r}(k) denote the r dimensional affine space over k. Viewing U simply as a k-vector space, we also consider the k-linear dual U^{∗} and its symmetric algebra S(U^{∗}). It is well known that there is a 1-1 order reversing correspondence between the irreducible affine subvarieties of U and the prime ideals in S(U^{∗}), obtained in the following way: Fixing a basis x_{1}, …, x_{r} of U, we let y_{1}, …, y_{r} denote the respective dual basis of U^{∗}, that is, each y_{i} is a k-linear map U → k such that y_{i}(x_{j}) = δ_{ij} for all 1 ≤ j ≤ r. We then have S(U^{∗}) ≅ k[y_{1}, …, y_{r}] as k-algebras, and under this isomorphism we can view any element of S(U^{∗}) as a polynomial function U → k. The required correspondence then assigns to a prime ideal 𝔭 in S(U^{∗}) the set

V (𝔭) = {u ∈ U | f(u) = 0 for all f ∈ 𝔭}.

It follows from an argument similar to that given in Proposition A.1 (below) that V (𝔭) is independent of the choice of basis x_{1}, …, x_{r} of U.

Now let V be an irreducible closed subvariety of U corresponding to a prime ideal 𝔭 in S(U^{∗}). The coordinate ring of V is defined to be the quotient algebra k[V] = S(U^{∗})/𝔭. Since 𝔭 is a prime ideal, k[V] is an integral domain. We define the function field of V to be the field of fractions k(V) = fof(k[V]).

We now fix a basis x_{1}, …, x_{r} of U with dual basis y_{1}, …, y_{r} as above. Letting ȳ_{i} denote the image of y_{i} in k[V], we define the generic point of V to be the element

ȳ_{1} ⊗ x_{1} + ⋯ + ȳ_{r} ⊗ x_{r} ∈ U ⊗_{k} k(V) ≅ 𝔸^{r}(k(V)).

Proposition A.1. The generic point of V is independent of the choice of basis x_{1}, …, x_{r} of U.

Appendix A.1 Page 82

Proof. Let x′_{1}, …, x′_{r} be another basis of U and let A = (λ_{ij}) ∈ GL_{n}(k) be the matrix corresponding to this change of basis, i.e., set x′_{j} = ∑_{i=1}^{r} λ_{ij}x_{i} for each 1 ≤ j ≤ r. If y′_{1}, …, y′_{r} is the dual basis corresponding to x′_{1}, …, x′_{r}, then the matrix corresponding to the change of basis y_{1}, …, y_{r} to y′_{1}, …, y′_{r} is given by A^{−1}. Letting A^{−1} = (μ_{ij}), we thus have

\begin{aligned}ȳ′_{1} ⊗ x′_{1} + ⋯ + ȳ′_{r} ⊗ x′_{r} & = ∑_{l=1}^{r} (∑_{j=1}^{r} μ_{lj}ȳ_{j} ⊗ ∑_{i=1}^{r} λ_{il}x_{i}) = ∑_{i,j} ∑_{l=1}^{r} λ_{il}μ_{lj}(ȳ_{j} ⊗ x_{i}) \\ & = ∑_{i,j} δ_{ij}(ȳ_{j} ⊗ x_{i}) = ȳ_{1} ⊗ x_{1} + ⋯ + ȳ_{r} ⊗ x_{r}. \end{aligned} □

If K/k is an algebraically closed field extension having transcendence degree at least r over k, then there exists an embedding k(V) ↪ K since k(V) has transcendence degree at most r over k and K is algebraically closed. Letting t_{1}, …, t_{r} denote the respective images of ȳ_{1}, …, ȳ_{r} in K under such an embedding, we call the point

t_{1} ⊗ x_{1} + ⋯ + t_{r} ⊗ x_{r} ∈ U ⊗_{k} K ≅ 𝔸^{r}(K)

a generic point of V.

Conversely, if K/k is a field extension, then any element α ∈ 𝔸^{r}(K) ≅ U ⊗_{k} K may be written in the form t_{1} ⊗ x_{1} + ⋯ + t_{r} ⊗ x_{r} with t_{1}, …, t_{r} ∈ K. If 𝔭 ⊆ k[y_{1}, …, y_{r}] is the set of polynomials annihilated by t_{1}, …, t_{r}, then 𝔭 is a prime ideal, hence it corresponds to an irreducible subvariety V of U. The homomorphism k(V) → K given by ȳ_{i} ↦ t_{i} then induces an extension map 𝔸^{r}(k(V)) → 𝔸^{r}(K) under which the generic point of V is mapped to α. We then say that α is a generic point of V. Under this correspondence, we may view every point of 𝔸^{r}(K) as the generic point of some irreducible closed subvariety of U.

The following proposition unifies the above definitions of generic points and illustrates the concept of generic properties.

Proposition A.2. If V is an irreducible closed subvariety of 𝔸^{r}(k) and T is any set of polynomials in k[y_{1}, …, y_{r}], then the following are equivalent.

(i) The generic point of V annihilates T.

(ii) Any generic point of V defined over a field extension K/k annihilates T. (iii) Every point in V annihilates T.

Moreover, if these conditions fail, then the points in V not annihilating T form a dense open subset of V.

Proof. Let 𝔭 be the prime ideal in k[y_{1}, …, y_{r}] corresponding to V and let

ȳ_{1} ⊗ x_{1} + ⋯ + ȳ_{r} ⊗ x_{r} ∈ 𝔸^{r}(k(V))

be the generic point of V. If f is any polynomial in k[y_{1}, …, y_{r}], then we may view f as a polynomial function 𝔸^{r}(k(V)) → k(V) by expanding the relation

y_{i}(ȳ_{j} ⊗ x_{l}) = ȳ_{j}δ_{il} for all 1 ≤ i, j, l ≤ r

using the distributive law. Under this assignment,

f(ȳ_{1} ⊗ x_{1} + ⋯ + ȳ_{r} ⊗ x_{r})

is simply the coset in k[y_{1}, …, y_{r}]/𝔭 containing f, thus the generic point of V annihilates f if and only if f ∈ 𝔭. This establishes (i) ⇔ (iii).

Now let α be a generic point of V defined over some field extension K/k as described above and let ι denote the injection k(V) ↪ K under which the induced map

ι̃: 𝔸^{r}(k(V)) → 𝔸^{r}(K)

sends ȳ_{1} ⊗ x_{1} + ⋯ + ȳ_{r} ⊗ x_{r} to α. If f is any polynomial in k[y_{1}, …, y_{r}], then

\begin{aligned}f(α) & = f(ι̃(ȳ_{1} ⊗ x_{1} + ⋯ + ȳ_{r} ⊗ x_{r})) \\ & = ι(f(ȳ_{1} ⊗ x_{1} + ⋯ + ȳ_{r} ⊗ x_{r})). \end{aligned}

Hence α annihilates f if and only if ȳ_{1} ⊗ x_{1} + ⋯ + ȳ_{r} ⊗ x_{r} does, and this establishes (i) ⇔ (ii).

Finally, the set of points in V that do not annihilate some element of T are precisely those in V \ V (T), which is clearly an open subset of V in the Zariski topology. If this subset is non-empty, then it is also dense in V since V is irreducible. □

A.2. Generic points in the language of schemes

The theory of generic points for affine varieties can be extended to the theory of schemes over a fixed field k with the appropriate modifications. If X → Spec k is a scheme over k and x is any point in X, then there exists an affine open neighbourhood U ≅ Spec A of x on which x is represented by a prime ideal 𝔭 of A. It follows from [29, Corollaire 10.4.7] that if X is locally of finite type over k, then x is closed in X if and only if 𝔭 is a maximal ideal in A. Recall that the residue field of x is defined to be the quotient k(x) = O_{X,x}/𝔪_{x}, where 𝔪_{x} is the unique maximal ideal in the local ring O_{X,x}. As outlined in [30, Section II.3], we may form the fibred product

X_{k(x)} = X ×_{Spec k} Spec k(x).

Note that X_{k(x)} contains the affine open subset

U_{k(x)} ≅ U_{i} ×_{Spec k} Spec k(x) ≅ Spec A ⊗_{k} k(x).

Letting 𝔪 be the ideal of A ⊗_{k} k(x) ≅ A ⊗_{k} A_{𝔭}/𝔭A_{𝔭} generated by the elements a ⊗ 1 − 1 ⊗ ā for all a ∈ A, we have

(A ⊗_{k} k(x))/𝔪 ≅ k(x),

hence 𝔪 is a maximal ideal of A ⊗_{k} k(x). Moreover, the image of 𝔪 under the projection U_{k(x)} → U is 𝔭. It follows that if y is the closed point in X_{k(x)} represented by 𝔪 on U_{k(x)}, then the image of y under the projection X_{k(x)} → X is x. We call the closed point y ∈ X_{k(x)} defined in this way the generic point of x.

Similarly, if K/k is any field extension and α is a closed point in X_{K} with K(α) = K, then we call α a generic point for its image x ∈ X under the projection X_{K} → X. The latter is a scheme morphism mapping α ↦ x, thus there exists a corresponding non-zero (hence injective) field homomorphism k(x) → K(α) = K. Observe that the image of α under the projection

X_{K} ≅ X ×_{Spec k} Spec k(x) ×_{Spec k(x)} Spec K ⟶ X ×_{Spec k} Spec k(x) = X_{k(x)}

necessarily maps α to the generic point of x in X_{k(x)}.

APPENDIX B Modules for ℤ/p

Let p be a prime number and k an algebraically closed field of characteristic p. In this appendix we briefly outline the classification of finite dimensional k(ℤ/p)-modules in terms of their Jordan types. This theory can be thought of as the primary motivation for studying modules over elementary abelian p-groups in terms of their restrictions to cyclic shifted subgroups. (See Section 2.2.)

Let ⟨g⟩ ≅ ℤ/p be a cyclic group of order p and define X = g − 1 ∈ J(k⟨g⟩). For a finite dimensional k⟨g⟩-module M, let ρ_{M} : k⟨g⟩ → Mat_{dim_{k}(M)}(k) be the matrix representation affording M. If M is indecomposable, then since X^{p} = 0, the Jordan canonical form ρ_{M}(X)′ of ρ_{M}(X) contains only one Jordan block of length dim_{k}(M) with eigenvalue zero. In particular, we may conclude that dim_{k}(M) ≤ p. The matrix representation defined by g ↦ I_{dim_{k}(M)} + ρ_{M}(X)′ is then a representation of k⟨g⟩ equivalent to ρ_{M}, having been obtained from ρ_{M} via conjugation by an element of GL_{dim_{k}(M)}(k). Conversely, if 1 ≤ i ≤ p and J_{i} ∈ Mat_{i}(k) is the Jordan block of length i with eigenvalue zero, then the map g ↦ I_{i} + J_{i} defines an indecomposable matrix representation k⟨g⟩ → Mat_{i}(k). This establishes a 1-1 correspondence between isomorphism types of indecomposable k⟨g⟩-modules and Jordan blocks having length 1 ≤ i ≤ p over k and eigenvalue zero. With a slight abuse of notation, we shall also denote the indecomposable representation of dimension 1 ≤ i ≤ p by J_{i}.

As in Section 2.2, we denote by M ˜⊗_{k} N the vector space M ⊗_{k} N endowed with the k⟨g⟩-module structure induced by the Lie theoretic comultiplication

X ⟼ X ⊗ 1 + 1 ⊗ X,

reserving the notation M ⊗_{k} N for the k⟨g⟩-module defined via the group theoretic comultiplication g ↦ g ⊗ g. Our immediate goal is to prove that these two structures are isomorphic for cyclic groups of order p.

Lemma B.1. If 1 ≤ i < p, then J_{2} ⊗_{k} J_{i} ≅ J_{2} ˜⊗_{k} J_{i} ≅ J_{i−1} ⊕ J_{i+1}. We also have J_{2} ⊗_{k} J_{p} ≅ J_{2} ˜⊗_{k} J_{p} ≅ J_{p} ⊕ J_{p}.

Appendix B.0 Page 86

Proof. For J_{2} ⊗_{k} J_{i} we have

ρ_{J_{2}⊗_{k}J_{i}}(g) = (I_{2} + J_{2}) ⊗ (I_{i} + J_{i}) = ( \begin{matrix}I_{i} + J_{i} & I_{i} + J_{i} \\ 0 & I_{i} + J_{i} \end{matrix} )

so that

ρ_{J_{2}⊗_{k}J_{i}}(X) = ρ_{J_{2}⊗_{k}J_{i}}(g) − I_{2i} = ( \begin{matrix}J_{i} & I_{i} + J_{i} \\ 0 & J_{i} \end{matrix} ) .

Recall that the product of 2 × 2 block matrices is given by

( \begin{matrix}A & B \\ C & D \end{matrix} ) ( \begin{matrix}E & F \\ G & H \end{matrix} ) = ( \begin{matrix}AE + BG & AF + BH \\ CE + DG & CF + DH \end{matrix} ) .

It follows that

ρ_{J_{2}⊗_{k}J_{i}}(X^{n}) = ( \begin{matrix}J_{i} & I_{i} + J_{i} \\ 0 & J_{i} \end{matrix} )^{n} = ( \begin{matrix}J_{i}^{n} & n(J_{i}^{n−1} + J_{i}^{n}) \\ 0 & J_{i}^{n} \end{matrix} ) .

Similarly, for J_{2} ˜⊗_{k} J_{i} we have

ρ_{J_{2} ˜⊗_{k} J_{i}}(X) = J_{2} ⊗ I_{i} + I_{2} ⊗ J_{i} = ( \begin{matrix}J_{i} & I_{i} \\ 0 & J_{i} \end{matrix} )

so that

ρ_{J_{2} ˜⊗_{k} J_{i}}(X^{n}) = ( \begin{matrix}J_{i} & I_{i} \\ 0 & J_{i} \end{matrix} )^{n} = ( \begin{matrix}J_{i}^{n} & nJ_{i}^{n−1} \\ 0 & J_{i}^{n} \end{matrix} ) .

From this we see that

rank(X^{n}, J_{2} ⊗_{k} J_{i}) = rank(X^{n}, J_{2} ˜⊗_{k} J_{i}) = \begin{cases}2i − 2n & n < i \\ 1 & n = i < p \\ 0 & n = i = p. \end{cases}

The proof now follows from the fact that the isomorphism types of J_{2}⊗_{k}J_{i} and J_{2} ˜⊗_{k}J_{i} are determined by the Jordan types of ρ_{J_{2}⊗_{k}J_{i}}(X) and ρ_{J_{2} ˜⊗_{k} J_{i}}(X), respectively, and the Jordan type of a nilpotent matrix is completely determined by the ranks if its powers. □

Recall that the representation ring of k(ℤ/p) is the ring a(k(ℤ/p)) = a(ℤ/p) whose elements are the isomorphism classes [M] of finite dimensional k(ℤ/p)-modules, with addition and multiplication defined by

[M] + [N] = [M ⊕ N], [M][N] = [M ⊗_{k} N].

We also define the Lie theoretic representation ring ã(ℤ/p) to be the ring having the same additive structure as a(ℤ/p), but with multiplication given via the Lie theoretic tensor product [M][N] = [M ˜⊗_{k} N].

Lemma B.2. For p odd, the isomorphism class [J_{2}] generates a(ℤ/p) and ã(ℤ/p).

Proof. We first prove the statement for a(ℤ/p). For this it suffices to show that [J_{2}] generates [J_{1}] = [k]. Note by Lemma B.1 that

[J_{2}][J_{i}] = [J_{2} ⊗_{k} J_{i}] = [J_{i−1} ⊕ J_{i+1}] = [J_{i−1}] + [J_{i+1}]

for all 2 ≤ i ≤ p − 1 and [J_{2}][J_{p}] = [J_{2} ⊗_{k} J_{p}] = [J_{p} ⊕ J_{p}] = 2[J_{p}]. It follows that

[J_{1}] = [J_{2}]([J_{2}] − [J_{4}] + [J_{6}] − ⋯ ± [J_{p−1}] ∓ \frac{1}{2}[J_{p}]).

A similar argument shows that [J_{2}] generates ã(ℤ/p). □

Proposition B.3. a(ℤ/p) and ã(ℤ/p) have equal ring structures.

Proof. The case p = 2 is obvious, and the proof for p > 2 follows directly from Lemmas B.1 and B.2. □

Corollary B.4. If M and N are finite dimensional k(ℤ/p)-modules, then

M ⊗_{k} N ≅ M ˜⊗_{k} N.

APPENDIX C A brief introduction to rank varieties

In this appendix we recall the definition of the rank variety of a finite dimensional kE-module. This material will be used in Section 2.8 to show that the rank variety is too coarse of an invariant to yield useful information about kE-modules of constant Jordan type. A standard reference for the following is Section 5.8 of Benson [4].

Studying kE-modules in terms of their restrictions to cyclic shifted subgroups was pioneered by Dade. The following theorem, which appeared as Lemma 11.8 of [21], is the starting point of our discussion. Recall that since kE is a local ring, a kE-module is projective if and only if it is free.

Theorem C.1 (Dade’s lemma). A finite dimensional kE-module M is projective if and only if M ↓_{⟨g_{α}⟩} is projective for all non-zero α ∈ 𝔸^{r}(k).

If ⟨g⟩ ≅ ℤ/p is a cyclic group of order p, then a k⟨g⟩-module M is projective if and only if the Jordan type of X = g − 1 on M contains only blocks of length p. In this case, writing M ≅ (k⟨g⟩)^{s}, we have rank(X, M) = s(p − 1). This shows that if M is a kE-module and α is a non-zero point in 𝔸^{r}(k), then M ↓_{⟨g_{α}⟩} is projective if and only if p divides dim_{k}(M) and the rank of X_{α} on M is (p − 1) dim_{k}(M)/p. Denoting the Jordan type of X_{α} on M by [p]^{a_{p}} … [1]^{a_{1}}, we also have

rank(X_{α}, M) = ∑_{j=1}^{p}(j − 1)a_{j} = dim_{k}(M) − ∑_{j=1}^{p} a_{j} ≤ dim_{k}(M) − (⌊\frac{dim_{k}(M)}{p}⌋ + 1)

≤ dim_{k}(M) − \frac{dim_{k}(M)}{p} = \frac{(p − 1) dim_{k}(M)}{p}

since ∑_{j=1}^{p} a_{j} is the number of Jordan blocks in the action of X_{α} on M, and each such block has length at most p. It follows that M ↓_{⟨g_{α}⟩} is not projective if and only if the rank of X_{α} on M is strictly less than (p − 1) dim_{k}(M)/p. As observed in Section 2.1, the points α ∈ 𝔸^{r}(k) satisfying the latter condition form a Zariski closed subset of 𝔸^{r}(k). This justifies the following definition.

Definition C.2 (Carlson [12]). If M is a kE-module, then the rank variety of M is the homogeneous subvariety

V_{E}^{♯}(M) = {0 ≠ α ∈ 𝔸^{r}(k) | M ↓_{⟨g_{α}⟩} is not projective} ∪ {0}

Appendix C.0 Page 89

of 𝔸^{r}(k). Combining the above definition with Dade’s lemma, we immediately obtain the following characterisation of projectivity. Corollary C.3. A kE-module M is projective if and only if V_{E}^{♯}(M) = {0}.

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