Varieties for Modules of Small Dimension

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

Fergus Reid

(MSci, University of Glasgow, 2009)

Declaration

I declare that this thesis has been entirely composed by myself and that the work contained within is my own. Quotations marks have been used to indicate quotations and sources of information have been specifically acknowledged. I also declare that this thesis has not been accepted in any previous application for a degree.

Signed:

Acknowledgements

There are several people whom I would like to thank. Firstly I would like to thank Dave Benson, my supervisor. You were always available for discussion and your advice was of great help to me throughout.

I would also like to thank the staff of the department for providing such a friendly, welcoming atmosphere.

My thanks also go out to my friends outside the department for the great deal of moral support they provided throughout the years.

Finally I would like to thank my family for all of their support and assistance. I am very fortunate to come from a very close family with such wonderful siblings.

Summary

This thesis focuses on the subject of varieties for modules for elementary abelian pgroups. Given a homogeneous polynomial we will give constructions for modules of small dimension having variety given by that polynomial. This is similar to an earlier construction given by Jon Carlson but our modules will in general be of considerably smaller dimension.

We present two main constructions. The first, restricted to working with quadratics and an algebraically closed field of characteristic 2, generalises Carlson’s technique of using cohomology elements to instead use matrices of cohomology elements and also replaces syzygies with relative syzygies. The second uses matrix factorisations to obtain modules with variety given by a desired homogeneous polynomial of arbitrary degree whilst working over an algebraically closed field of characteristic 2. We then use the invariant of Loewy length to show that, when both defined, the two constructions give non-isomorphic modules.

Finally we investigate the connection between the variety of a module and its Loewy length. We show that working over an algebraically closed field of characteristic 2 with modules of Loewy length 2 allows us to find modules with any hypersurface as their variety. On the other hand we also demonstrate that in odd characteristic p, with modules of Loewy length p, the only possible varieties are finite unions of linear hypersurfaces.

Contents

1 Introduction 3 1.1 Representations of Elementary Abelian Groups . . . . . . . . . . . . 3 1.2 Varieties for Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Preliminaries 6 2.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.2 Modular Representation Theory . . . . . . . . . . . . . . . . . . . . . 6 2.3 Diagrams for Modules . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.4 Elementary Abelian p-groups . . . . . . . . . . . . . . . . . . . . . . 10

2.4.1 Group Cohomology . . . . . . . . . . . . . . . . . . . . . . . . 12 2.5 Varieties for Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.5.1 Cohomological Varieties . . . . . . . . . . . . . . . . . . . . . 22 2.5.2 Varieties for Modules . . . . . . . . . . . . . . . . . . . . . . . 23 2.5.3 Rank Varieties . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.5.4 The Structure of the Cohomological Variety . . . . . . . . . . 25 2.5.5 Computing the Rank Variety . . . . . . . . . . . . . . . . . . 27

2.6 L_{ζ} Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 2.7 Loewy Length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.8 Matrix Factorisations . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3 Modules of Small Dimension 35

3.1 Quadratics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.1.1 Modules for general quadratics . . . . . . . . . . . . . . . . . 35 3.2 General Homogeneous Polynomials . . . . . . . . . . . . . . . . . . . 47

3.2.1 Polynomials of form Z_{1} ⋯ Z_{n} + Z_{(n+1)} ⋯ Z_{2n} . . . . . . . . . 48 3.2.2 Modules from Matrix Factorisations . . . . . . . . . . . . . . . 50 3.2.3 Comparison with L_{n} . . . . . . . . . . . . . . . . . . . . . . . 56

3.3 Modules of Loewy Length p . . . . . . . . . . . . . . . . . . . . . . . 59 3.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 3.3.2 Characteristic 2 . . . . . . . . . . . . . . . . . . . . . . . . . . 60 3.3.3 Odd Characteristic . . . . . . . . . . . . . . . . . . . . . . . . 62 3.3.4 No Other Modules . . . . . . . . . . . . . . . . . . . . . . . . 63 3.3.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 4 Further Research 66

4.1 Module Constructions . . . . . . . . . . . . . . . . . . . . . . . . . . 66 4.2 Minimal Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 4.3 Other Invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

Chapter 1

Introduction

1.1 Representations of Elementary Abelian Groups

Throughout this thesis we will work almost exclusively with representations of elementary abelian p-groups. This has been a thriving subject for many years now, with perhaps the first key result being Chouinard’s Theorem [16] in 1976. This states that a representation of a finite group is projective if and only if its restriction to every elementary abelian p-subgroup is projective. This is just one of many results that demonstrate how a better understanding of the representation theory of elementary abelian p-groups can be of great help in understanding the representation theory of finite groups in general. In fact we will see later that Chouinard’s Theorem provides a strong motivation for the definition of the rank variety of a module.

Two areas of research that are currently particularly active are modules of constant Jordan type and the theory of support varieties. In a sense studying modules of constant Jordan type involves studying an entirely different collection of modules from studying the rank variety of a module. This is due to the fact that a module with constant Jordan type must have trivial rank variety.

Support varieties, on the other hand, are a slightly more general notion than the rank varieties that we will be studying here and appear to be a very powerful tool. For example in [6] support varieties are used to classify tensor ideal localising subcategories of the stable module category of a finite group.

1.2 Varieties for Modules

This thesis will focus on the subject of varieties for modules, in particular studying the rank variety of a representation of an elementary abelian p-group. The idea of associating a variety to a module as an invariant first began with Quillen [26] in 1971. Given a group G and a field k he studied the maximal ideal spectrum of the cohomology ring H^{∗}(G, k) as an affine variety.

It was not until 1981 that Carlson developed the idea of varieties for modules [12]. Initially this involved two seemingly different definitions. The first, the cohomological variety of a module, is defined for a representation M of any finite group G. If we denote the kernel of the map Ext_{kG}(k, k) ⟶^{⊗M} Ext_{kG}(M, M) by I then the cohomological variety of M is given by considering the maximal ideal spectrum of H^{∗}(G, k)/I.

The rank variety of a module, however, is only defined for a rank r elementary abelian p-group E and is determined by considering whether restriction to certain cyclic shifted subgroups is projective. The motivation for this definition comes from a theorem of Dade [18] stating that a representation M of E is projective if and only if the restriction of M to every cyclic shifted subgroup is projective.

Carlson showed that when working over an elementary abelian p-group both notions of a variety for a module had the same dimension, conjecturing that the two varieties were in fact equal. This was later proved in 1982 by Avrunin and Scott [1].

In 1984 Carlson produced a construction that, given a homogeneous polynomial f ∈ k[Z_{1}, …, Z_{n}], an elementary abelian p-group E of rank n and k an algebraically closed field of strictly positive characteristic, would produce a kE-module L_{f} whose variety was given by the polynomial f. The existence and possible dimensions of such modules is one of the central ideas studied in this thesis.

One of the difficulties with Carlson’s original construction is that the modules produced tend to be rather large. For example if we take the polynomial f = Z_{1}Z_{2} + Z_{3}Z_{4} + Z_{5}Z_{6} and work over k an algebraically closed field of characteristic 2 then the corresponding L_{f} has dimension approximately 15(2^{6}) = 960.

1.3 Results

The results in this thesis can be divided into three categories: Constructing modules of the smallest dimension we can with particular variety, discussing the minimal possible dimension for such modules and the connection between a module’s variety and its Loewy length.

Our construction of modules of small dimension will rely on the idea of matrices of cohomology elements. As we shall see it proves possible to generalise the idea from Carlson’s L_{ζ} construction of constructing modules from individual elements of the group cohomology to using matrices of cohomology elements of smaller degree. This will allow us to produce smaller modules than those obtained from the L_{ζ} construction.

Discussion of the constructions themselves will divide into two distinct halves. In the first half we restrict our attention to modules with variety an irreducible quadratic in characteristic 2. This helps to keep the dimensions of the various objects we work with small, making it easier to perform computations. Given a polynomial f = Z_{1}Z_{n+1} +⋯+Z_{n}Z_{2n}, k an algebraically closed field of characteristic 2 and E the elementary abelian 2-group of rank 2n we will show how to construct a kE-module of dimension 2^{n} whose variety is given by f.

Having done this we will then detail a more general construction for modules with variety given by an arbitrary irreducible homogeneous polynomial over an algebraically closed field of characteristic 2. The key tool we use here is that of matrix factorisations (see Section 2.8). Given a polynomial f = Z_{1}Z_{(n+1)} ⋯ Z_{((d−1)n+1)} + ⋯ + Z_{n}Z_{2n} ⋯ Z_{dn}, for some n, d ∈ ℕ, k an algebraically closed field of characteristic 2 and E an elementary abelian 2-group of rank dn, we will construct a kE-module of dimension 2d^{n−1} having variety given by f. Of particular interest is that if we set d = 2 here then the modules we obtain are very different from those obtained before (in particular we will show that they have different Loewy lengths) but in fact have the same dimension.

Finally we shall describe a relation between the Loewy length of a module and the possible varieties for that module. Of particular interest is that the situation is very different depending on whether we are working in characteristic 2 or odd characteristic. If k is an algebraically closed field of characteristic 2, E an elementary abelian 2-group of rank r and M a kE-module of Loewy length 2, then the variety of M may be any hypersurface. If however our field k has odd characteristic and M has Loewy length p then we will show that the variety of M must be a finite union of linear hypersurfaces.

Before we proceed with these results we will first provide a detailed account of the required background information.

Preliminaries

In this chapter we will present a summary of the minimal information necessary for what follows. There are several areas we will assume prior knowledge of, including the cohomology of a group and the definition of an affine variety. A good introduction to group cohomology can be found in [22] and a detailed introduction to varieties may be found in [28]. Note that the latter gives a far more comprehensive introduction than we will require here — in fact the first chapter is likely to suffice.

2.1 Notation

Our essential first step is to establish the minimal necessary notation for what follows. Unless explicitly stated otherwise all modules will be finitely generated left-modules, k will denote an algebraically closed field of strictly positive characteristic and all rings R will be unitary. Given a group G and a field k we will use kG to denote the group algebra and write kG-mod to denote the category of left kG-modules. We will use Z_{1}, …, Z_{n} to indicate the co-ordinate functions on affine n-space.

2.2 Modular Representation Theory

We begin with a brief introduction to modular representation theory. As we will be making heavy use of results particular to the elementary abelian p-groups we will work with throughout, we will merely give a small number of motivational results. Modular representation theory is itself an enormous area of mathematics and we shall see that restricting to the case of elementary abelian p-groups leaves us with a great deal of interesting questions to investigate.

One of the major differences in studying modular as opposed to ordinary representation theory comes from the lack of Maschke’s Theorem (see [19], Theorem 3.1), presented below.

Theorem 2.2.1 (Maschke’s Theorem). Let G be a finite group of order n, k a field such that char(k) ∤ n. Then the group algebra kG is a semisimple ring.

The assumption that the characteristic of the field does not divide the order of the group proves essential. In general the group algebra will not be semisimple over a field of strictly positive characteristic. Maschke’s theorem and ordinary representation theory also allow us the following extremely useful theorem (Theorem 3.2 of [19]).

Theorem 2.2.2. Let G be a finite group of order n and k an algebraically closed field of characteristic not dividing n. Then

kG ≅ Mat_{n_{1}}(k) ⊕ ⋯ ⊕ Mat_{n_{s}}(k),

where n = n_{1}^{2}+⋯+n_{s}^{2}. The ring kG has exactly s nonisomorphic irreducible modules, of dimensions n_{1}, n_{2}, …, n_{s}, where s is the number of conjugacy classes of G.

This fails when the characteristic of our field does divide n. Attempts to classify modular representations will in fact inevitably run up against the so-called ‘trichotomy theorem’ of Drozd — see [20] and [9] — on representation types. We provide a brief sketch of this as an important motivation for what follows. Firstly we must define exactly what we mean by ‘representation type’ (our interpretation is that given in [3]).

Definition 2.2.3 (Representation Types). Let A be a finite-dimensional algebra over an algebraically closed field k. Then we will define three different representation types:

1. Finite representation type: There are finitely many indecomposable A-modules (up to isomorphism).

2. Tame representation type: A is not of finite representation type and for any fixed dimension d the d-dimensional A-modules are contained in a finite number of 1-parameter families (with finitely many exceptions).

3. Wild representation type: The category of finite-dimensional A-modules ‘contains’ the category of k⟨x, y⟩-modules. This means that there exists an Ak⟨x, y⟩-bimodule M that is free as a right k⟨x, y⟩-module such that the functor ϕ from finite-dimensional k⟨x, y⟩-modules to finite-dimensional A-modules given by,

ϕ(N) = M ⊗_{k⟨x,y⟩} N,

sends indecomposable and isomorphic modules to indecomposable and isomorphic modules, respectively.

We now find ourselves in a position to state the Trichotomy Theorem, allowing us to partition finite-dimensional algebras into three distinct classes.

Theorem 2.2.4 (Trichotomy Theorem). All finite-dimensional algebras are of either finite, tame or wild representation type.

The Trichotomy theorem leads us to the following theorem on representations of finite groups, demonstrating that the group algebra of ‘most’ groups will have wild representation type.

Theorem 2.2.5. Suppose that G is a finite group, k an algebraically closed field of characteristic p and P a Sylow-p-subgroup of G. The representation type of kG can then be determined as follows:

1. Finite: Occurs if and only if P is cyclic.

2. Tame: Occurs if and only if p = 2 and P is either dihedral, semi-dihedral or generalised quaternion.

3. Wild: All other groups.

In particular we see that even a group as small as C_{3} × C_{3} has wild representation type (in characteristic 3). For this reason it is interesting to select smaller classes of kG-modules for further study.

2.3 Diagrams for Modules

Before we begin our discussion of elementary abelian p-groups we must first make an important digression. The use of diagrams for modules will serve as an extremely powerful tool for us. When we shortly come to define the functors Ω^{n} and Ω^{−n} we will see that they allow computations to be performed with a minimal amount of work. Unfortunately the diagrams work best for groups of ranks two and three, although they may occasionally be carefully used in higher ranks.

Remark 2.3.1. What follows is an informal introduction to the use of diagrams for modules. It should also be noted that a wide variety of different conventions and techniques are used throughout the literature. We will follow the notation of Benson and Carlson [4].

Definition 2.3.2. Given an elementary abelian p-group

E ≅ (ℤ/pℤ)^{r} ≅ ⟨g_{1}, …, g_{r}; g_{i}^{p} = 1, g_{i}g_{j} = g_{j}g_{i}⟩

over an algebraically closed field k of positive characteristic p we have that kE ≅ k[x_{1}, …, x_{r}]/(x_{1}^{p}, …, x_{r}^{p}), where x_{i} = g_{i} − 1 (we will see how to prove this later in Theorem 2.4.2). We define the diagram of a kE-module M to be a directed graph Γ with the following properties:

1. Γ has vertex set V, where V is a basis for M.

2. There is an edge (v_{i}, v_{j}) if and only if x_{k}(v_{i}) = v_{j} for some 1 ≤ k ≤ r.

3. There is at most one edge between any pair of vertices and if x_{k}(v_{i}) = v_{j} then the edge (v_{i}, v_{j}) has label x_{k}.

4. If there is no edge (v_{i}, v_{j}) for some fixed choice of v_{i} and all v_{j} ∈ V, then x_{k}(v_{i}) = 0 for all 1 ≤ k ≤ r.

Given this definition we now give diagrams for some small and hopefully familiar modules that will come in useful throughout this thesis.

Example 2.3.3. We will begin by describing a small example, namely we will look at the group algebra kE where k is an algebraically closed field of characteristic 2 and E is a product of two copies of the cyclic group of order two, i.e. E = (ℤ/2ℤ)^{2}. Recall from earlier that kE is now simply the truncated polynomial algebra k[x_{1}, x_{2}]/(x_{1}^{2}, x_{2}^{2}).

We can represent kE as a kE-module using the following diagram:

\xymatrix{ & 1 \ar[dl]_{x_{1}} \ar[dr]^{x_{2}} \\ x_{1} \ar[dr]_{x_{2}} &  & x_{2} \ar[dl]^{x_{1}} \\  & x_{1}x_{2}}

The vertices represent the basis elements and the arrows represent actions. So for example x_{2} acting on x_{1} results in the element x_{2}x_{1}, which, by commutativity, equals x_{1}x_{2}. It is usual to have parallel arrows representing the action of a particular element, e.g., here arrows pointing left represent acting by x_{1}, arrows pointing to the right represent acting by x_{2}. This usually works well for ranks 2 and 3 but the situation is a little more complex in higher ranks.

Example 2.3.4. For our next example we again work with two copies of the cyclic group of order two but this time over an algebraically closed field of characteristic 3. Recall that this means the group algebra now has form kE ≅ k[x_{1}, x_{2}]/(x_{1}^{3}, x_{2}^{3}). The diagram is very similar to that above but we must add extra vertices and edges to reflect the fact that x_{i}^{2} is no longer 0, giving the following:

\xymatrix{ &  & 1 \ar[dl]_{x_{1}} \ar[dr]^{x_{2}} \\  & x_{1} \ar[dl]_{x_{1}} \ar[dr]^{x_{2}} &  & x_{2} \ar[dl]_{x_{1}} \ar[dr]^{x_{2}} \\ x_{1}^{2} \ar[dr]_{x_{2}} &  & x_{1}x_{2} \ar[dl]^{x_{1}} \ar[dr]_{x_{2}} &  & x_{2}^{2} \ar[dl]^{x_{1}} \\  & x_{1}^{2}x_{2} \ar[dr]_{x_{2}} &  & x_{1}x_{2}^{2} \ar[dl]^{x_{1}} \\  &  & x_{1}^{2}x_{2}^{2}}

Hopefully it is now fairly clear how this idea extends to other odd characteristics. The next step is to consider a rank three example.

Example 2.3.5. Returning to our algebraically closed field of characteristic 2, we now let our group E be a product of three copies of the cyclic group of order two. The group algebra therefore becomes a truncated polynomial algebra in three variables, i.e., kE ≅ k[x_{1}, x_{2}, x_{3}]/(x_{1}^{2}, x_{2}^{2}, x_{3}^{2}). The addition of x_{3} requires us to extend the diagram to three dimensions, giving us the following:

\xymatrix{ & x_{1}x_{2} \ar@{.>}[ddd]_{x_{3}} \\ x_{1} \ar[ur]_{x_{2}} \ar[ddd]^{x_{3}} &  &  & x_{2} \ar[ull]^{x_{1}} \ar[ddd]_{x_{3}} \\  &  & 1 \ar[ull]_{x_{1}} \ar[ur]^{x_{2}} \ar[ddd]_{x_{3}} \\  & x_{1}x_{2}x_{3} \\ x_{1}x_{3} \ar@{.>}[ur]_{x_{2}} &  &  & x_{2}x_{3} \ar@{.>}[ull]^{x_{1}} \\  &  & x_{3} \ar[ull]_{x_{1}} \ar[ur]^{x_{2}}}

Note that the direction of an arrow still allows us to see at a glance which element is acting.

It is worth remarking that these diagrams can sometimes be used for higher ranks. Although we would often require a four-dimensional diagram there are many modules for which we can work around this problem by means of different arrow styles. As this will not be required for our purposes we omit the details.

Finally we note that even in the rank 2 case most modules do not have a diagram of this form. In this thesis our use of diagrams will focus on copies of the group algebra and the modules Ω_{kE}^{n}(k) we will define later (see Definition 2.4.13).

Now that we are equipped with this tool we may discuss more results on elementary abelian p-groups leading to a definition of varieties for modules, the central objects of this thesis.

2.4 Elementary Abelian p-groups

We now come to one of our key objects. As we will be working with elementary abelian p-groups throughout the rest of this thesis there are many facts it will prove useful to present here. We will see many examples, such as group cohomology, where knowing that our finite group is an elementary abelian p-group will simplify matters considerably.

Definition 2.4.1 (Elementary Abelian p-group). A group E is an elementary abelian p-group of rank r if E ≅ (ℤ/pℤ)^{r}.

Working with elementary abelian p-groups also allows us to give a nice description of the group algebra.

Theorem 2.4.2. Let E ≅ (ℤ/pℤ)^{r} ≅ ⟨g_{1}, …, g_{r}; g_{i}g_{j} = g_{j}g_{i}, g_{i}^{p} = 1⟩, k an algebraically closed field of characteristic p. Then kE ≅ k[x_{1}, …, x_{r}]/(x_{1}^{p}, …, x_{r}^{p}), where deg(x_{i}) = 1 and x_{i} = g_{i} − 1.

Remark 2.4.3. The above result is easily proved by determining the case r = 1 and then using the fact that, given two finite groups G, H and a field k, k(G × H) ≅ kG ⊗ kH.

There is one final property of the group algebra that will come in useful throughout. In particular we will see that it helps greatly when discussing projective resolutions and the functors Ω, Ω^{−1} we will soon define.

Theorem 2.4.4. Let M be a kE-module, where E is an elementary abelian p-group. Then the following are equivalent:

1. M is projective.

2. M is free.

3. M is injective.

Proof. The equivalence ‘projective ⟺ injective’ is proven in [17, Theorem 58.14] and works for E an arbitrary finite group.

The equivalence ‘projective ⟺ free’ is a consequence of a theorem of Kaplansky [24] as the group algebra kE is a local ring.

We now describe several results that will not only be useful here but also provide further motivation for studying elementary abelian p-groups in general. Before doing so we will first need the definition of a cyclic shifted subgroup (taken from page 81 of [15]).

Definition 2.4.5 (Cyclic Shifted Subgroup). Let k be an algebraically closed field and E an elementary abelian p-group of rank r such that E ≅ ⟨g_{1}, …, g_{r}⟩ and kE ≅ k[x_{1}, …, x_{r}]/(x_{1}^{p}, …, x_{r}^{p}), where x_{i} = g_{i} − 1. Now let 0 ≠ α = α_{1}x_{1} + ⋯ + α_{r}x_{r}, where the α_{i} are elements of k. Then a cyclic shifted subgroup of E is a subgroup of the group of units of kE of form ⟨α + 1⟩.

Given this definition we are now able to state Dade’s lemma, appearing as Lemma 11.8 of [18], which allows us to determine projectivity by considering cyclic shifted subgroups. We will see later that this plays an important part in the definition of a rank variety.

Lemma 2.4.6. A finitely generated kE-module M, where k is an algebraically closed field and E an elementary abelian p-group, is projective if and only if M ↓_{E_{1}} is projective for every cyclic shifted subgroup E_{1} of E.

A further motivation for the study of representations of elementary abelian pgroups comes from a similar theorem of Chouinard [16].

Theorem 2.4.7. Let G be a finite group. A kG-module M is projective if and only if its restriction to every elementary abelian p-subgroup of G is projective.

Having covered some general results on elementary abelian p-groups we now move on to looking at their cohomology. The nature and properties of this will prove vital to defining the variety of a module later in this chapter.

2.4.1 Group Cohomology

Minimal Resolutions

We begin our discussion of group cohomology by considering minimal resolutions. These resolutions have particularly nice behaviour that will allow us to define projective covers and the functors Ω, Ω^{−1}.

The following information on minimal resolutions comes from Section 2.4 of [22].

Definition 2.4.8. If G is a finite group and k an algebraically closed field then the projective resolution

\xymatrix{⋯ \ar[r] & P_{n} \ar[r]^{d_{n}} & P_{n−1} \ar[r]^{d_{n−1}} & ⋯ \ar[r] & P_{1} \ar[r]^{d_{1}} & P_{0} \ar[r]^{ϵ} & M \ar[r] & 0}

is said to be minimal if Im(d_{i+1}) ⊆ J(kG)P_{i} for all i, where J(kG) denotes the Jacobson radical of kG.

Note that minimal resolutions are unique up to isomorphism of complexes. Given a minimal projective resolution the maps d_{i}, ϵ are often called ‘projective covers’ of Ker(d_{i−1}) (with d_{1} being a projective cover of Ker(ϵ)).

We may also define minimal injective resolutions as the dual of this notion, where we replace projectives with injectives and projective covers with ‘injective hulls’.

Theorem 2.4.9. Let k be an algebraically closed field of characteristic p and E an elementary abelian p-group of rank r. Then the minimal projective resolution of k over kE has the following form:

\xymatrix{⋯ \ar[r] & d_{2}kE \ar[r]^{δ_{2}} & d_{1}kE \ar[r]^{δ_{1}} & kE \ar[r]^{δ_{0}} & k \ar[r] & 0}

where d_{i} is the number of monomials of degree i in r variables, i.e., \binom{r+i−1}{i}.

The Stable Module Category, Ω and Ω^{−1}

The notions of projective cover and the Ω^{n}, Ω^{−n} provide us with a useful invariant that will be used throughout. Before we begin we must first establish a more suitable category, the stable module category, in which to define them. The reason for this is essentially that, as we shall see, using the stable module category allows us to define Ω and Ω^{−1} as functors.

Definition 2.4.10 (Stable Module Category). Let _{R}Mod denote the category of left R-modules, for some ring R. Then the stable module category StMod(R) is the category whose objects are given by those in _{R}Mod with morphisms a quotient of those in _{R}Mod by the morphisms factoring through a projective. So in particular any projective module P ∈ _{R}Mod will be isomorphic to 0 in StMod(R).

We will also use the notation stmod(R) to denote the subcategory of StMod(R) consisting of finitely-generated modules.

Before proceeding to the definitions of Ω and Ω^{−1} we first give a small example.

Example 2.4.11. Let k be an algebraically closed field of characteristic 2, with E the elementary abelian 2-group of rank 2. Then the minimal projective resolution of k as a kE-module has the following form:

\xymatrix{⋯ \ar[r]^{δ_{2}} & 3kE \ar[r]^{δ_{1}} & 2kE \ar[r]^{δ_{0}} & kE \ar[r] & k \ar[r] & 0}

We now use diagrams for modules to give a more concrete description of the maps δ_{0} and δ_{1}.

Firstly fix a basis of kE to be given by {1, x_{1}, x_{2}, x_{1}x_{2}}. Then the map δ_{0} : kE → k sends 1_{kE} to 1_{k} and all other basis elements of kE are sent to zero. We may represent this with the following diagram:

[figure: ·; δ_{0}; ·; ·; ·^{k}; ·]

The module Ω_{kE}(k) is defined to be the kernel of this map. As the kernel consists of all generators of kE except 1_{kE} we may obtain the diagram for Ω_{kE}(k) by simply removing the appropriate vertex from the diagram for kE.

\xymatrix{· \ar[dr]_{x_{2}} &  & · \ar[dl]^{x_{1}} \\  & ·}

Similarly we define Ω_{kE}^{−1}(k) by looking at the co-kernel of the following map f : k → kE:

[figure: ·; f; ·; ·; ·]

In this case we remove the ‘lowest’ vertex from the diagram for kE, so Ω_{kE}^{−1}(k) is described by the following diagram:

\xymatrix{ & · \ar[dl]_{x_{1}} \ar[dr]^{x_{2}} \\ · &  & ·}

Having worked through these two examples we are now in a position to give a more general definition of the functors Ω and Ω^{−1}. The well-definedness of Ω and Ω^{−1} depends upon Schanuel’s Lemma (Lemma 1.5.3 of [3]), so we first present this.

Lemma 2.4.12. Suppose that 0 → M_{1} → P_{1} → M → 0 and 0 → M_{2} → P_{2} → M → 0 are short exact sequences of modules with P_{1} and P_{2} projective. Then M_{1} ⊕ P_{2} ≅ P_{1} ⊕ M_{2}.

Definition 2.4.13 (Ω, Ω^{−1}). As we used no special properties of δ_{0} compared to an arbitrary δ_{n} it is straightforward to generalise the definitions of Ω_{kE}(k) and Ω_{kE}^{−1}(k). We define Ω_{kE}^{n}(k) to be the kernel of δ_{n−1} and similarly Ω_{kE}^{−n}(k) is the n-th cokernel in a minimal injective resolution of k.

Given an arbitrary kE-module M we define Ω_{kE}^{n}(M) and Ω_{kE}^{−n}(M) to be the n-th kernel in a minimal projective resolution of M and the n-th cokernel in a minimal injective resolution of M, respectively. We will also define Ω^{0}(M) to be Ω^{−1}(Ω(M)) as this gives us that M ≅ Ω^{0}(M) ⊕ P, for some projective kE-module P.

As Ω, Ω^{−1} are functors (on the stable category) we must also define them on morphisms. Given a map f : M → N we define Ω(f) to be the map Ω(f) : Ω(M) → Ω(N) making the following diagram commute:

\xymatrix{Ω(M) \ar[d]^{Ω(f)} \ar[r]^{δ_{1}} & P_{1} \ar[d] \ar[r]^{ϵ_{1}} & M \ar[d]^{f} \\ Ω(N) \ar[r]^{δ_{2}} & P_{2} \ar[r]^{ϵ_{2}} & N}

where P_{1}, P_{2} are the projective covers of M, N respectively. Note that Ω(f) is unique only up to addition of maps factoring through a projective.

Similarly for Ω^{−1}, given a map f : M → N we define Ω^{−1}(f) to be the map Ω^{−1}(f) : Ω^{−1}(M) → Ω^{−1}(N) making the following diagram commute:

\xymatrix{M \ar[d]^{f} \ar[r]^{δ_{1}} & I_{1} \ar[d] \ar[r]^{ϵ_{1}} & Ω^{−1}(M) \ar[d]^{Ω^{−1}(f)} \\ N \ar[r]^{δ_{2}} & I_{2} \ar[r]^{ϵ_{2}} & Ω^{−1}(N)}

This time I_{1}, I_{2} are the injective hulls of M, N and again Ω^{−1}(f) is unique only up to addition of maps factoring through a projective.

Example 2.4.14. Returning to our previous example we now compute Ω_{kE}^{2}(k). Consider the map δ_{1} : 2kE → kE. This map sends (1_{kE}, 0) to x_{1} and (0, 1_{kE}) to x_{2}. The diagram is therefore as follows:

[figure: ·; ·; 1; ·; ·; ·; ·; x_{1}; x_{2}; ·; ·; x_{1}x_{2}]

In this case the kernel is slightly more difficult to picture. There are several vertices that clearly go to zero but we must pay special attention to the vertex x_{1}x_{2} in the image. As δ_{1}(x_{2}, 0) = x_{1}x_{2} = δ_{1}(0, x_{1}) we have that δ_{1}(ax_{2}, −ax_{1}) = 0 and so the diagram for Ω_{kE}^{2}(k) is in fact given by:

\xymatrix{· \ar[dr]_{x_{2}} &  & · \ar[dl]^{x_{1}} \ar[dr]_{x_{2}} &  & · \ar[dl]^{x_{1}} \\  & · &  & ·}

Example 2.4.15. We now give a very brief description of Ω^{−2}(k) using the above example. Performing the appropriate computations gives us a module with the following diagram:

\xymatrix{ & · \ar[dl]_{x_{1}} \ar[dr]^{x_{2}} &  & · \ar[dl]_{x_{1}} \ar[dr]^{x_{2}} \\ · &  & · &  & ·}

So we see that Ω^{−2}(k) may be obtained simply by ‘turning the diagram upside down’. In fact this process will in general give a simple method of computing Ω(k) from Ω^{−1}(k) and vice-versa.

One of the main reasons Ω and Ω^{−1} will prove so useful to us is the following theorem, which appears as Theorem 4.5 of [15].

Theorem 2.4.16. Let M, N ∈ kG-mod and let n be a positive integer. Then

1. Every cohomology element ζ ∈ Ext_{kG}^{n}(M, N) is represented by a homomorphism ζ̂: Ω^{n}(M) → N.

2. Every homomorphism ζ̂ : Ω^{n}(M) → N represents a cohomology element class (ζ̂) ∈ Ext_{kG}^{n}(M, N).

3. Two such homomorphisms ζ̂, ζ̃ represent the same class if and only if ζ̂− ζ̃ factors through a projective kG-module. Note that this implies the two morphisms are equivalent in the stable category.

We also note the following useful properties of Ω and Ω^{−1} (see Proposition 4.4 of [15] for details):

Proposition 2.4.17. Let M, N ∈ kG-mod, P a projective kG-module, H a subgroup of G, L ∈ kH-mod and n, m integers. Then

1. Ω^{n}(P) = 0.

2. Ω^{n}(M) has no non-zero projective submodules.

3. Ω^{n}(M ⊕ N) ≅ Ω^{n}(M) ⊕ Ω^{n}(N).

4. Ω^{n}(M)^{∗} ≅ Ω^{−n}(M^{∗}).

5. Ω^{n}(Ω^{m}(M)) ≅ Ω^{n+m}(M).

6. Ω^{m}(M) ⊗ Ω^{n}(N) ≅ Ω^{n+m}(M ⊗ N) ⊕ Q, for some projective kG-module Q.

Relative Projectivity

The use of relative projectivity will prove to be the key step in our search for modules of small dimension with variety a quadratic. Throughout this section G will be a finite group, k an algebraically closed field and H a subgroup of G.

Our main use for relative projectivity will be to combine these ideas with Ω and Ω^{−1} but before considering this relation we must first define relative projectivity in general. The contents of this section come from Chapter 3 of [3].

Definition 2.4.18. A module M in kG-mod is termed H-projective or relatively H-projective if, given kG-modules M_{1}, M_{2}, a map f : M → M_{1} and an epimorphism g : M_{2} → M_{1} such that there exists a map of kH-modules h : M ↓_{H}→ M_{2} ↓ H with f = g ∘ h, then there exists a map of kG-modules h′ : M → M_{2} with f = g ∘ h′. This is illustrated by the following diagram.

\xymatrix{ & M \ar[d]^{f} \ar@{-->}[dl]_{h′} \\ M_{2} \ar@{->>}[r]^{g} & M_{1}}

A kG-module M is said to be injective relative to H or relatively H-injective if whenever we are given modules M_{1} and M_{2}, a map f : M_{1} → M and a monomorphism g : M_{1} → M_{2} such that there exists a map of kH-modules h : M_{2} ↓_{H}→ M ↓_{H} with f = h ∘ g, then there exists a map of kG-modules h′ : M_{2} → M with f = h′ ∘ g. We can again represent this using the following diagram.

\xymatrix{M \\ M_{1} \ar[u]^{f} \ar[r]^{g} & M_{2} \ar@{-->}[ul]_{h′}}

We say that a short exact sequence of kG-modules is H-split if it splits on restriction to H.

Definition 2.4.19. A relatively H-projective resolution of a kG-module M is a long exact sequence of kG-modules

\xymatrix{⋯ \ar[r] & P_{2} \ar[r]^{δ_{2}} & P_{1} \ar[r]^{δ_{1}} & P_{0}}

such that the following conditions hold:

1. P_{0}/Im(δ_{1}) ≅ M

2. Each P_{i} is projective relative to H

3. Each sequence

\xymatrix{0 \ar[r] & P_{n+1}/ Ker(δ_{n+1}) \ar[r] & P_{n} \ar[r] & Im(δ_{n}) \ar[r] & 0}

as well as

\xymatrix{0r \ar[r] & Im(δ_{1}) \ar[r] & P_{0} \ar[r] & M \ar[r] & 0}

splits on restriction to H.

Given these definitions it is important to know that, if M is a kG module, such objects will always exist.

Proposition 2.4.20. Given H a subgroup of a group G, every module M in kG-mod has a relatively H-projective resolution.

Proposition 2.4.21. Every module M in kG-mod has a minimal H-projective resolution and a minimal H-injective resolution, where a H-projective resolution

\xymatrix{⋯ \ar[r] & P_{2} \ar[r]^{δ_{2}} & P_{1} \ar[r]^{δ_{1}} & P_{0} \ar@{->>}[r]^{ϵ} & M \ar[r] & 0}

if minimal if and only if Im δ_{i} has no nonzero H-projective direct summands for all δ_{i}.

We are now able to give the definitions of relative Ω and Ω^{−1}.

Definition 2.4.22. For M in kG-mod and n ∈ ℤ define Ω_{(kG,kH)}^{n}(M) as in the definition of Ω_{kG}^{n}(M), but using H-projective and H-injective resolutions instead of projective and injective resolutions.

We also set Ω_{(kG,kH)}^{0}(M) = Ω_{(kG,kH)}^{n}(Ω_{(kG,kH)}^{−n}(M)).

Having defined relative Ω and Ω^{−1} we now give a result that will allow us to make useful computations later (see Section 3.1.1). We first prepare for this with the following proposition.

Proposition 2.4.23. Let M be a kG-module and H a subgroup of G. Then the following are equivalent:

1. M is projective relative to H, 2. Every H-split epimorphism of kG-modules splits,

3. M is injective relative to H,

4. M is a direct summand of M ↓_{H}↑^{G},

5. M is a direct summand of some module induced from H.

Proposition 2.4.24. Suppose that k is an algebraically closed field of characteristic p and E = E_{1} × E_{2} is an elementary abelian p-group of rank r, i.e. E ≅ (ℤ/pℤ)^{r}. Then Ω_{(kE,kE_{2})}(k) ↓_{kE_{1}} ≅ Ω_{kE_{1}}(k) and Ω_{(kE,kE_{2})}(k) ↓_{kE_{2}} has trivial action.

Proof. Suppose we have the following minimal projective resolution of k as a kE_{1}- module:

\xymatrix{⋯ \ar[r] & a_{2}kE_{1} \ar[r] & a_{1}kE_{1} \ar[r] & kE_{1} \ar[r] & k \ar[r] & 0} .

Recalling that kE_{1} ⊗ kE_{2} ≅ kE and using Proposition 2.4.23 we have that kE_{1} and each of the a_{i}kE_{1} are E_{2}-projective. This follows from the fact that kE_{1} is a direct summand of kE_{1} ↓_{kE_{2}}↑ kE. We also have that the above complex is a minimal E_{2}- projective resolution of k as a kE-module. If this were not the case then there must exist a smaller E_{2} projective resolution and we may use the fact that kE_{1} ⊗kE_{2} ≅ kE to produce a minimal resolution of kE contradicting Theorem 2.4.9, so the result holds.

Example 2.4.25. Let k be an algebraically closed field of characteristic 2, with E being a product of 6 copies of the cyclic group of order 2. So we have E ≅ (ℤ/2ℤ)^{6} ≅ ⟨g_{1}, …, g_{6}⟩ and kE ≅ k[x_{1}, …, x_{6}]/(x_{1}^{2}, …, x_{6}^{2}), where x_{i} = g_{i} − 1. Now we must take two subgroups each corresponding to different sets of generators, so let E_{1} = ⟨g_{1}, g_{2}, g_{3}⟩, E_{2} = ⟨g_{4}, g_{5}, g_{6}⟩.

Considering k as a kE-module we obtain a minimal E_{2}-projective resolution of k:

\xymatrix{⋯ \ar[r] & 6kE_{1} \ar[r] & kE_{1} \ar[r] & k \ar[r] & 0}

By Proposition 2.4.24 we have that Ω_{(kE,kE_{2})}^{n}(k) ↓_{kE_{1}} ≅ Ω_{kE_{1}}^{n}(k).

Note that the resulting Ω_{(kE,kE_{2})}^{n}(k) will be substantially smaller than the corresponding Ω_{kE}^{n}(k). In this case we have that Ω_{(kE,kE_{2})}(k) is a 7-dimensional kE-module, whereas Ω_{kE}(k) is a 63-dimensional kE-module. We will see later (See Section 3.1.1) that this will prove a key technique for constructing modules of small dimension with a given variety.

Cohomology of Elementary Abelian p-Groups

We now reach the heart of this section, a brief reminder of group cohomology in the context of our elementary abelian p-groups. The cohomology behaves particularly nicely in this case and we will see later that it is extremely useful both when discussing rank varieties and in our search for modules of small dimension having a particular variety.

For our purposes the following proposition, appearing in Section 3.5 of [22], will be sufficient to describe the cohomology of any group we are working with.

Proposition 2.4.26. Let E ≅ (ℤ/pℤ)^{r} be an elementary abelian p-group of rank r. Then the cohomology of E with coefficients in k has two forms, depending on whether the characteristic of k is even or odd.

If k has characteristic 2 then the cohomology is a polynomial ring in generators of degree 1, i.e. H^{∗}(E, k) ≅ k[X_{1}, …, X_{r}], where the X_{i} are in degree 1.

If however k has odd characteristic then the cohomology is the tensor product of an exterior algebra with degree 1 generators and a polynomial ring with degree 2 generators, i.e. H^{∗}(E, k) ≅ Λ[Y_{1}, …, Y_{r}] ⊗ k[X_{1}, …, X_{r}], where the Y_{i} are in degree 1 and the X_{i} in degree 2.

When we eventually come to look at modules with variety a quadratic in characteristic two we will find that we need even more information on the cohomology ring. We present it now to allow clarity later. Note that the information here comes from [3].

Suppose we are given E an elementary abelian 2-group of rank r and k an algebraically closed field of characteristic 2. We know from above that H^{∗}(E, k) ≅ k[X_{1}, …, X_{r}], with the X_{i} elements in degree one.

Recalling Theorem 2.4.16, if we set M = N = k we recover a very interesting result. The first statement of the theorem now says that every cohomology element in Ext_{kE}^{n}(k, k), i.e. the group cohomology H^{n}(E, k), is represented by a homomorphism from Ω^{n}(k) to k. Setting n = 1 we have that each of the X_{i} is represented by a homomorphism X̂_{i} : Ω(k) → k.

The maps X̂_{i} : Ω(k) → k are defined as follows:

X̂_{i} : Ω(k) → k,
x_{i} ↦ 1,
b ↦ 0 for all other basis elements b.

Now, if we write kE ≅ k[x_{1}, …, x_{r}]/(x_{1}^{2}, …, x_{r}^{2}) then a basis for Ω_{kE}(k) is given by

B = {x_{1}^{a_{1}} ⋯ x_{r}^{a_{r}} : a_{i} ∈ {0, 1}, ∑_{i=1}^{r} a_{i} > 0}.

As it will be useful later we quickly mention here that a basis for Ω_{kE}^{−1}(k) is given by

B_{2} = {1} ∪ {x_{1}^{a_{1}} ⋯ x_{r}^{a_{r}} : a_{i} ∈ {0, 1}, r > ∑_{i=1}^{r} a_{i} > 0}.

So we see that in fact the maps X̂_{i} are in duality with the x_{i} and therefore so are the original X_{i} themselves. Note that in characteristic 2 we will often identify Z_{i}, the co-ordinate functions on affine n-space, and X_{i}.

We also need to investigate the effect of the functors Ω and Ω^{−1} on these maps. A concrete description of the maps Ω(X̂_{i}) can be obtained by completing the following commutative diagram:

\xymatrix{0 \ar[r] & Ω(M_{1}) \ar[r] \ar[d]^{Ω(X̂_{1})} & P_{1} \ar[r] \ar[d] & M_{1} \ar[r] \ar[d]^{X̂_{1}} & 0 \\ 0 \ar[r] & Ω(M_{2}) \ar[r] & P_{2} \ar[r] & M_{2} \ar[r] & 0}

Recall that Ω(M_{1}), Ω(M_{2}) are unique only up to addition of projective summands. In particular they are unique if we pass to the stable module category. As it will come in extremely useful later we now also provide two quick examples of this computation:

Example 2.4.27. Let E be the elementary abelian 2-group of rank 2, with k an algebraically closed field of characteristic 2. We will describe explicitly the map Ω(X̂_{1}).

Firstly recall the structures of the group algebra kE, Ω(k) and Ω^{2}(k):

kE : \xymatrix{ & 1 \ar[dl]_{x_{1}} \ar[dr]^{x_{2}} \\ x_{1} \ar[dr]_{x_{2}} & & x_{2} \ar[dl]^{x_{1}} \\ & x_{1}x_{2}}

Ω(k) : \xymatrix{x_{1} \ar[dr]_{x_{2}} & & x_{2} \ar[dl]^{x_{1}} \\ & x_{1}x_{2}}

Ω^{2}(k) : \xymatrix{(x_{1}, 0) \ar[dr]_{x_{2}} & & (x_{2}, x_{1}) \ar[dl]^{x_{1}} \ar[dr]_{x_{2}} & & (0, x_{2}) \ar[dl]^{x_{1}} \\ & (x_{1}x_{2}, 0) & & (0, x_{1}x_{2})}

The appropriate commutative diagram is as follows:

\xymatrix{0 \ar[r] & Ω^{2}(k) \ar[d]^{Ω(X̂_{1})} \ar[r]^{f_{1}} & 2kE \ar[r]^{f_{0}} \ar[d]^{h} & Ω(k) \ar[d]^{X̂_{1}} \ar[r] & 0 \\ 0 \ar[r] & Ω(k) \ar[r]^{g_{1}} & kE \ar[r]^{g_{0}} & k \ar[r] & 0}

We begin by determining h using the commutativity of the diagram. We have

X̂_{1} ∘ f_{0}(1, 0) = 0,
X̂_{1} ∘ f_{0}(x_{1}, 0) = 1,
X̂_{1} ∘ f_{0}(x_{2}, 0) = 0,
X̂_{1} ∘ f_{0}(x_{1}x_{2}, 0) = 0.
X̂_{1} ∘ f_{0}(0, 1) = 0,
X̂_{1} ∘ f_{0}(0, x_{1}) = 0,
X̂_{1} ∘ f_{0}(0, x_{2}) = 0,
X̂_{1} ∘ f_{0}(0, x_{1}x_{2}) = 0.

Therefore defining h : 2kE → kE by

(1, 0) ↦ 1, (x_{1}, 0) ↦ x_{1}, (x_{2}, 0) ↦ x_{2},

(x_{1}x_{2}, 0) ↦ x_{1}x_{2}, (0, 1) ↦ 0, (0, x_{1}) ↦ 0, (0, x_{2}) ↦ 0,

(0, x_{1}x_{2}) ↦ 0,

gives us that X̂_{1} ∘ f_{0} = g_{0} ∘ h. We now use h ∘ f_{1} to determine Ω(X̂_{1}). We have

h ∘ f_{1}(x_{1}, 0) = x_{1},
h ∘ f_{1}(x_{1}x_{2}, 0) = x_{1}x_{2},
h ∘ f_{1}(x_{2}, x_{1}) = x_{2},
h ∘ f_{1}(0, x_{1}x_{2}) = 0,
h ∘ f_{1}(0, x_{2}) = 0.

Finally we may define Ω(X̂_{1}) : Ω^{2}(k) → Ω(k) by

(x_{1}, 0) ↦ x_{1},
(x_{1}x_{2}, 0) ↦ x_{1}x_{2},
(x_{2}, x_{1}) ↦ x_{2},
(0, x_{1}x_{2}) ↦ 0,
(0, x_{2}) ↦ 0.

Example 2.4.28. We now give a brief example similar to that above but this time we determine the map Ω^{−1}(X̂_{1}). The procedure is essentially the same as that for Ω(X̂_{1}) but reversed. We take the following commutative diagram:

\xymatrix{0 \ar[r] & Ω(k) \ar[r]^{f_{1}} \ar[d]^{X̂_{1}} & kE \ar[r]^{f_{0}} \ar[d]^{h} & k \ar[r] \ar[d]^{Ω^{−1}(X̂_{1})} & 0 \\ 0 \ar[r] & k \ar[r]^{g_{1}} & kE \ar[r]^{g_{0}} & Ω^{−1}(k) \ar[r] & 0}

Commutativity of the diagram again allows us to uniquely determine the maps h and Ω^{−1}(X̂_{1}).

2.5 Varieties for Modules

We now find ourselves in a position to describe the key objects we will be working with, varieties for modules. We will begin by defining the standard cohomological variety of a module. Having done this we will then move on to the case of elementary abelian p-groups, for which there is a variety far easier to compute, the rank variety. Finally we will show that the rank variety and the cohomological variety, when both defined, in fact coincide.

Note that the results in this section come from Chapter 5 of [5]. Our definition of the cohomological variety is slightly different (we will give a precise comparison later) but the variety itself is unaffected. The details of this now follow.

2.5.1 Cohomological Varieties

The first issue we have to tackle with a cohomological definition of a variety comes from looking at cohomology in odd characteristic. Recall from earlier that, given a finite group over an algebraically closed field of odd characteristic, the cohomology ring is graded-commutative. We would like to work with a strictly commutative ring. Thankfully there is a very nice trick we can use, appearing in section 2.3.1 of [7], to show that graded-commutativity is in fact enough.

Lemma 2.5.1. A graded-commutative ring R is commutative modulo its nil radical.

Proof. We begin by noting that even degree elements are commutative so it suffices to take x ∈ R of odd degree and show that x ≡ −x modulo the nil radical.

Firstly graded-commutativity gives us that x.x = −x.x. Therefore 2x^{2} = 0, so (2x)^{2} = 0.

Now suppose that y, z ∈ R. Then (y(2x)z)^{2} = ±y^{2}(2x)^{2}z^{2} = 0, where the first equality comes from using graded commutativity and the second from the fact that (2x)^{2} = 0. We therefore have that 2x is in the nil radical, therefore x ≡ −x modulo the nil radical.

We now find ourselves with a very natural way to define a variety given by the group cohomology.

Definition 2.5.2. We define V_{G} = max spec H^{∗}(G, k), i.e. we take the maximal ideal spectrum of the group cohomology.

Remark 2.5.3. The definition given in [5] uses H^{ev}(G, k), the subring of even degree cohomology elements, in the place of H^{∗}(G, k). This is to resolve the issue of only having graded-commutativity rather than commutativity. As the odd degree elements all lie within the nil radical of H^{∗}(G, k) for p odd the resulting varieties are the same. Working with H^{ev}(G, k) has several drawbacks. For example the Kunneth formula no longer holds and H^{ev}(E, k) of an elementary abelian group may not be Gorenstein. Note that this will not impact what follows but our definition is chosen to match that more commonly used in the literature.

2.5.2 Varieties for Modules

Now that we have our definition of cohomological variety we wish to provide some way to associate subvarieties to particular modules. The following definition associates the trivial module to the standard cohomological variety. It also has the rather nice property that every closed homogeneous subvariety of V_{G} is given by V_{G}(M) for some kG-module M (see Corollary 5.9.2 of [5] for details).

Definition 2.5.4. Let M ∈ kG-Mod.

There is a natural map

Φ : H^{∗}(G, k) = Ext_{kG}(k, k) ⟶^{⊗M} Ext_{kG}(M, M).

We denote the kernel of Φ by I_{G}(M).

Then V_{G}(M) = max spec H^{∗}(G, k)/I_{G}(M).

We now briefly mention some important results on the effect of subgroups, tensor products and direct sums of modules on their corresponding varieties. Note that no useful results on general submodules currently appear to exist. These results appear as Proposition 5.7.5, Proposition 5.7.10 and Theorem 5.7.11 of [5].

Proposition 2.5.5. Let G, G_{1}, G_{2} be finite groups, k an algebraically closed field, M, N finitely-generated kG-modules, M_{1}, M_{2} finitely-generated kG_{i}-modules and P a projective kG-module. Then:

1. V_{G}(M ⊕ N) ≅ V_{G}(M) ∪ V_{G}(N).

2. V_{G}(M ⊗ N) ≅ V_{G}(M) ∩ V_{G}(N).

3. V_{G_{1}×G_{2}} ≅ V_{G_{1}} × V_{G_{2}}.

4. V_{G_{1}×G_{2}}(M_{1} ⊗ M_{2}) ≅ V_{G_{1}}(M_{1}) ⊗ V_{G_{2}}(M_{2}).

5. V_{G}(M ⊕ P) ≅ V_{G}(M).

Before proceeding to define rank varieties we first provide an example of the cohomological variety of a module. It should be noted that we choose a particularly simple example as the computation is in general a very difficult one. We will see shortly that a theorem of Quillen allows us to use rank varieties to make such computations far easier.

Example 2.5.6. Let E be an elementary abelian 2-group of rank r and k an algebraically closed field of characteristic 2. We then have that H^{∗}(E, k) ≅ k[X_{1}, …, X_{r}]. The maximal ideal spectrum can be described in the following manner.

Suppose α = (α_{1}, …, α_{r}) ∈ k^{r} is an arbitrary element of affine r-space. Now let I_{α} = {f ∈ k[X_{1}, …, X_{r}] : f(α) = 0}. Then I_{α} is a maximal ideal in H^{∗}(E, k) and all maximal ideals are of this form. We therefore have that V_{E}(k) ≅ k^{r}.

2.5.3 Rank Varieties

We now move on to defining rank varieties themselves. It is worth noting that while at first glance the rank variety of a module may appear rather difficult to compute we shall see from some examples that in general it is far easier to compute than the ‘standard’ cohomological variety.

We first prepare for the definition with the following lemma.

Lemma 2.5.7. Let E = (ℤ/pℤ)^{r} ≅ ⟨g_{1}, …, g_{r}; g_{i}^{p} = 1, g_{i}g_{j} = g_{j}g_{i}⟩ and k an algebraically closed field of characteristic p. Then the linear subspace V_{E}^{♯} of J(kE) (where J(kE) denotes the Jacobson radical) spanned by (g_{1} − 1), …, (g_{r} − 1) has dimension r and is isomorphic to J(kE)/J(kE)^{2}.

Definition 2.5.8. Let E = (ℤ/pℤ)^{r}, k an algebraically closed field of characteristic p and M a finitely-generated kE-module. Define V_{E}^{♯}(M), the rank variety of M, by

V_{E}^{♯}(M) = {0} ∪ {v ∈ V_{E}^{♯} : M_{↓⟨1+v⟩} not free}.

We now come to our key results in the theory of rank varieties. Firstly we must take a brief moment to discuss the Bockstein map. Further details can be found in Section 4.3 of [5].

Definition 2.5.9. The Bockstein map is a particular connecting homomorphism β : H^{r}(G, k) → H^{r+1}(G, k). We first define the maps β : H^{r}(G, ℤ_{p}) → H^{r+1}(G, ℤ_{p}) before generalising.

The maps β : H^{r}(G, ℤ_{p}) → H^{r+1}(G, ℤ_{p}) are defined to be the connecting homomorphisms associated to the following short exact sequence:

\xymatrix{0 \ar[r] & ℤ_{p} \ar[r] & ℤ/p^{2} \ar[r] & ℤ_{p} \ar[r] & 0}

Given a field k we now define the Bockstein maps β : H^{r}(G, k) → H^{r+1}(G, k). Noting that we may write any element in the form ∑_{i} λ_{i}x_{i}, with λ_{i} ∈ k, x_{i} ∈ H^{n}(G, ℤ_{p}), we then set β(∑_{i} λ_{i}x_{i}) = ∑_{i} λ_{i}^{p}β(x_{i}), for degree one elements, i.e. elements of H^{1}(G, ℤ_{p}). In particular when p = 2 we have that β(x) is just the cup product square.

We now describe the relationship between the rank variety and the cohomological variety, appearing as Lemma 5.8.2 and Theorem 5.8.3 of [5].

Theorem 2.5.10.

V_{E} ≅ V_{E}^{♯}.
V_{E}(M) ≅ V_{E}^{♯}(M).

For p = 2 this is an isomorphism of varieties, for p > 2 it is a bijection but no longer invertible as a map of varieties.

Remark 2.5.11. The problem with the p > 2 case is that the isomorphism involves the Bockstein and therefore the Frobenius map. In the case p = 2 the proof uses the fact that the group cohomology is a polynomial algebra with generators in degree 1. For p odd the group cohomology is the tensor product of an exterior algebra with generators in degree 1 and a polynomial algebra with generators in degree 2, so the Bockstein β : H^{1}(G, k) → H^{2}(G, k) provides us with a means of working with the polynomial algebra.

Our second key result, appearing as Corollary 8.33 in [22], provides further motivation for studying elementary abelian p-groups. It shows that understanding of varieties for modules over elementary abelian p-groups can in fact be used to understand varieties for modules over arbitrary finite groups.

Theorem 2.5.12 (Quillen’s Dimension Theorem). Let E_{p}(G) be the set of all elementary abelian p-subgroups of G. Then

V_{G}(k) = ⋃_{E∈E_{p}(G)} res_{G,E}^{∗}(V_{E}(k)),

where res_{G,E}^{∗} indicates the map from V_{E}(k) to V_{G}(k) induced by the restriction map from G to E.

2.5.4 The Structure of the Cohomological Variety

The previous result, whilst useful, does little to describe exactly how the various components fit together. The details are given in a theorem of Avrunin and Scott [1]. We will first need some preparatory definitions and results.

The Avrunin-Scott Theorem is a varieties for modules analogue of the Quillen Stratification Theorem for the cohomological variety of a group. We will therefore begin by detailing the Stratification Theorem. The information here comes from [26] and [27]. Throughout this section we will use the language of [5].

The Quillen Stratification Theorem

The Quillen Stratification theorem shows how the cohomological variety of a finite group G may be constructed using the cohomological varieties of elementary abelian subgroups of G. Before we may proceed we must first define some related varieties, giving us a nicer way of considering the information retained by varieties of restrictions from G to elementary abelian subgroups.

Definition 2.5.13.

V_{G,E} := res_{G,E}^{∗} V_{E} ⊆ V_{G}
V_{G,E}(M) := res_{G,E}^{∗} V_{E}(M) ⊆ V_{G}(M).
V_{E}^{+} := V_{E}\ ⋃_{E′<E} res_{E,E′}^{∗} V_{E′}.
V_{G,E}^{+} = res_{G,E}^{∗} V_{E}^{+}.
V_{E}^{+}(M) := V_{E}^{+} ∩ V_{E}(M).
V_{G,E}^{+}(M) := res_{G,E}^{∗} V_{E}^{+}(M).

We will also require the following proposition to understand how the cohomological variety is altered by restrictions from elementary abelian p-groups to subgroups.

Proposition 2.5.14. If E′ ≤ E are elementary abelian p-groups and M is a kEmodule, then regarding V_{E′} as embedded in V_{E} via res_{E,E′}^{∗}, we have

V_{E′}(M) = V_{E′} ∩ V_{E}(M).

Remark 2.5.15. The proof given in [5] utilises the following two equalities that may help to explain the Stratification Theorem.

V_{E}^{+}(M) = V_{E}(M)\ ⋃_{E′<E} res_{E,E′}^{∗} V_{E′}(M).

V_{G}(M) = ⋃_{E≤G} res_{G,E}^{∗} V_{E}^{+}(M).

Theorem 2.5.16 (Quillen’s Theorem). The variety V_{G} is the disjoint union of locally closed subvarieties V_{G,E}^{+}, one for each conjugacy class of elementary abelian subgroups E ≤ G. Let W_{G}(E) = N_{G}(E)/C_{G}(E). Then W_{G}(E) acts on V_{E} and V_{E}^{+}, and res_{G,E}^{∗} induces an inseparable isogeny

V_{E}^{+}/W_{G}(E) → V_{G,E}^{+}.

Remark 2.5.17. In our case being an ‘inseparable isogeny’ can be thought of merely as being a homeomorphism under the Zariski topology. It should be noted that a far more general definition exists but we omit the details as it will not be required here.

The Avrunin-Scott Theorem

We now state the Avrunin-Scott Theorem in a manner that demonstrates its similarity to the original Stratification Theorem.

Theorem 2.5.18. Suppose M is a finitely generated kG-module. Then the variety V_{G}(M) is the disjoint union of locally closed subvarieties V_{G,E}^{+}(M), one for each conjugacy class of elementary abelian subgroups E ≤ G. The group W_{G}(E) acts on V_{E}(M) and V_{E}^{+}(M), and res_{G,E}^{∗} induces an inseparable isogeny

V_{E}^{+}(M)/W_{G}(E) → V_{G,E}^{+}(M).

2.5.5 Computing the Rank Variety

We now give a small example of computing a rank variety. The computation is made considerably easier by using the following Lemma, which is a restatement of Lemma 4.2 of [13]. Lemma 2.5.19. Let E ≅ (ℤ/pℤ)^{r} ≅ k[x_{1}, …, x_{r}]/(x_{1}^{p}, …, x_{r}^{p}) and M be a kEmodule. Suppose α = α_{1}x_{1} + ⋯ + α_{r}x_{r}, where α_{i} ∈ k and A denotes the matrix of the action of α on M. Then rank(A) ≤ \frac{p−1}{p} dim M, with equality if and only if α ∉ V_{E}^{♯}(M). Proof. Since ⟨α+1⟩ is a cyclic group of order p, we know that M ↓_{⟨α+1⟩}≅ L_{1}⊕⋯⊕L_{t}, where each L_{i} is a cyclic, indecomposable k⟨α + 1⟩-module. Each L_{i} has dimension at most p and is projective if and only if its dimension is p.

Also dim(αL_{i}) = (dim L_{i}) − 1 ≤ \frac{p−1}{p} dim L_{i}. The result follows.

Example 2.5.20. Let E = (ℤ/2ℤ)^{4} ≅ ⟨g_{1}, …, g_{4}; g_{i}^{2} = 1, g_{i}g_{j} = g_{j}g_{i}⟩, k be a field of characteristic 2, M be the module defined by the following matrices:

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

g_{3} ↦ (\begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 1 \end{matrix}) , g_{4} ↦ (\begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \end{matrix}) .

An arbitrary element v ∈ V_{E}^{♯} has form

v = λ_{1}(g_{1} − 1) + λ_{2}(g_{2} − 1) + λ_{3}(g_{3} − 1) + λ_{4}(g_{4} − 1)

and is represented by the matrix

(\begin{matrix}0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ λ_{1} & λ_{2} & 0 & 0 \\ λ_{3} & λ_{4} & 0 & 0 \end{matrix}) .

So the restriction M_{↓⟨1+v⟩} is free/projective (Recall from Theorem 2.4.4 that the two terms are interchangeable here) if and only if the matrix has rank two. To see this recall that the maximal rank of such a matrix over an algebraically closed field of characteristic p is given by \frac{(p−1) dim M}{p} = \frac{4}{2} = 2. Therefore V_{E}^{♯}(M) is given by the following equation in 𝔸^{4}(k)

Z_{1}Z_{4} + Z_{2}Z_{3} = 0,

where the Z_{i} are the co-ordinate functions on 𝔸^{4}(k). This follows as working with E an elementary abelian p-group gives us that being free and projective are equivalent, by Theorem 2.4.4. Therefore the restriction to a cyclic shifted subgroup is not projective if and only if it is not free.

2.6 L_{ζ} Modules

This section describes one of our key motivations. Given any homogeneous polynomial f ∈ k[X_{1}, …, X_{r}], where k is an algebraically closed field of positive characteristic p and r ∈ ℕ, Carlson’s L_{ζ} construction (originally detailed in [11]) provides us with a module having f as variety (meaning V_{E}^{♯}(L_{ζ}) = V (f)). The modules obtained are often extremely large. As we will show later when working over elementary abelian p-groups we can almost always produce modules of far smaller dimension having the same variety.

The construction of the L_{ζ} relies upon the cohomological variety of a group, defined earlier. It is worth noting that although the resulting modules are large the process can be performed for any finite group, giving it far greater generality than some constructions we will show later, which are limited to elementary abelian p-groups. The construction relies on the isomorphism H^{n}(G, k) ≅ Hom_{kG}(Ω^{n}k, k) (see Theorem 2.4.16).

The following results come from Chapter 5, Section 8 of [5] and throughout this section G will be a finite group.

Definition 2.6.1. Suppose that 0 ≠ ζ ∈ H^{n}(G, k) and write ζ̂ for the image of ζ under the above isomorphism. As ζ̂ maps onto a submodule of k and k is simple, we have that ζ̂ is surjective. Define L_{ζ} to be the kernel of ζ̂.

Remark 2.6.2. Though unimportant for our purposes, we should note that in the event that ζ = 0 we define L_{ζ} = Ω^{n}k ⊕ Ωk.

Our main use for the modules L_{ζ} comes from the following theorem:

Theorem 2.6.3. The variety V_{G}(L_{ζ}) is the hypersurface V_{G}⟨ζ⟩ in V_{G} determined by ζ (in other words, the set of maximal ideals containing ζ).

Equipped with this construction we will now provide some examples in the case of elementary abelian p-groups. It should become apparent that the size of L_{ζ} grows extremely quickly with respect to the degree of ζ.

Example 2.6.4. We will work over an algebraically closed field k of characteristic 2, choosing our elementary abelian p-group E to be a product of two copies of the cyclic group of order two. This gives us that kE is the truncated polynomial algebra k[x_{1}, x_{2}]/(x_{1}^{2}, x_{2}^{2}), where the x_{i} each have degree 1. The cohomology ring H^{∗}(E, k) is simply the polynomial ring in two variables k[X_{1}, X_{2}].

For our first example we will take the degree 2 function f = X_{1}X_{2}. The minimal projective resolution of k over kE has the following form:

\xymatrix{⋯ \ar[r] & 2kE \ar[r]^{δ} & kE \ar[r]^{ϵ} & k \ar[r] & 0}

As f has degree 2 it will correspond to a surjection from Ω^{2}(k) to k. Now Ω(k) is the kernel of a surjective map from dimension p^{2} to dimension 1. Therefore it will have dimension p^{2} − 1. As the resolution is a complex, we must now have that δ maps onto Ω(k), giving a map from dimension 2p^{2} to dimension p^{2} − 1. So Ω^{2}(k) has dimension p^{2} + 1. Finally, since f corresponds to a surjection from dimension p^{2} + 1 to dimension 1, we conclude that the corresponding L_{ζ} has dimension p^{2} = 4.

A natural question now arises of how to obtain dimensions of L_{ζ} modules in odd characteristic and with higher ranks. We shall now provide an example of each of these in turn.

Example 2.6.5. For this example we shall remain in rank two but our field will instead have characteristic 3. We will therefore choose E to be a product of two copies of the cyclic group of order 3. So we now have that kE is a different truncated polynomial algebra, namely k[x_{1}, x_{2}]/(x_{1}^{3}, x_{2}^{3}). The cohomology ring has a very different structure, being a tensor product of an exterior algebra and a polynomial algebra. We have H^{∗}(E, k) ≅ Λ(X_{1}, X_{2}) ⊗_{k} k[Y_{1}, Y_{2}], where the X_{i} have degree 1 and the polynomial generators Y_{i} now have degree 2. This means that our polynomial f = Y_{1}Y_{2} will now correspond to a degree 4 element of cohomology and therefore to a map from Ω^{4}(k) to k.

The minimal projective resolution of k over kE has form

\xymatrix{⋯ \ar[r] & 4kE \ar[r]^{δ_{3}} & 3kE \ar[r]^{δ_{2}} & 2kE \ar[r]^{δ_{1}} & kE \ar[r]^{ϵ} & k \ar[r] & 0}

We then use the following diagram to obtain dimensions for the Ω^{n}(k):

\xymatrix{⋯ \ar[r] & 5kE \ar@{^{(}->}[dr] \ar[rr]^{δ_{4}} &  & 4kE \ar[rr]^{δ_{3}} \ar@{^{(}->}[dr] &  & 3kE \ar[rr]^{δ_{2}} \ar@{^{(}->}[dr] &  & 2kE \ar[rr]^{δ_{1}} \ar@{^{(}->}[dr] &  & kE \ar[r]^{ϵ} & k \ar[r] & 0 \\  &  & Ω^{4}(k) \ar@{->>}[ur] &  & Ω^{3}(k) \ar@{->>}[ur] &  & Ω^{2}(k) \ar@{->>}[ur] &  & Ω(k) \ar@{->>}[ur]}

Computing the dimensions iteratively we obtain:

dim(kE) = p^{2},
dim(Ω(k)) = p^{2} − 1,
dim(2kE) = 2p^{2},
dim(Ω^{2}(k)) = p^{2} + 1,
dim(3kE) = 3p^{2},
dim(Ω^{3}(k)) = 2p^{2} − 1,
dim(4kE) = 4p^{2},
dim(Ω^{4}(k)) = 2p^{2} + 1.

So we eventually find that L_{ζ} must have dimension 2p^{2} = 2 × 9 = 18.

In fact there exists a 6 dimensional kE-module having f as variety. It arises from defining the action of α_{1}x_{1} + α_{2}x_{2} by the following matrix:

( \begin{matrix}0 & 0 & 0 & 0 & 0 & 0 \\ α_{1} & 0 & 0 & 0 & 0 & 0 \\ 0 & α_{1} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & α_{2} & 0 & 0 \\ 0 & 0 & 0 & 0 & α_{2} & 0 \end{matrix} ) .

As we are working over an algebraically closed field of characteristic 3 the maximal rank of such a matrix is given by \frac{(p−1) dim M}{p} = \frac{2 dim M}{3} = 4, so we determine the variety by computing the 4 × 4 minors. We therefore see that the L_{ζ} modules are already far from the minimal dimension even in this very small example.

Remark 2.6.6. In the previous example it may be tempting to think there is a 3dimensional kE-module also having f as variety, with the action of α_{1}x_{1} + α_{2}x_{2} given by the matrix

( \begin{matrix}0 & 0 & 0 \\ α_{1} & 0 & 0 \\ 0 & α_{2} & 0 \end{matrix} ) .

The problem is that the matrices for the action of x_{1} and the action of x_{2} do not commute.

Example 2.6.7. Our final example will look at a group of rank 4. For clarity we return to our algebraically closed field of characteristic 2. We now choose E to be a product of four copies of the cyclic group of order 2, which leads kE to be the truncated polynomial algebra k[x_{1}, x_{2}, x_{3}, x_{4}]/(x_{1}^{2}, x_{2}^{2}, x_{3}^{2}, x_{4}^{2}). The cohomology ring is now simply the polynomial ring k[X_{1}, X_{2}, X_{3}, X_{4}], where the X_{i} all have degree 1.

We choose our polynomial to be f = X_{1}X_{2} + X_{3}X_{4}, which will now correspond to a degree 2 element of cohomology and therefore to a map from Ω^{2}(k) to k. The minimal resolution is as follows:

\xymatrix{⋯ \ar[r] & 10kE \ar[rr]^{δ_{1}} \ar@{^{(}->}[dr] &  & 4kE \ar[rr]^{δ_{1}} \ar@{^{(}->}[dr] &  & kE \ar[r]^{ϵ} & k \ar[r] & 0 \\  &  & Ω^{2}(k) \ar@{->>}[ur] &  & Ω(k) \ar@{->>}[ur]}

We again compute the dimensions iteratively, giving us:

dim(kE) = p^{4},
dim(Ω(k)) = p^{4} − 1,
dim(4kE) = 4p^{4},
dim(Ω^{2}(k)) = 3p^{4} + 1.

So we eventually find that L_{ζ} must have dimension 3p^{4} = 3 × 16 = 48.

Again note that there exists a 4-dimensional module with the same variety. It is obtained by letting α_{1}x_{1} + ⋯ + α_{4}x_{4} act via the following matrix:

( \begin{matrix}0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ α_{1} & α_{3} & 0 & 0 \\ α_{4} & α_{2} & 0 & 0 \end{matrix} ) .

2.7 Loewy Length

We now come to a brief introduction to Loewy Length. The Loewy Length of a module is an interesting invariant and we will see later that there is in fact a connection between a module’s Loewy Length and its variety.

The context for the remainder of this section is as follows. We work with a rank r elementary abelian p-group E ≅ (ℤ/pℤ)^{r} ≅ ⟨g_{1}, …, g_{r}⟩. Recall that the group algebra kE is then a truncated polynomial algebra k[x_{1}, …, x_{r}]/(x_{1}^{p}, …, x_{r}^{p}), where x_{i} = g_{i} − 1 for 1 ≤ i ≤ r.

We begin by defining Loewy Length itself. The definition appears as Definition 8.1 in [25].

Definition 2.7.1 (Loewy Length). Let R be an Artinian ring and M an R-module with radical given by Rad(M). Then the Loewy length of M is the minimal n ∈ ℕ such that Rad^{n}(M) = 0.

The following two lemmas will prove extremely useful later, the first appearing as a remark on pages 28, 29 of [25].

Lemma 2.7.2. Let M be a kE-module of Loewy length p, where k also has characteristic p, with A being the matrix representing the action of x_{α} = α_{1}x_{1} + ⋯ + α_{r}x_{r} on M. Then there exists a choice of basis such that A is lower triangular.

Lemma 2.7.3. If M is a kE-module of Loewy length p whose variety is not the whole of affine space, with k being an algebraically closed field of characteristic p, then there exists a choice of basis such that the the restriction to some (fixed) α_{1}x_{1} + ⋯ + α_{r}x_{r} is given by the following matrix:

T = ( \begin{matrix}0 & 0 \\ I & 0 \end{matrix} ), where dim(I) = dim(M)(p − 1)/p.

Proof. Since the variety of M is not the whole of affine space we may choose a point α = (α_{1}, …, α_{r}) not in the variety of M. Let A be the matrix representing the action of x_{α} = α_{1}x_{1} + ⋯ + α_{r}x_{r} on M. Then A^{p} = 0 and if we put A into Jordan canonical form then all of its Jordan blocks have length p. This follows from the fact that α was not in the variety of M, therefore M ↓_{⟨1+x_{α}⟩} is projective. To see this, recall that Theorem 2.4.4 gives us that M ↓_{⟨1+x_{α}⟩} is projective if and only if it is free, which occurs when the matrix A has maximal rank. So A is conjugate to T.

Remark 2.7.4. Note that our matrix T in fact induces an isomorphism between M/Rad^{(p−1)}(M) and Rad(M).

We will also later require some results on the socle of a module, a definition closely linked to the radical. We use the definition appearing as Definition 8.1 of [25].

Definition 2.7.5. If M is a kE-module then let S_{1}(M) denote the maximal semisimple submodule of M and define S_{i}(M) successively by

S_{i}(M)/S_{i−1}(M) = S_{1}(M/S_{i−1}(M)).

We call S_{1}(M) the socle of M, denoted Soc(M), and say S_{i} is the ith socle layer of M.

The following result, appearing as Lemma 8.2 of [25], gives a useful relation between the radical and socle of a module.

Lemma 2.7.6. Let m be the Loewy length of M, so that Rad^{m}(M) = 0. Then Rad^{r}(M) ⊆ S_{m−r}(M) for all 0 ≤ r ≤ m if we set S_{0}(M) = 0. This also gives us that S_{m}(M) = S_{m−0}(M) = Rad^{0}(M) = M.

2.8 Matrix Factorisations

We now give a brief introduction to matrix factorisations. We shall see later that these provide us with a means of constructing modules of small dimension with a particular variety.

Please note that the definitions in this section are those given in [2]. The fundamental definition we will be using is that of a matrix factorisation.

Definition 2.8.1 (Matrix Factorisation). Let f ∈ k[y_{1}, …, y_{n}] be a homogeneous polynomial of degree d ≥ 2, where y_{1}, …, y_{n} are indeterminates. A (linear) matrix factorisation of f (of size m) is an equation (f)I = α_{1} ⋯ α_{d}, where the α_{i} are m × m matrices whose entries are linear forms in y_{1}, …, y_{n} with coefficients in k and I is the m × m identity matrix.

Definition 2.8.2 (Equivalence of Matrix Factorisations). Two matrix factorisations α_{1}, …, α_{d} and β_{1}, …, β_{d} of a polynomial f are said to be equivalent if there exists a collection of invertible matrices γ_{1}, …, γ_{d} such that γ_{i}α_{i}γ_{i+1}^{−1} = β_{i} for all 1 ≤ i ≤ d, where we set γ_{d+i} = γ_{i} for all i ∈ ℕ.

Remark 2.8.3. There are several different definitions of matrix factorisation in the literature and great care must be taken to ensure consistency. For example many of the earlier papers on the subject, including Eisenbud’s seminal paper [21], restrict the factorisation to a pair of matrices.

We now give two brief examples of small matrix factorisations.

Example 2.8.4. Suppose k is an algebraically closed field of characteristic p and f = y_{1}y_{2} − y_{3}y_{4}. Then the following pair of matrices gives a factorisation of f:

α_{0} = ( \begin{matrix}y_{1} & y_{3} \\ y_{4} & y_{2} \end{matrix} ) , α_{1} = ( \begin{matrix}y_{2} & −y_{3} \\ −y_{4} & y_{1} \end{matrix} ) .

To prove this we simply observe that

α_{0}α_{1} = α_{1}α_{0} = ( \begin{matrix}f & 0 \\ 0 & f \end{matrix} ) .

Example 2.8.5. Suppose k is an algebraically closed field of characteristic 2 and f = y_{1}y_{3}y_{5} + y_{2}y_{4}y_{6}. Then the following three matrices give a matrix factorisation of f:

A_{0} = ( \begin{matrix}y_{5} & ζ^{2}y_{2} & 0 \\ 0 & y_{3} & ζy_{4} \\ y_{6} & 0 & y_{1} \end{matrix} ) , A_{1} = ( \begin{matrix}y_{1} & y_{2} & 0 \\ 0 & y_{5} & ζ^{2}y_{4} \\ ζy_{6} & 0 & y_{3} \end{matrix} ) ,

A_{2} = ( \begin{matrix}y_{3} & ζy_{2} & 0 \\ 0 & y_{1} & y_{4} \\ ζ^{2}y_{6} & 0 & y_{5} \end{matrix} ),where ζ is a cube root of unity in k.

Note that in this case the order of the matrices becomes important. We have

A_{0}A_{1}A_{2} = A_{1}A_{2}A_{0} = A_{2}A_{0}A_{1} = (f)I_{3×3},

whereas A_{0}A_{2}A_{1} ≠ (f)I_{3×3}.

Given a polynomial we will need to construct a matrix factorisation corresponding to it. There will usually be a great many options so in general we want to try and choose our matrices to be as small as possible. The constructions we use rely heavily on the use of Clifford Algebras.

Definition 2.8.6 (Universal Clifford Algebra). Given a homogeneous polynomial f ∈ k[x_{1}, …, x_{n}] of degree d ≥ 2 over an algebraically closed field k of positive characteristic we define the universal Clifford algebra C(f) of f by

C(f) := T (V)/⟨\{\underbrace{x ⊗ ⋯ ⊗ x}_{d times} −f_{V} (x) : x ∈ V \}⟩, where T denotes the tensor algebra.

Here V is an n-dimensional k-vectorspace with basis e_{1}, …, e_{n} and f_{V} : V → k is the map defined by

f_{V} (x_{1}e_{1} + ⋯ + x_{n}e_{n}) = f(x_{1}, …, x_{n}).

Definition 2.8.7 (Generalised Clifford Algebra). Suppose that C(f) is the universal Clifford algebra for some polynomial f. Then for any homogeneous two-sided ideal I in C(f) such that I ∩ C_{1} = 0 we call the quotient C(f)/I a generalised Clifford algebra for f (where C_{1} denotes the degree 1 part of C(f)). Note that A generalised Clifford algebra is ℤ/dℤ-graded by the grading induced from that on T (V).

We now come to the definition of twisted tensor product. The definition is followed by the motivating theorem, appearing as Theorem 3.1 of [2]. We shall see that twisted tensor products allow us to combine generalised Clifford algebras in a manner that corresponds simply with their associated matrix factorisations.

Definition 2.8.8 (Twisted Tensor Product). Let f_{1}, f_{2} be degree d polynomials over an algebraically closed field k in disjoint sets of variables, with C_{1}, C_{2} generalised Clifford algebras for f_{1}, f_{2} respectively. Let ζ ∈ k be a primitive d′-th root of unity (where char(k) = p and d′ is the p′-part of d).

Then we define the ℤ/dℤ-graded tensor product C_{1} ˆ⊗C_{2} as the ordinary tensor product with multiplication defined by

(a ˆ⊗b)(c ˆ⊗d) = ζ^{(degb)(degc)}ac ˆ⊗bd

where b ∈ C_{2}, c ∈ C_{1} are homogeneous.

In the case that char(k) = d we have that ˆ⊗ is the same as the standard tensor product. As we shall see later (See Section 3.2.2) the reason for the twisted tensor product is to allow us to use properties of primitive d′ roots of unity. If char(k) = d we can simply use the fact that ∑_{i=1}^{d} 1 = 0.

Theorem 2.8.9. Let f_{1} and f_{2} be homogeneous polynomials of degree d in disjoint sets of variables. If C_{i} is a generalised Clifford algebra for f_{i} (i = 1, 2), then C := C_{1} ˆ⊗C_{2} is a generalised Clifford algebra for f := f_{1} + f_{2}.

Before we proceed we need one last definition, that of a Clifford module (given in [23]).

Definition 2.8.10. A Clifford module (for f a degree d homogeneous polynomial) is a ℤ/dℤ-graded C(f)-module M = ⨁_{z∈ℤ/dℤ} M_{z} for which dim_{k}(M) < ∞.

Given these definitions we are now able to describe the correspondence between matrix factorisations and Clifford algebras. The following theorem is the key result, appearing as an unnumbered theorem in Section 1 of [23].

Theorem 2.8.11. If f(x) ≠ 0 for some x ∈ k^{n}, then the equivalence classes of matrix factorisations α = (α_{1}, …, α_{d}) of f, where the entries of the α_{i} are linear forms, correspond bijectively to the isomorphism classes of finite-dimensional Clifford modules for f.

When it comes to making use of this correspondence later it will prove very useful to know when the universal Clifford algebra is of finite dimension and when there exists at least one finite matrix factorisation. The following results appear as Theorem 1.8 of [2] and Corollary 1.7 of [23].

Theorem 2.8.12. Let f be a homogeneous polynomial of degree d ≥ 2, consisting of a sum of n terms. Then we have

1. The natural map V → C(f) is injective. In particular, C(f) ≠ 0.

2. dim_{k}(C(f)) < ∞ if and only if n ≤ 1 or d = 2.

Theorem 2.8.13. Every homogeneous form f ≠ 0 has a Clifford module M ≠ 0.

The above results show that we will in general want to work with Clifford modules rather than universal Clifford algebras, where possible.

We will see later that understanding matrix factorisations and their correspondence with Clifford algebras/modules provides us with a means of constructing modules of small dimension with variety given by an irreducible homogeneous polynomial.

Modules of Small Dimension

This chapter is split into three distinct sections. Firstly we restrict our attention to looking at modules with variety an irreducible quadratic. Using a construction similar to that of Carlson’s L_{ζ} construction (Recall Section 2.6) we find modules that are in general far smaller than the corresponding L_{ζ} modules.

In the next section we generalise to looking at modules with variety given by irreducible polynomials of arbitrary degree whilst working in arbitrary (strictly positive) characteristic. The construction we give relies heavily on matrix factorisations. Having described this construction we then compare it to that obtained in the previous section and find that, restricting to the case where both constructions are defined, they give non-isomorphic modules (with the exception of the two smallest cases).

Finally we then present a result giving a relation between the variety of a module and its possible Loewy length. Note that our attention will be restricted to looking at kE-modules of Loewy length p with k an algebraically closed field of characteristic p and E an elementary abelian p-group.

3.1 Quadratics

We begin by constructing modules with variety an irreducible quadratic. This allows us to work with fairly small modules and perform more explicit computations than when we later look at polynomials of greater degree.

Given a polynomial f = Z_{1}Z_{(n+1)} + Z_{2}Z_{(n+2)} + ⋯ + Z_{n}Z_{2n} over an algebraically closed field k of characteristic 2, with E ≅ (ℤ/2ℤ)^{2n} ≅ ⟨g_{1}, …, g_{2n}⟩, we will construct a kE-module L_{n} of dimension 2^{n} such that the variety of L_{n} is given by the polynomial f, i.e. V (L_{n}) = V (f).

3.1.1 Modules for general quadratics

Introduction

In this section we will investigate a generalisation of Carlson’s L_{ζ} construction to provide matrices of small (perhaps minimal) dimension with variety a chosen quadratic.

As we wish to work with an arbitrary number of terms we will first establish some suitable notation.

Firstly let E_{n} be the elementary abelian 2-group of rank 2n, so that E_{n} ≅ (ℤ/2ℤ)^{2n} ≅ ⟨g_{1}, …, g_{2n}⟩ and kE_{n} ≅ k[x_{1}, …, x_{2n}]/(x_{1}^{2}, …, x_{2n}^{2}). We define f = X_{1}X_{(n+1)} + ⋯ + X_{n}X_{2n} and set E_{(n,1)} and E_{(n,2)} to be the subgroups generated by g_{1}, …, g_{n} and g_{n+1}, …, g_{2n} respectively. Recall that H^{∗}(E, k) ≅ k[X_{1}, …, X_{2n}], where the X_{i} are in duality with the x_{i}.

Our desired construction will also benefit from a more concrete description of Ω_{kE_{(n,1)}}(k). Ω_{kE_{(n,1)}}(k) has dimension 2^{n} − 1 with basis given by

B = {x_{1}^{a_{1}}x_{2}^{a_{2}} ⋯ x_{n}^{a_{n}} : a_{1}, …, a_{n} ∈ {0, 1}, ∑_{i=1}^{n} a_{i} > 0}.

The action of kE_{(n,1)} on Ω_{kE_{(n,1)}}(k) is given by

x_{i}(x_{1}^{a_{1}}x_{2}^{a_{2}} ⋯ x_{i}^{a_{i}} ⋯ x_{n}^{a_{n}}) = (x_{1}^{a_{1}}x_{2}^{a_{2}} ⋯ x_{i}^{a_{i}+1} ⋯ x_{n}^{a_{n}}) (Note x_{i}^{2} = 0).

Before we proceed we note that, whilst the characteristic of k is 2, we may safely identify Z_{i}, the co-ordinate functions on affine 2n-space, and X_{i}. However when the characteristic of k is odd the structure of the group cohomology changes and we will need to be more careful (See Section 2.5).

We are now in a position to give details of the construction.

Construction of the L_{n}

The idea here is essentially a generalisation of that seen earlier for Carlson’s L_{ζ} construction. Whereas that uses single elements of the group cohomology, we instead use matrices whose entries are these cohomology elements, with determinant the desired polynomial. This provides us with a large number of modules to study, many of which prove to be considerably smaller than the corresponding L_{ζ} modules. It should also be noted that our construction can not at first glance be guaranteed to give modules with the correct variety. We will see later that some considerable work is involved in proving that we do obtain the varieties we want.

The idea for our general construction comes from working through a small example. Before giving this it may prove helpful to recall the definition of some crucial maps. Remark 3.1.1. Suppose k is an algebraically closed field of characteristic 2 and we are working with E_{3} ≅ (ℤ/2ℤ)^{6} ≅ ⟨g_{1}, …, g_{6}⟩, giving us kE_{3} ≅ k[x_{1}, …, x_{6}]/(x_{1}^{2}, …, x_{6}^{2}), where x_{i} = g_{i} − 1. Recall E_{(3,2)} ≅ (ℤ/2ℤ)^{3} ≅ ⟨g_{4}, g_{5}, g_{6}⟩.

The maps X̂_{i}, Ω_{kE_{3}}^{−1}(X̂_{i}) and Ω_{kE_{(3,2)}}^{−1}(X̂_{i}) are then defined as follows (for an example of a similar computation, see Section 2.4.27):

X̂_{i} : Ω_{kE_{3}}(k) → k,
X̂_{i}(x_{i}) = 1_{k},
X̂_{i}(b) = 0 for all other basis elements b.

Ω_{kE_{3}}^{−1}(X̂_{i}) : k → Ω_{kE}^{−1}(k),
Ω_{kE_{3}}^{−1}(X̂_{1})(1) = x_{2}x_{3}x_{4}x_{5}x_{6},
Ω_{kE_{3}}^{−1}(X̂_{2})(1) = x_{1}x_{3}x_{4}x_{5}x_{6},
Ω_{kE_{3}}^{−1}(X̂_{3})(1) = x_{1}x_{2}x_{4}x_{5}x_{6},
Ω_{kE_{3}}^{−1}(X̂_{4})(1) = x_{1}x_{2}x_{3}x_{5}x_{6},
Ω_{kE_{3}}^{−1}(X̂_{5})(1) = x_{1}x_{2}x_{3}x_{4}x_{6},
Ω_{kE_{3}}^{−1}(X̂_{6})(1) = x_{1}x_{2}x_{3}x_{4}x_{5}.

Ω_{kE_{(3,2)}}^{−1}(X̂_{i}) : k → Ω_{kE_{(3,2)}}^{−1}(k),
Ω_{kE_{(3,2)}}^{−1}(X̂_{4})(1) = x_{5}x_{6},
Ω_{kE_{(3,2)}}^{−1}(X̂_{5})(1) = x_{4}x_{6},
Ω_{kE_{(3,2)}}^{−1}(X̂_{6})(1) = x_{4}x_{5}.

Example 3.1.2. The smallest interesting example is the polynomial f = X_{1}X_{4} + X_{2}X_{5} + X_{3}X_{6}, with our choice of group being E_{3}.

We divide the construction into two distinct halves. The first is to generalise Carlson’s L_{ζ} construction to use matrices of elements of cohomology rather than elements of cohomology themselves. The second is to use relative syzygies rather than syzygies to reduce the dimension of the module we construct.

1. Matrices of elements of cohomology: Given the polynomial f = X_{1}X_{4} + X_{2}X_{5} + X_{3}X_{6} Carlson’s L_{ζ} construction (See Section 2.6) considers f as a degree 2 element of cohomology. Recalling from Section 2.4.1 that each of the X_{i} is represented by a homomorphism X̂_{i} : Ω(k) → k, the next step in the L_{ζ} construction would be to consider f as the surjective linear map

f̂: Ω^{2}(k) → k,
f̂ = X̂_{1} ∘ Ω(X̂_{4}) + X̂_{2} ∘ Ω(X̂_{5}) + X̂_{3} ∘ Ω(X̂_{6}),

then consider the kernel of this map. The variety of the module defined by the kernel is then given by our polynomial f.

Instead of using a single element of cohomology we now consider a matrix of elements with determinant f,

A = ( \begin{matrix}1 & 0 & 0 & X̂_{1} \\ 0 & 1 & 0 & X̂_{2} \\ 0 & 0 & 1 & X̂_{3} \\ X̂_{4} & X̂_{5} & X̂_{6} & 0 \end{matrix} ) .

We wish to interpret A as a linear map, say ϕ, with our goal being to consider the kernel as in the L_{ζ} construction. Recalling that the X̂_{i} are linear maps from Ω(k) → k our first attempt may be to try the following:

ϕ : k^{3} ⊕ Ω(k) → k^{4}.

This doesn’t quite work as the fourth row of our matrix doesn’t make sense. We therefore apply the functor Ω^{−1} to the bottom row of our matrix, giving us:

A′ = ( \begin{matrix}1 & 0 & 0 & X̂_{1} \\ 0 & 1 & 0 & X̂_{2} \\ 0 & 0 & 1 & X̂_{3} \\ Ω^{−1}(X̂_{4}) & Ω^{−1}(X̂_{5}) & Ω^{−1}(X̂_{6}) & 0 \end{matrix} ) ,

where Ω^{−1}(X̂_{i}) is a map from k to Ω^{−1}(k).

Working with this modified matrix we may now define a linear map with matrix A′

ϕ′ : k^{3} ⊕ Ω(k) → k^{3} ⊕ Ω^{−1}(k).

Writing d = λ_{1}x_{1} + λ_{2}x_{2} + λ_{3}x_{3} + ⋯ (other terms) for an arbitrary element of Ω(k) we have:

ϕ′(a, b, c, d) = (a + X̂_{1}(d), b + X̂_{2}(d), c + X̂_{3}(d), Ω^{−1}(X̂_{4})(a) + Ω^{−1}(X̂_{5})(b) + Ω^{−1}(X̂_{6})(c)),

= (a + λ_{1}, b + λ_{2}, c + λ_{3}, ax_{1}x_{2}x_{3}x_{5}x_{6} + ax_{1}x_{2}x_{3}x_{4}x_{6} + cx_{1}x_{2}x_{3}x_{4}x_{5}).

Returning to Carlson’s L_{ζ} construction we see that we want to work with a surjective linear map. Our map ϕ′ fails to be surjective as the fourth component of the image does not contain every basis element of Ω^{−1}(k). Fortunately the variety of a module is unchanged if we add a projective as a direct summand (see Proposition 2.5.5). We therefore add a copy of kE_{3} to the domain (Recall that free and projective are equivalent for kE_{3}-modules by Theorem 2.4.4) to give the following map:

ϕ̄: k^{3} ⊕ Ω(k) ⊕ kE_{3} → k^{3} ⊕ Ω^{−1}(k),
ϕ̄(a, b, c, d, e) = (a + X̂_{1}(d), b + X̂_{2}(d), c + X̂_{3}(d),
Ω^{−1}(X̂_{4})(a) + Ω^{−1}(X̂_{5})(b) + Ω^{−1}(X̂_{6})(c) + e).

We now have a surjective map. We will prove later that taking the kernel of this map gives us a kE_{3}-module with variety V (f).

2. Relative syzygies: We now move on to the second half of our construction. The rough idea is that as a relative syzygy will be smaller than a standard syzygy we can reduce the dimension of the kernel of our map and therefore the dimension of the module we construct. Returning to the previous part, we had a map

ϕ′ : k^{3} ⊕ Ω_{kE_{3}}(k) → k^{3} ⊕ Ω_{kE_{3}}^{−1}(k),
ϕ′(a, b, c, d) = (a + X̂_{1}(d), b + X̂_{2}(d), c + X̂_{3}(d),
Ω_{kE_{3}}^{−1}(X̂_{4})(a) + Ω_{kE_{3}}^{−1}(X̂_{5})(b) + Ω_{kE_{3}}^{−1}(X̂_{6})(c)).

Looking at the definition of the map ϕ′ we see that the first three components contain no occurrences of X̂_{4}, X̂_{5} or X̂_{6}, whereas the final component contains no occurrence of X̂_{1}, X̂_{2} or X̂_{3}. Recalling Proposition 2.4.24 we see that we can use the fact that our group is a direct product E_{3} ≅ E_{(3,1)}×E_{(3,2)} to replace syzygies with relative syzygies. In particular we have that Ω_{(kE_{3},kE_{(3,1)})}(k) ↓_{kE_{(3,2)}} ≅ Ω_{kE_{(3,2)}}(k) and Ω_{(kE_{3},kE_{(3,2)})}(k) ↓_{kE_{(3,1)}} ≅ Ω_{kE_{(3,1)}}(k).

We therefore replace the Ω_{kE_{3}}(k) in the domain with Ω_{(kE_{3},kE_{(3,2)})}(k) and the Ω_{kE_{3}}^{−1}(k) in the co-domain with Ω_{(kE_{3},kE_{(3,1)})}^{−1}(k). This gives us a new map

ϕ_{rel} : k^{3} ⊕ Ω_{(kE_{3},kE_{(3,2)})}(k) → k^{3} ⊕ Ω_{(kE_{3},kE_{(3,1)})}^{−1}(k).

We choose bases B_{1} of Ω_{(kE_{3},kE_{(3,2)})}(k) and B_{2} of Ω_{(kE_{3},kE_{(3,1)})}^{−1}(k) as follows (Recall Section 2.4.1):

B_{1} = {x_{1}, x_{2}, x_{3}, x_{1}x_{2}, x_{1}x_{3}, x_{2}x_{3}, x_{1}x_{2}x_{3}},
B_{2} = {1, x_{4}, x_{5}, x_{6}, x_{4}x_{5}, x_{4}x_{6}, x_{5}x_{6}},

where the x_{i} represent cosets.

If we let b = λ_{1}x_{1} + λ_{2}x_{2} + λ_{3}x_{3} + λ_{4}x_{1}x_{2} + λ_{5}x_{1}x_{3} + λ_{6}x_{2}x_{3} + λ_{7}x_{1}x_{2}x_{3}

then we may write ϕ_{rel} as the following map:

ϕ_{rel}(a_{1}, a_{2}, a_{3}, b) = (a_{1} + X̂_{1}(b), a_{2} + X̂_{2}(b), a_{3} + X̂_{3}(b),
Ω^{−1}(X̂_{4})(a_{1}) + Ω^{−1}(X̂_{5})(a_{2}) + Ω^{−1}(X̂_{6})(a_{3}))

= (a_{1} + λ_{1}, a_{2} + λ_{2}, a_{3} + λ_{3}, a_{1}x_{5}x_{6} + a_{2}x_{4}x_{5} + a_{3}x_{4}x_{6}).

It is important to pause here to remark that we cannot be certain at this stage that replacing Ω, Ω^{−1} with relative Ω, Ω^{−1} will not alter the variety of the module given by the kernel of ϕ_{rel}. We will prove later that the module we construct does indeed have the desired variety.

We again find that our map is not surjective as the fourth component of the image does not contain every basis element of Ω_{(kE_{3},kE_{(31)})}^{−1}(k). In our earlier construction we solved this problem by adding a direct summand of kE to the domain. However as we are now using Ω_{(kE_{3},kE_{(3,1)})}^{−1}(k) in place of Ω_{kE_{3}}^{−1}(k) in the co-domain we instead add a copy of kE_{(3,1)}.

Again it is not clear that adding a relative projective will not alter the variety of the module given by the kernel. However use of a relative projective results in a smaller kernel and we will show later that the variety does indeed remain unchanged.

We now have a map

ϕ̄_{rel} : k^{3} ⊕ Ω_{(kE_{3},kE_{(3,2)})}(k) ⊕ kE_{(3,1)} → k^{3} ⊕ Ω_{(kE_{3},kE_{(3,1)})}^{−1}(k).

If we let

c = c_{0} + c_{1}x_{4} + c_{2}x_{5} + c_{3}x_{6} + c_{4}x_{4}x_{5} + c_{5}x_{4}x_{6} + c_{6}x_{5}x_{6} + c_{7}x_{4}x_{5}x_{6},

then this map is given by:

ϕ̄_{rel}(a_{1}, a_{2}, a_{3}, b, c) = (a_{1} + X̂_{1}(b), a_{2} + X̂_{2}(b),

a_{3} + X̂_{3}(b), Ω^{−1}(X̂_{4})(a_{1}) + Ω^{−1}(X̂_{5})(a_{2}) + Ω^{−1}(X̂_{6})(a_{3}) + c)

= (a_{1} + λ_{1}, a_{2} + λ_{2}, a_{3} + λ_{3},
c_{0} + c_{1}x_{4} + c_{2}x_{5} + c_{3}x_{6}
+ (a_{2} + c_{4})x_{4}x_{5} + (a_{3} + c_{5})x_{4}x_{6} + (a_{1} + c_{6})x_{5}x_{6}).

Now that we have a surjective map we may consider the kernel. Looking for solutions to the four equations

a_{1} + λ_{1} = 0,
a_{2} + λ_{2} = 0,
a_{3} + λ_{3} = 0,
c_{0} + c_{1}x_{4} + c_{2}x_{5} + c_{3}x_{6} + (a_{2} + c_{4})x_{4}x_{5} + (a_{3} + c_{5})x_{5}x_{6} + (a_{1} + c_{6})x_{5}x_{6} = 0,

we obtain the following equalities:

a_{1} = λ_{1} = c_{6},
a_{2} = λ_{2} = c_{4},
a_{3} = λ_{3} = c_{5},
c_{0} = c_{1} = c_{2} = c_{3} = 0.

Therefore we see that a typical element of Ker(ϕ̄_{rel}) has form

(c_{6}, c_{5}, c_{4},
c_{6}x_{1} + c_{5}x_{2} + c_{4}x_{3} + λ_{4}x_{1}x_{2} + λ_{5}x_{1}x_{3} + λ_{6}x_{2}x_{3} + λ_{7}x_{1}x_{2}x_{3},
c_{4}x_{4}x_{5} + c_{5}x_{4}x_{6} + c_{6}x_{5}x_{6} + c_{7}x_{4}x_{5}x_{6}).

We may re-name our variables to make this easier to read, writing a typical element of Ker(ϕ̄_{rel}) as:

(a_{1}, a_{2}, a_{3},
a_{1}x_{1} + a_{2}x_{2} + a_{3}x_{3} + a_{4}x_{1}x_{2} + a_{5}x_{1}x_{3} + a_{6}x_{2}x_{3} + a_{7}x_{1}x_{2}x_{3},
a_{3}x_{4}x_{5} + a_{2}x_{4}x_{6} + a_{1}x_{5}x_{6} + a_{8}x_{4}x_{5}x_{6}).

We now define L_{3} to be the 8 = 2^{3}-dimensional kE_{3}-module given by the kernel of ϕ̄_{rel}. As a more explicit description of L_{3} will help us to determine the general case, we note that a basis of L_{3} is given by the following:

b_{1} = (1, 0, 0, x_{1}, x_{5}x_{6}),  b_{5} = (0, 0, 0, x_{1}x_{3}, 0),
b_{2} = (0, 1, 0, x_{2}, x_{4}x_{6}),  b_{6} = (0, 0, 0, x_{2}x_{3}, 0),
b_{3} = (0, 0, 1, x_{3}, x_{4}x_{5}),  b_{7} = (0, 0, 0, x_{1}x_{2}x_{3}, 0),
b_{4} = (0, 0, 0, x_{1}x_{2}, 0),  b_{8} = (0, 0, 0, 0, x_{4}x_{5}x_{6}),

where each of the b_{i} is obtained by setting a_{i} = 1 and a_{j} = 0 for j ≠ i in the above description of a typical element of Ker(ϕ̄_{rel}). Keeping the basis in this form it is still not clear what the general construction should look like. We therefore produce the diagram for the module

\xymatrix{ & b_{4} \ar[dd]^{x_{3}} \\ b_{1} \ar[dd]^{x_{3}} \ar@{-->}[ur]_{x_{2}} \ar[drrr]^{x_{4}} &  & b_{2} \ar@{-->}[ul]^{x_{1}} \ar[dd]_{x_{3}} \ar[dr]^{x_{5}} \\  & b_{7} &  & b_{8} \\ b_{5} \ar@{-->}[ur]_{x_{2}} &  & b_{6} \ar@{-->}[ul]^{x_{1}} \\  & b_{3} \ar@{-->}[ul]_{x_{1}} \ar@{-->}[ur]^{x_{2}} \ar[uurr]_{x_{6}}}

Working solely from the diagram we are able to give a far nicer description of L_{3}. We can consider it as the kE_{3}-module with basis

{u_{1}^{a_{1}}u_{2}^{a_{2}}u_{3}^{a_{3}} : a_{1}, a_{2}, a_{3} ∈ {0, 1}, ∑_{i=1}^{3} a_{i} > 0} ∪ {v}

and kE_{3}-module action defined as follows:

x_{1}(u_{1}^{a_{1}}u_{2}^{a_{2}}u_{3}^{a_{3}}) = u_{1}^{(a_{1}+1)}u_{2}^{a_{2}}u_{3}^{a_{3}},
x_{2}(u_{1}^{a_{1}}u_{2}^{a_{2}}u_{3}^{a_{3}}) = u_{1}^{a_{1}}u_{2}^{(a_{2}+1)}u_{3}^{a_{3}},
x_{3}(u_{1}^{a_{1}}u_{2}^{a_{2}}u_{3}^{a_{3}}) = u_{1}^{a_{1}}u_{2}^{a_{2}}u_{3}^{(a_{3}+1)},
x_{1}(v) = 0,
x_{2}(v) = 0,
x_{3}(v) = 0,
If 1 ≤ i ≤ 3 : x_{i+3}(u_{i}) = v,
x_{i}(b) = 0 for all other basis elements,

where u_{i}^{a_{i}} = 0 if a_{i} ≥ 2.

We now wish to generalise this construction to quadratics with an arbitrary number of terms. Looking back to the previous example we see that the diagram for L_{3} is very similar to the diagram for kE_{(3,1)} itself, i.e.,

\xymatrix{ & x_{1}x_{2} \ar@{.>}[ddd]^{x_{3}} \\ x_{1} \ar[ur]_{x_{2}} \ar[ddd]^{x_{3}} &  &  & x_{2} \ar[ull]^{x_{1}} \ar[ddd]_{x_{3}} \\  &  & 1 \ar[ull]^{x_{1}} \ar[ur]^{x_{2}} \ar[ddd]_{x_{3}} \\  & x_{1}x_{2}x_{3} \\ x_{1}x_{3} \ar@{.>}[ur]_{x_{2}} &  &  & x_{2}x_{3} \ar@{.>}[ull]^{x_{1}} \\  &  & x_{3} \ar[ull]_{x_{1}} \ar[ur]^{x_{2}}}

Considering Ω_{kE_{(3,1)}}(k) takes us even closer to L_{3}

\xymatrix{ & x_{1}x_{2} \ar[dd]^{x_{3}} \\ x_{1} \ar[ur]^{x_{2}} \ar[dd]_{x_{3}} &  & x_{2} \ar[ul]_{x_{1}} \ar[dd]^{x_{3}} \\  & x_{1}x_{2}x_{3} \\ x_{1}x_{3} \ar[ur]^{x_{2}} &  & x_{2}x_{3} \ar[ul]_{x_{1}} \\  & x_{3} \ar[ul]^{x_{1}} \ar[ur]_{x_{2}}}

So we now see that L_{3} is essentially Ω_{kE_{(3,1)}}(k) with an added vertex reachable by x_{4}, x_{5}, x_{6}. This observation leads us to the generalised construction.

Definition 3.1.3. Given a quadratic f = X_{1}X_{(n+1)} + ⋯ + X_{n}X_{2n} we will mirror the above and define L_{n} to be obtained by taking Ω_{kE_{(n,1)}}(k), adding an additional basis element and extending the action in a similar fashion.

We therefore define L_{n} to be the 2^{n}-dimensional kE_{n}-module with basis

{u_{1}^{a_{1}}u_{2}^{a_{2}} ⋯ u_{n}^{a_{n}} : a_{1}, …, a_{n} ∈ {0, 1}, ∑_{i=1}^{n} a_{i} > 0} ∪ {v}.

The kE_{n}-module action on L_{n} is as follows

• If 1 ≤ i ≤ n:

x_{i}(u_{1}^{a_{1}}u_{2}^{a_{2}} ⋯ u_{n}^{a_{n}}) = u_{1}^{a_{1}}u_{2}^{a_{2}} ⋯ u_{i}^{a_{i}+1} ⋯ u_{n}^{a_{n}} (where u_{i}^{2} = 0),
x_{i}(v) = 0.

• If n + 1 ≤ i ≤ 2n:

x_{i}(u_{i−n}) = v,
x_{i}(b) = 0 for all other basis elements b.

The fact that this is well-defined follows from the fact that x_{i}^{2}(b) = 0 for all basis elements b, as u_{i}^{2} = 0 and, for n + 1 ≤ i ≤ 2n, x_{i}^{2}(u_{i−n}) = x_{i}(v) = 0. Note that restricting to the action of x_{1}, …, x_{n} gives us a description of Ω_{kE_{(n,1)}}(k), ignoring the basis element v.

Having defined the L_{n} our goal is now to prove the following theorem.

Theorem 3.1.4. The kE_{n}-module L_{n} has variety given by f.

The Varieties

Let n ∈ ℕ, n ≥ 3 and A be the matrix of the action of λ_{1}x_{1} + ⋯ + λ_{2n}x_{2n} on L_{n}.

Note that we may choose any ordering of the basis elements. As our field has characteristic two we may swap rows and columns without altering the determinant.

Remark 3.1.5. We ignore the case n = 1 as X_{1}X_{2} is reducible. The case n = 2 may be dealt with quite easily. We have X_{1}X_{3} + X_{2}X_{4} = 0. We may take the 4-dimensional kE_{2}-module given by the following matrices (where x_{i} = g_{i} − 1):

x_{1} ↦ (\begin{matrix}0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix}) ,  x_{2} ↦ (\begin{matrix}0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix}) ,

x_{3} ↦ (\begin{matrix}0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \end{matrix}) ,  x_{4} ↦ (\begin{matrix}0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 \end{matrix}) .

Before we may proceed with the proof of our main theorem we first need to prove a technical lemma.

Lemma 3.1.6. For 1 ≤ i ≤ n there exists a 2^{(n−1)} × 2^{(n−1)} minor M of A such that det(M) = λ_{i}^{(2^{(n−1)}−2)}(λ_{1}λ_{(n+1)} + ⋯ + λ_{n}λ_{2n}).

Proof. We begin by choosing two submatrices A_{1}, A_{2} of A such that det(A_{1}) =

λ_{i}^{(n−1)}(λ_{1}λ_{(n+1)} + ⋯ + λ_{n}λ_{2n}), A_{2} = ( \begin{matrix}λ_{i} & & 0 \\ & · \\ 0 & & λ_{i} \end{matrix} ).

Firstly we construct A_{1}, giving an ordering of the basis elements. Choose the columns {c_{k}} to be u_{1}, …, u_{n} and u_{1} ⋯ û_{i} ⋯ u_{n} (where û_{i} indicates we omit the element u_{i}). Then choose the rows {r_{k}} to be given by

{v} ∪ {u_{1} ⋯ u_{n}} ∪ {u_{i}u_{j} : j ≠ i}.

This gives A_{1} the following form:

( \begin{matrix}λ_{(n+1)} & · & · & · & · & λ_{2n} & 0 \\ 0 & & & & & & λ_{i} \\ λ_{1} & λ_{i} \\ · & & · \\ λ̂_{i} & & & · \\ · & & & & · \\ λ_{n} & & & & & λ_{i} & 0 \end{matrix} ) .

As this matrix is a little difficult to present clearly we will also provide an example. If we set n = 3 and i = 1 then A_{1} is the following matrix:

( \begin{matrix}λ_{4} & λ_{5} & λ_{6} & 0 \\ 0 & 0 & 0 & λ_{1} \\ λ_{2} & λ_{1} & 0 & 0 \\ λ_{3} & 0 & λ_{1} & 0 \end{matrix} ) .

We now need to construct A_{2}. The key consideration is that we wish to ensure the rows we choose are not reachable from the columns in A_{1}, i.e. if b_{1} is a basis element labelling a row in A_{1} and b_{2} a basis element labelling a row we choose, there should not exist any a ∈ kE_{n} such that a(b_{1}) = b_{2}. This makes combining the two matrices later a considerably easier task.

To this end we define A_{2} by ordering our basis as follows. Take columns c_{1}, …, c_{n} given by

{u_{1}^{a_{1}}u_{2}^{a_{2}} ⋯ u_{n}^{a_{n}} : a_{i} = 0, a_{j} ∈ {0, 1}, 1 < ∑_{j=1}^{n} a_{j} < n},

i.e. elements of degree 2, …, n−1 not containing u_{i}. Next choose the rows to be given

by r_{j} = u_{i}c_{j}. Then as all r_{j} and no c_{j} contain u_{i} we have that A_{2} = ( \begin{matrix}λ_{i} & & 0 \\ & · \\ 0 & & λ_{i} \end{matrix} ).

Now let M have rows {r_{j}}∪{r_{k}} and columns {c_{j}}∪{c_{k}}. Then considering degrees

of elements gives us that M = ( \begin{matrix}A_{1} & 0 \\ 0 & A_{2} \end{matrix} ). The result is now immediate.

We now find ourselves in a position where we may prove that our modules L_{n} do indeed have the desired varieties.

Theorem 3.1.4. The kE_{n}-module L_{n} has variety given by f.

Proof. We prove this result by showing containment of each variety within the other.

• V (L_{n}) ⊆ V (X_{1}X_{(n+1)} + ⋯ + X_{n}X_{2n}):

Given a = (a_{1}, …, a_{2n}) ∈ V (L_{n}), suppose it is not contained in V (X_{1}X_{(n+1)} + ⋯ + X_{n}X_{2n}), i.e., a_{1}a_{(n+1)} + ⋯ + a_{n}a_{2n} ≠ 0. Lemma 3.1.6 and the fact that a ∈ V (L_{n}) give us that a_{i}^{(2^{n−1}−2)}(a_{1}a_{(n+1)} + ⋯ + a_{n}a_{2n}) = 0 for 1 ≤ i ≤ n, as a ∈ V (L_{n}) means that all 2^{(n−1)} × 2^{(n−1)} minors of A must be zero (See Section 2.5.5). Since a_{1}a_{(n+1)} + ⋯ + a_{n}a_{2n} ≠ 0 we have a_{i} = 0 for all 1 ≤ i ≤ n. This contradicts our assumption that a_{1}a_{(n+1)} + ⋯ + a_{n}a_{2n} ≠ 0.

• V (X_{1}X_{(n+1)} + ⋯ + X_{n}X_{2n}) ⊆ V (L_{n}):

Let α = (α_{1}, …, α_{2n}) ∈ 𝔸^{2n}(k) be such that α ∈ V (X_{1}X_{(n+1)} + ⋯ + X_{n}X_{2n}).

If we had taken Ω and Ω^{−1} in the construction of the L_{n} instead of relative Ω_{(kE_{n},kE_{(n,1)})} and Ω_{(kE_{n},kE_{(n,2)})}^{−1} then the result would follow from Carlson’s L_{ζ} construction. Therefore it is sufficient to show that the use of relatives does not alter the variety of the resulting module, i.e. that α ∈ E_{(n,1)} or α ∈ E_{(n,2)} ⇒ α ∈ V (L_{n}). Note that by α ∈ E_{(n,1)} (and similarly α ∈ E_{(n,2)}) we mean that α ∈ kE_{(n,1)} is in the linear span of 1E_{(n,1)}.

To clarify, if we had not used relative projectives then any element of V (X_{1}X_{(n+1)} + ⋯ + X_{n}X_{2n}) would be contained in V (L_{n}) (this follows from Carlson’s L_{ζ} construction). If the use of relative projectives meant that this was no longer the case then it would mean that changing from Ω and Ω^{−1} to Ω_{(kE_{n},kE_{(n,1)})} and Ω_{(kE_{n},kE_{(n,2)})}^{−1}, respectively, had caused some element of V (X_{1}X_{(n+1)}+⋯+X_{n}X_{2n}) to be not an element of V (L_{n}), as these were the only alterations made during this step. Recalling Proposition 2.4.24 we have that Ω_{(kE_{n},kE_{(n,1)})}(k) ↓_{kE_{(n,2)}} ≅ Ω_{kE_{(n,2)}}(k) and Ω_{(kE_{n},kE_{(n,2)})}^{−1}(k) ↓_{kE_{(n,1)}} ≅ Ω_{kE_{(n,1)}}^{−1}(k). Considering each of these we see that the only elements for which this could happen would be elements of E_{(n,1)} or elements of E_{(n,2)}. We now prove that all such elements are in fact contained in V (L_{n}).

1. α ∈ E_{(n,1)}: The action of L_{n} ↓_{⟨α_{1}x_{1}+···+α_{n}x_{n}⟩} is

x_{i}(u_{1}^{a_{1}} ⋯ u_{n}^{a_{n}}) = u_{1}^{a_{1}} ⋯ u_{i}^{(a_{i}+1)} ⋯ u_{n}^{a_{n}},
x_{i}(v) = 0.

So L_{n} ↓_{⟨α_{1}x_{1}+···+α_{n}x_{n}⟩}= Ω_{(kE_{n},kE_{(n,2)})}(k) ⊕ k, which is not projective. To see this recall that the terms free, projective and injective are equivalent when working with kE_{n}-modules (See Theorem 2.4.4). So if Ω_{(kE_{n},kE_{(n,2)})}(k) ⊕ k was projective it would be free, therefore isomorphic to a direct sum of copies of kE. The lack of projectivity gives us that α_{1}x_{1} + ⋯ + α_{n}x_{n} ∈ V (L_{n}).

2. α ∈ E_{(n,2)}:

The kE_{n}-module action on L_{n} ↓_{⟨α_{(n+1)}x_{(n+1)}+···+α_{2n}x_{2n}⟩} is given by

x_{(n+i)}(u_{i}) = v,

x_{s}(b) = 0 for all other basis elements b, for all other x_{s}.

In particular the matrix of this action has only one non-zero row. Now \frac{dim L_{n}}{2} = \frac{2^{n}}{2} = 2^{n−1} > 1 (as n ≥ 3) so the rank of this matrix is less than \frac{dim L_{n}}{2} . Therefore L_{n} ↓_{⟨α_{(n+1)}x_{(n+1)}+···+α_{2n}x_{2n}⟩} is not projective. This follows from the fact that E is an elementary abelian 2-group, therefore being projective is equivalent to being free (Recall Theorem 2.4.4). So if L_{n} ↓_{⟨α_{(n+1)}x_{(n+1)}+···+α_{2n}x_{2n}⟩} were projective then it would be free and isomorphic to a direct sum of copies of kE. However the dimension is less than that of the dimension of kE. So (α_{(n+1)}, …, α_{2n}) ∈ V (L_{n}), i.e. V (X_{1}X_{(n+1)} + ⋯ + X_{n}X_{2n}) ⊆ V (L_{n}).

Remark 3.1.7. Having produced this theorem there are two obvious possible generalisations. The first is to allow our polynomials to have higher degree and the second is to allow the characteristic of our field to be arbitrary. We briefly discuss these each in turn.

The problem with considering polynomials of higher degree comes down to the difficulty of performing the calculations. For example suppose we have the polynomial Z_{1}Z_{2}Z_{3} + Z_{4}Z_{5}Z_{6} + Z_{7}Z_{8}Z_{9}. A possible matrix to begin with would then be

(\begin{matrix}1 & 0 & 0 & X_{1}X_{2} \\ 0 & 1 & 0 & X_{4}X_{5} \\ 0 & 0 & 1 & X_{7}X_{8} \\ X_{3} & X_{6} & X_{9} & 0 \end{matrix}) .

The difficulty comes from interpreting the X_{1}X_{2}, X_{4}X_{5} and X_{7}X_{8}. We could, for example, consider the first as X_{1} ∘ Ω(X_{2}). This would give us a map from Ω_{kE_{(4,1)}}^{2}(k) to k. However Ω_{kE_{(4,1)}}^{2}(k) is a 49-dimensional kE-module. So even in this fairly small case computations will become extremely difficult.

Working over an algebraically closed field of odd characteristic leads to a similar problem. If we return to our earlier example of Z_{1}Z_{4} + Z_{2}Z_{5} + Z_{3}Z_{6} then recall we used the following matrix:

(\begin{matrix}1 & 0 & 0 & X_{1} \\ 0 & 1 & 0 & X_{2} \\ 0 & 0 & 1 & X_{3} \\ X_{4} & X_{5} & X_{6} & 0 \end{matrix}) .

At first glance the problem appears no different than the characteristic 2 case. Unfortunately several of our objects are in fact rather different in characteristic 3. Firstly we will need to choose our elementary abelian p-group to be E ≅ (ℤ/3ℤ)^{6}, giving us a group algebra kE ≅ k[x_{1}, …, x_{6}]/(x_{1}^{3}, …, x_{6}^{3}). We also have that H^{∗}(E, k) ≅ k[X_{1}, …, X_{6}] ⊗ Λ(Y_{1}, …, Y_{6}), where the X_{i} are now generators in degree 2.

So the X_{i} therefore correspond to maps from Ω^{2}(k) to k and, in characteristic 3, Ω^{2}(k) will be notably bigger than in characteristic 2 (due to the different structure of the group algebra). Again the dimensions of the modules we need to work with make the computation considerably more difficult.

An alternative strategy may be to somehow generalise the fact that our modules are merely Ω_{kE_{(n,1)}}(k) with an additional basis element. The difficulty is in knowing what the appropriate generalisation should be. There seems to be no obvious method to adapt this idea for a different characteristic of field or degree of polynomial.

3.2 General Homogeneous Polynomials

We now move on to looking at modules with variety an irreducible homogeneous polynomial of arbitrary degree. Firstly we consider polynomials Z_{1} ⋯ Z_{n} + Z_{(n+1)} ⋯ Z_{2n}, where small modules prove particularly simple to construct. We then generalise to polynomials of form Z_{1}Z_{(n+1)} ⋯ Z_{((d−1)n+1)} + ⋯ + Z_{n}Z_{2n} ⋯ Z_{dn} by using matrix factorisations and show that, given E ≅ (ℤ/2ℤ)^{dn} and k an algebraically closed field of characteristic 2, there exists a 2d^{n−1} dimensional kE-module with variety given by f = Z_{1}Z_{(n+1)} ⋯ Z_{((d−1)n+1)} + ⋯ + Z_{n}Z_{2n} ⋯ Z_{dn}.

Finally we will compare the construction for quadratics obtained in the previous section with the construction in this section when d = 2. Using Loewy length we see that the modules are in fact non-isomorphic for all n ≥ 3.

3.2.1 Polynomials of form Z_{1} ⋯ Z_{n} + Z_{(n+1)} ⋯ Z_{2n}

The following brief section will deal with finding modules with variety given by particularly nice polynomials, namely polynomials of form Z_{1} ⋯ Z_{n} +Z_{(n+1)} ⋯ Z_{2n}. In this case we are able to prove that the modules we construct do have minimal dimension with corresponding variety. As the modules are particularly easy to define we will give one short example and then leap straight to the definition.

Before giving this example we first establish some notation. E will be an elementary abelian p-group of rank r, so E ≅ (ℤ/pℤ)^{r} ≅ ⟨g_{1}, …, g_{r}⟩ and k an algebraically closed field of characteristic p, so kE is the truncated polynomial algebra given by kE ≅ k[x_{1}, …, x_{r}]/(x_{1}^{p}, …, x_{r}^{p}), where x_{i} = g_{i} − 1.

Example 3.2.1. Suppose that k has characteristic 2 and E is the elementary abelian 2-group of rank 8. Choose f to be the polynomial Z_{1}Z_{2}Z_{3}Z_{4} + Z_{5}Z_{6}Z_{7}Z_{8}. Use the following matrix to define the action of α_{1}x_{1} + ⋯ + α_{8}x_{8} on a module M:

( \begin{matrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ α_{1} & α_{5} & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & α_{2} & α_{6} & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & α_{3} & α_{7} & 0 & 0 & 0 & 0 \\ α_{8} & 0 & 0 & α_{4} & 0 & 0 & 0 & 0 \end{matrix} ) .

Then M is an 8-dimensional module with variety given by f. This follows from using Lemma 2.5.19 to see that the variety of M is determined by considering the 4 × 4 minors of the above matrix.

Proposition 3.2.2. Given a polynomial f = Z_{1} ⋯ Z_{n} + Z_{(n+1)} ⋯ Z_{2n}, an algebraically closed field k of characteristic 2 and an elementary abelian 2-group E ≅ (ℤ/2ℤ)^{2n}, there exists a 2n-dimensional kE-module with variety given by f.

Proof. The proof proceeds by quickly defining a matrix with linear entries and determinant f, then using the fact that we are in characteristic 2 to construct an appropriate representation.

We begin by taking the following matrix:

A = ( \begin{matrix}Z_{1} & Z_{(n+1)} & 0 & 0 & 0 & 0 \\ 0 & Z_{2} & Z_{(n+2)} & 0 & 0 & 0 \\ 0 & 0 & · & · & 0 & 0 \\ 0 & 0 & 0 & · & · & 0 \\ 0 & 0 & 0 & 0 & Z_{(n−1)} & Z_{(2n−1)} \\ Z_{2n} & 0 & 0 & 0 & 0 & Z_{n} \end{matrix} ) .

Now let A_{α} denote the matrix given by taking A and replacing each occurrence of Z_{i} with the corresponding α_{i}. As A has determinant f and all entries are of degree 1 we now define a representation M of kE by using the following 2n × 2n matrix to define the action of α_{1}x_{1} + ⋯ + α_{2n}x_{2n} on M:

( \begin{matrix}0 & 0 \\ A_{α} & 0 \end{matrix} ) .

M now gives the desired representation of kE. This is where our argument only works in characteristic 2 as we use Lemma 2.5.19 to see that the variety of M is determined by considering the n × n minors of the above matrix.

Having constructed these modules we now proceed to prove their minimality. The result is a corollary of the following result of Carlson (given in [14]).

Theorem 3.2.3. Suppose that E is an elementary abelian p-group of rank r, i.e. E ≅ (ℤ/pℤ)^{r}, k is an algebraically closed field of characteristic p and M is a kEmodule. If we use d and s to denote the dimension and the degree of the rank variety of M, (i.e. the degree of f, where V (M) = V (f)), respectively, then the following inequality holds:

dim_{k}(M)/p^{r−d} ≥ s.

Corollary 3.2.4. Suppose that E is an elementary abelian p-group of rank r, i.e. E ≅ (ℤ/pℤ)^{r}, k is an algebraically closed field of characteristic p and f = Z_{1} ⋯ Z_{n} + Z_{n+1} ⋯ Z_{2n}. If M is a kE-module with rank variety given by f then dim(M) ≥ 2n.

Proof. Applying Theorem 3.2.3 in the case p = 2, r = 2n, d = 2n − 1 (as the rank variety is a hypersurface), s = n we obtain:

(\frac{dim(M)}{2^{r−d}}) ≥ s,
(\frac{dim(M)}{2^{2n−(2n−1)}} ) ≥ 2n,
(\frac{dim(M)}{2}) ≥ n,
dim(M) ≥ 2n.

3.2.2 Modules from Matrix Factorisations

Overview

Given a homogeneous irreducible polynomial f = Z_{1}Z_{(n+1)} ⋯ Z_{((d−1)n+1)} + ⋯ + Z_{n}Z_{2n} ⋯ Z_{dn} of degree d over an algebraically closed field k of characteristic 2 we will show how to construct a 2d^{n−1}-dimensional module whose variety is given by f (where n is the number of terms). We also give a brief discussion of the difficulties constructing such modules when the characteristic of k is odd. The key to this construction lies in the use of matrix factorisations, seen earlier.

The main theorem we will prove is as follows:

Theorem 3.2.10

Let f = Z_{1}Z_{(n+1)} ⋯ Z_{((d−1)n+1)} + ⋯ + Z_{n}Z_{2n} ⋯ Z_{dn} (for some n ∈ ℕ) be a homogeneous polynomial of degree d over an algebraically closed field of characteristic 2 and let E = (ℤ/2ℤ)^{dn} = ⟨g_{1}, …, g_{dn}⟩, so kE ≅ k[x_{1}, …, x_{dn}]/(x_{1}^{2}, …, x_{dn}^{2}), with x_{i} = g_{i} − 1. Then there exists a kE-module M of dimension 2d^{n−1} such that V (M) = V (f).

Having constructed these modules we are also faced with a further interesting question: Are these modules of minimal dimension with the corresponding variety? Unfortunately we are unable to provide a definite answer. It is however worth noting that setting d = 2 in the above theorem gives modules with variety a quadratic of the same dimension we found earlier. As the constructions are very different this seems to support the idea that the modules may well be minimal with such variety.

Square Matrix Factorisations

Before we begin it is worth explaining why the definition of matrix factorisation restricts us to working solely with square matrices. Backelin et al [2] explain an extremely simple method to obtain rectangular matrix factorisations, requiring far less effort than the method for square matrices used below. However allowing non-square matrices would not seem to be useful in finding modules with particular varieties anyway. The problem can be demonstrated by use of a small example.

Example 3.2.5. Let k be an algebraically closed field of characteristic 2 and E be the elementary abelian 2-group of rank 4, i.e. E ≅ (ℤ/2ℤ)^{4}. If we suppose E ≅ ⟨g_{1}, …, g_{4}⟩ then kE ≅ k[x_{1}, …, x_{4}]/(x_{1}^{2}, …, x_{4}^{2}), where x_{i} = g_{i} − 1. If f = X_{1}X_{2} + X_{3}X_{4} then we can obtain a factorisation of the 1×1 matrix (X_{1}X_{2}+X_{3}X_{4}) consisting of the following two matrices:

α_{0} = ( X_{1} X_{3} ) ,

α_{1} = ( \begin{matrix}X_{2} \\ X_{4} \end{matrix} ) .

Using these matrices we may now consider f as a composition of two linear maps f_{1}, f_{2}. Unfortunately this proves to be of very little use.

To continue the analogue with Carlson’s L_{ζ} construction we would consider f_{1} and f_{2} as the following maps (Note that the construction we suggest here is similar to that seen in Example 3.1.2):

f_{1} : k → (Ω_{kE}^{−1}(k))^{⊕2},
f_{2} : (Ω_{kE}^{−1}(k))^{⊕2} → (Ω_{kE}^{−2}(k))^{⊕2}

We now wish to consider the map f_{2} ∘ f_{1}. As f_{1} is a map from k to (Ω_{kE}^{−1}(k))^{⊕2} it cannot possibly be surjective. We must therefore add several copies of kE to the domain. At first glance the resulting modules would seem to be no smaller than those produced using Carlson’s L_{ζ} construction. Instead of adding copies of kE we may try to keep the dimension of the kernel minimal by using relative projectives rather than projectives.

For example, looking at f_{1} we find only X_{2} and X_{4} appear in the defining matrix. It would be nice to make use of relative projectives here, for example restricting using the subgroup ⟨X_{1}, X_{3}⟩. However composition with f_{2} prevents us from taking advantage of this as f_{2} ∘ f_{1} = f.

In fact the maps f_{1} and f_{2} are such that we cannot make use of any relative projectives and if we attempt to compute the corresponding L_{ζ} we obtain a module of the same dimension as if we had worked solely with f. The calculations involved are also no easier.

It may be that there is some method of using such matrix factorisations but this seems unlikely. We therefore have to take a more sophisticated approach using square matrices.

Approach

The obvious question to ask at this point is why square matrices should prove any more useful than those we obtained above. The key fact is that the polynomials we are working with are all irreducible.

Suppose that we are working over an algebraically closed field of characteristic p and have obtained a matrix factorisation of f, some degree d irreducible polynomial.

Our matrix factorisation will have form fI = α_{1} ∘ α_{2} ∘ ⋯ ∘ α_{d}, for some n × n matrices α_{i} (where n ∈ ℕ is fixed among the α_{i} but, importantly, may differ for another choice of matrix factorisation).

We know that det(fI) = f^{n} = det(α_{1} ⋯ α_{d}). Since f is irreducible we must have det(α_{i}) = f^{m_{i}} for some m_{i} ∈ ℕ, for all 1 ≤ i ≤ d (where ∑_{i=1}^{d} m_{i} = n). We will show later that we can therefore use any of the α_{i} in constructing a module with variety given by f. Since the α_{i} all have the same dimension we will see that any choice results in a module of the same dimension.

We begin with an example of finding a matrix factorisation for f = y_{1}y_{2}y_{3} +y_{4}y_{5}y_{6} (where y_{1}, …, y_{6} ∈ k[y_{1}, …, y_{6}], with k an algebraically closed field of characteristic 2). Firstly we need to obtain a matrix factorisation for g = y_{1}y_{2}y_{3}. Having done this we then use an inductive construction (inductive on the number of terms) to extend this to a matrix factorisation of f.

This example completed we will then find a matrix factorisation for y_{1}y_{2}y_{3} + y_{4}y_{5}y_{6} + y_{7}y_{8}y_{9} over an algebraically closed field of characteristic 3. The process is similar to the previous example but we must take into account the fact that the characteristic of the field divides the degree of our polynomial. This difference comes from the different structure of the underlying Clifford algebras (see Section 2.8).

Having given some small examples we will then describe a general construction for finding a module with a particular variety using matrix factorisations.

Obtaining Matrix Factorisations

The key result we will need is the following, appearing as Lemma 1.5 of [23].

Theorem 3.2.6. The polynomial f = y_{1}y_{(n+1)} ⋯ y_{((d−1)n+1)} + ⋯ + y_{n}y_{2n} ⋯ y_{dn}, where y_{1}, …, y_{dn} ∈ k[y_{1}, …, y_{dn}], has a matrix factorisation α = (α_{1}, …, α_{d}) of size d^{n−1} with entries in ℤ[ζ][y_{1}, …, y_{dn}], where ζ is a primitive dth root of unity. More precisely, the entries of α_{i} are all of the form 0 or ζ^{k}y_{j}.

Remark 3.2.7. The form of this matrix factorisation is described inductively in the proof of the above result. For the moment suppose that the characteristic of our field k does not divide the degree d of our polynomial (we will see shortly how to modify the argument in this case).

The initial step is to observe that the polynomial y_{1} ⋯ y_{d} has a factorisation α_{m} = (y_{m}), where 1 ≤ m ≤ d.

Now if we suppose that f − (y_{n} ⋯ y_{dn}) has factorisation α̃ = (α̃_{1}, …, α̃_{d}) of size d^{n−2} then f has factorisation α = (α_{1}, …, α_{d}), given by

α_{i} = (\begin{matrix}α̃_{i−1} & ζ^{i−1}y_{n}I & 0 & 0 & 0 & 0 \\ 0 & α̃_{i−2} & ζ^{i−2}y_{2n}I & 0 & 0 & 0 \\ 0 & 0 & α̃_{i−3} & ... & 0 & 0 \\ 0 & 0 & 0 & ... & ... & 0 \\ 0 & 0 & 0 & 0 & ... & ζ^{i−d+1}y_{(d−1)n}I \\ ζ^{i−d}y_{dn}I & 0 & 0 & 0 & 0 & α̃_{i−d} \end{matrix}) ,

where i ∈ ℤ_{d} and I denotes the d^{n−2} × d^{n−2} identity matrix.

We now discuss the case that char(k) | d. The difficulty is that we cannot find primitive d-th roots of unity but in fact the process works if we simply replace ζ by a primitive d′-th root of unity, where d′ is the p′-part of d.

If we restrict our attention to matrix factorisations alone then it is not clear why this should work but the reasoning comes from the underlying Clifford algebra ideas.

Recall that Theorem 2.8.9 shows that given two homogeneous polynomials f, g of equal degree and generalised Clifford algebras C(f), C(g) then C(f) ˆ⊗C(g) is a generalised Clifford algebra for f + g. Part of the proof that C(f) ˆ⊗C(g) is indeed a generalised Clifford algebra for f + g is to show that ((f + g)(x))^{d} = (f + g)(x^{d}).

The proof given in [2] writes ((f + g)(x))^{d} in the form (f + g)(x^{d}) + a, for some additional term a, then demonstrates that a can be written in the form (ζ +ζ^{2} +⋯+ ζ^{d′−1})b for some primitive d′th root of unity ζ in k and some b. When char(k) = d we have the equality ((f + g)(x))^{d} = (f + g)(x^{d}) without the need to use roots of unity. This also explains our definition of ˆ⊗.

The inductive construction given above is an explicit description of how Theorem 2.8.9 corresponds to matrix factorisations. Therefore using 1 in place of ζ corresponds to the difference in the definition of ˆ⊗ and we still obtain the desired matrix factorisation.

Example 3.2.8. For our first example we consider the polynomial f = y_{1}y_{2}y_{3}+y_{4}y_{5}y_{6} whilst working over k an algebraically closed field of characteristic 2. Following the inductive construction given above we first take the factorisation of y_{1}y_{2}y_{3} by 1 × 1 matrices, i.e. α̃_{1}α̃_{2}α̃_{3} = (y_{1})(y_{2})(y_{3}).

We then choose ζ to be a primitive 3rd root of unity and obtain a matrix factorisation of f consisting of the following three matrices:

α_{1} = (\begin{matrix}α̃_{3} & y_{4} & 0 \\ 0 & α̃_{2} & ζy_{5} \\ ζ^{2}y_{6} & 0 & α̃_{1} \end{matrix}) = (\begin{matrix}y_{3} & y_{4} & 0 \\ 0 & y_{2} & ζy_{5} \\ ζ^{2}y_{6} & 0 & y_{1} \end{matrix}) ,

α_{2} = (\begin{matrix}α̃_{1} & y_{4} & 0 \\ 0 & α̃_{3} & y_{5} \\ y_{6} & 0 & α̃_{2} \end{matrix}) = (\begin{matrix}y_{1} & y_{4} & 0 \\ 0 & y_{3} & ζy_{5} \\ ζ^{2}y_{6} & 0 & y_{2} \end{matrix}) ,

α_{3} = (\begin{matrix}α̃_{2} & y_{4} & 0 \\ 0 & α̃_{1} & ζy_{5} \\ ζ^{2}y_{6} & 0 & α̃_{3} \end{matrix}) = (\begin{matrix}y_{2} & y_{4} & 0 \\ 0 & y_{1} & ζy_{5} \\ ζ^{2}y_{6} & 0 & y_{3} \end{matrix}) .

Some simple computations confirm that α_{1}α_{2}α_{3} = (f)I_{3×3}.

Example 3.2.9. For our second example we will expand upon that above and take our polynomial to be f = y_{1}y_{2}y_{3} + y_{4}y_{5}y_{6} + y_{7}y_{8}y_{9}, with k an algebraically closed field of characteristic 3. The computations are similar but now we set ζ = 1. This gives us the following factorisation of y_{1}y_{2}y_{3} + y_{4}y_{5}y_{6}:

α̃_{1} = (\begin{matrix}y_{3} & y_{4} & 0 \\ 0 & y_{2} & y_{5} \\ y_{6} & 0 & y_{1} \end{matrix}) , α̃_{2} = (\begin{matrix}y_{1} & y_{4} & 0 \\ 0 & y_{3} & y_{5} \\ y_{6} & 0 & y_{2} \end{matrix}) , α̃_{3} = (\begin{matrix}y_{2} & y_{4} & 0 \\ 0 & y_{1} & y_{5} \\ y_{6} & 0 & y_{3} \end{matrix}) .

We now use the same inductive construction to obtain a matrix factorisation for f. As the matrices are now fairly large we present only the matrix α_{1}, noting that we have the desired property that the determinant of α_{1} is given by some power of our polynomial f:

α_{1} = (\begin{matrix}α̃_{3} & y_{7}I_{3×3} & 0 \\ 0 & α̃_{2} & y_{8}I_{3×3} \\ y_{9}I_{3×3} & 0 & α̃_{1} \end{matrix})

= ( \begin{matrix}y_{3} & y_{4} & 0 & y_{7} & 0 & 0 & 0 & 0 & 0 \\ 0 & y_{2} & y_{5} & 0 & y_{7} & 0 & 0 & 0 & 0 \\ y_{6} & 0 & y_{1} & 0 & 0 & y_{7} & 0 & 0 & 0 \\ 0 & 0 & 0 & y_{1} & y_{4} & 0 & y_{8} & 0 & 0 \\ 0 & 0 & 0 & 0 & y_{3} & y_{5} & 0 & y_{8} & 0 \\ 0 & 0 & 0 & y_{6} & 0 & y_{2} & 0 & 0 & y_{8} \\ y_{9} & 0 & 0 & 0 & 0 & 0 & y_{2} & y_{4} & 0 \\ 0 & y_{9} & 0 & 0 & 0 & 0 & 0 & y_{1} & y_{5} \\ 0 & 0 & y_{9} & 0 & 0 & 0 & y_{6} & 0 & y_{3} \end{matrix} ) .

Modules from Matrix Factorisations

Now that we are able to find small matrix factorisations relating to our polynomials we are finally in a position to prove our main result on varieties for modules.

Theorem 3.2.10. Let f = Z_{1}Z_{(n+1)} ⋯ Z_{((d−1)n+1)}+⋯+Z_{n}Z_{2n} ⋯ Z_{dn} (for some n ∈ ℕ) be a homogeneous polynomial of degree d over an algebraically closed field of characteristic 2 and let E = (ℤ/2ℤ)^{dn} = ⟨g_{1}, …, g_{dn}⟩, so kE ≅ k[x_{1}, …, x_{dn}]/(x_{1}^{2}, …, x_{dn}^{2}), with x_{i} = g_{i} − 1. Then there exists a kE-module M of dimension 2d^{n−1} such that V (M) = V (f).

Proof. We begin by letting f_{λ} denote the polynomial given by taking f and replacing all occurrences of Z_{i} with λ_{i}, for all 1 ≤ i ≤ dn, where λ_{i} ∈ k.

Now application of Theorem 3.2.6 gives us matrices A_{1}, A_{2}, …, A_{d} of size d^{n−1} such that det((f)I) = det(A_{1}) det(A_{2}) ⋯ det(A_{d}) = f_{λ}^{d^{n−1}}. So, since f is irreducible, det(A_{i}) = f_{λ}^{m_{i}}, for some m_{i} ∈ ℕ.

Choose one of the A_{i} and denote this simply by A. Our matrix A is a d^{n−1} × d^{n−1} matrix with determinant a power of f_{λ}. Recall that the variety only depends on the radical of ideals, not upon ideals themselves. Therefore we are able to work with a power of f_{λ} in place of f_{λ}. Now define a 2d^{n−1} ×2d^{n−1} matrix A′ to have the following form:

(\begin{matrix}0 & 0 \\ A & 0 \end{matrix}) .

A′ is now a 2d^{n−1} ×2d^{n−1} matrix of rank d^{n−1}. We can now finally use A′ to construct the desired module.

Define M to be a 2d^{n−1} dimensional kE-module with basis denoted {b_{1}, …, b_{(2d^{n−1})}}. Given some x_{i} ∈ kE we define its action to be given by taking A′ and setting each λ_{i} to 1 and all other λ_{j}, with 1 ≤ j ≤ dn, j ≠ i, to 0. This then gives us that the action of λ_{1}x_{1} + λ_{2}x_{2} + ⋯ + λ_{dn}x_{dn} is given by our original matrix A′. As the determinant of A is a power of f_{λ} and f is irreducible we now have that V (M) = V (f). This follows from Lemma 2.5.19, which gives us that the variety of M is determined by considering the d^{n−1} × d^{n−1} minors of A′.

We now give two examples of this construction, firstly a sum of two terms of degree 3, followed by a sum of three terms of degree 2.

Example 3.2.11. Consider the polynomial f = Z_{1}Z_{3}Z_{5} + Z_{2}Z_{4}Z_{6} and let k be an algebraically closed field of characteristic 2. We now take the matrix A_{1} found as part of our matrix factorisation of f_{λ} in Example 2.8.5, i.e.,

( \begin{matrix}λ_{1} & λ_{2} & 0 \\ 0 & λ_{5} & ζ^{2}λ_{4} \\ ζλ_{6} & 0 & λ_{3} \end{matrix} ) ,

where ζ is a primitive cube root of unity. Now let E = (ℤ/2ℤ)^{6} = ⟨g_{1}, …, g_{6}⟩, so kE ≅ k[x_{1}, …, x_{6}]/(x_{1}^{2}, …, x_{6}^{2}), with x_{i} = g_{i} − 1. Then define a kE-module M so that the action of λ_{1}x_{1} + ⋯ + λ_{6}x_{6} is given by the following matrix (call this A):

( \begin{matrix}0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ λ_{1} & λ_{2} & 0 & 0 & 0 & 0 \\ 0 & λ_{5} & ζ^{2}λ_{4} & 0 & 0 & 0 \\ ζλ_{6} & 0 & λ_{3} & 0 & 0 & 0 \end{matrix} ) .

Since the determinant of A_{1} was a power of f_{λ} and Size(A) = 2 Size(A_{1}) we have V (M) = V (f).

Example 3.2.12. This time let k be an algebraically closed field of characteristic 2 and consider the polynomial f = Z_{1}Z_{4} + Z_{2}Z_{5} + Z_{3}Z_{6}. Take the following matrix as our matrix A_{1}:

( \begin{matrix}λ_{1} & λ_{2} & λ_{3} & 0 \\ λ_{5} & λ_{4} & 0 & λ_{3} \\ λ_{6} & 0 & λ_{4} & λ_{2} \\ 0 & λ_{6} & λ_{5} & λ_{1} \end{matrix} ) .

This matrix is obtained by using the earlier inductive construction to find a matrix factorisation of λ_{1}λ_{2} + λ_{3}λ_{4} then extending it to λ_{1}λ_{2} + λ_{3}λ_{4} + λ_{5}λ_{6}, with the λ_{i} relabeled at each stage. Now let E = (ℤ/2ℤ)^{6} = ⟨g_{1}, …, g_{6}⟩, so kE ≅ k[x_{1}, …, x_{6}]/(x_{1}^{2}, …, x_{6}^{2}), with x_{i} = g_{i} − 1. Then define a kE-module M so that the action of λ_{1}x_{1} + ⋯ + λ_{6}x_{6} is given by the following matrix (call this A):

( \begin{matrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ λ_{1} & λ_{2} & λ_{3} & 0 & 0 & 0 & 0 & 0 \\ λ_{5} & λ_{4} & 0 & λ_{3} & 0 & 0 & 0 & 0 \\ λ_{6} & 0 & λ_{4} & λ_{2} & 0 & 0 & 0 & 0 \\ 0 & λ_{6} & λ_{5} & λ_{1} & 0 & 0 & 0 & 0 \end{matrix} ) .

Since the only non-zero 4 × 4 (\frac{(p−1)dim(M)}{p} = \frac{1×8}{2} = 4) minor of A is f_{λ}^{2} we have V (M) = V (f).

We now conclude this section with a brief discussion of the difficulty extending these ideas to k an algebraically closed field of odd characteristic.

Remark 3.2.13. As we are able to construct matrix factorisations over any positive characteristic, at first glance it seems that the above result should easily generalise. Unfortunately this proves more difficult than it may first appear.

Suppose that k is an algebraically closed field of characteristic 3, f = Z_{1}Z_{3} +Z_{2}Z_{4} and E = (ℤ/3ℤ)^{4} ≅ ⟨g_{1}, …, g_{4}⟩. Recall that this gives kE ≅ k[x_{1}, …, x_{4}]/(x_{1}^{3}, …, x_{4}^{3}), where x_{i} = g_{i} − 1. We will also set f_{λ} = λ_{1}λ_{3} + λ_{2}λ_{4}.

Using Theorem 3.2.6 we may obtain a matrix factorisation of f given by:

A = (\begin{matrix}λ_{1} & λ_{2} \\ λ_{4} & λ_{3} \end{matrix}) .

The difficulty is in using this matrix factorisation to construct a kE-module M with variety given by f (i.e. such that V (M) = V (f)). If M is an m-dimensional kE-module and A′ is the matrix representing the action of α_{1}x_{1} + ⋯ + α_{4}x_{4} on M then recall Lemma 2.5.19 gives us that the variety of M is determined by considering the (\frac{p−1}{p})dim(M) = (\frac{2m}{3})×(\frac{2m}{3}) minors of A′. This is because an element (α_{1}, …, α_{4}) is in V_{E}^{♯}(M) if and only if rank(A′) = \frac{2m}{3}.

This means that the construction given in Theorem 3.2.10 will not give us a module with the desired variety. An alternative may be to try working with A′ as the following matrix:

(\begin{matrix}0 & 0 & 0 \\ A & 0 & 0 \\ 0 & A & 0 \end{matrix}) .

Although at first glance this seems to give us a module with variety V (M) = V (f) it does not in fact give a representation of kE. If we let A_{1}, A_{2} be the matrices representing the actions of x_{1}, x_{2} respectively then we see that A_{1}A_{2} ≠ A_{2}A_{1}.

We will see later that such a result would directly contradict Theorem 3.3.10, so it seems unlikely that any minor adjustment could fix this problem.

The root cause of the difficulty would seem to be the difference in the group cohomology. Recall (see Section 2.4.1) that if the characteristic of k is 2 then H^{∗}(E, k) is the polynomial algebra k[X_{1}, …, X_{r}], where the X_{i} are in degree 1 but if k has odd characteristic then H^{∗}(E, k) is the tensor product of an exterior and polynomial algebra, λ(Y_{1}, …, Y_{r}) ⊗ k[X_{1}, …, X_{r}], where the Y_{i} are in degree 1 but this time the polynomial generators X_{i} are in degree 2.

3.2.3 Comparison with L_{n}

We now pause for a moment to compare the two constructions we have given for modules with variety given by a quadratic. They have the same dimension but we will see that they are in fact non-isomorphic, having different Loewy lengths.

We begin with a small example comparing the constructions for the polynomial f = Z_{1}Z_{4} + Z_{2}Z_{5} + Z_{3}Z_{6}.

Example 3.2.14. Suppose that E = (ℤ/2ℤ)^{6} ≅ ⟨g_{1}, …, g_{6}⟩ and k is an algebraically closed field of characteristic 2, so that kE ≅ k[x_{1}, …, x_{6}]/(x_{1}^{2}, …, x_{6}^{2}), where x_{i} = g_{i} − 1. Now let L_{3}, M_{3} be the modules produced from our L_{n} construction and our matrix factorisation construction, respectively. We begin by computing the Loewy length of M_{3}.

Denote the matrix for the action of α_{1}x_{1} + ⋯ + α_{6}x_{6} on M_{3} by A. This gives us the following matrix (by Theorem 3.2.10):

A = ( \begin{matrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ α_{4} & α_{2} & α_{3} & 0 & 0 & 0 & 0 & 0 \\ α_{5} & α_{1} & 0 & α_{3} & 0 & 0 & 0 & 0 \\ α_{6} & 0 & α_{1} & α_{2} & 0 & 0 & 0 & 0 \\ 0 & α_{6} & α_{5} & α_{4} & 0 & 0 & 0 & 0 \end{matrix} ) .

As A^{2} = 0 we have that M_{3} must have Loewy length 2.

Next we show that the Loewy length of L_{3} is greater than 2. It is an 8-dimensional kE-module and Definition 3.1.3 gives us that it has the following basis:

{u_{1}^{a_{1}}u_{2}^{a_{2}}u_{3}^{a_{3}} : a_{1}, a_{2}, a_{3} ∈ {0, 1}, ∑_{i=1}^{3} a_{i} > 0} ∪ {v}.

The kE-module action on L_{3} is given by:

x_{1}(u_{1}^{a_{1}}u_{2}^{a_{2}}u_{3}^{a_{3}}) = u_{1}^{a_{1}+1}u_{2}^{a_{2}}u_{3}^{a_{3}},
x_{2}(u_{1}^{a_{1}}u_{2}^{a_{2}}u_{3}^{a_{3}}) = u_{1}^{a_{1}}u_{2}^{a_{2}+1}u_{3}^{a_{3}},
x_{3}(u_{1}^{a_{1}}u_{2}^{a_{2}}u_{3}^{a_{3}}) = u_{1}^{a_{1}}u_{2}^{a_{2}}u_{3}^{a_{3}+1},
x_{4}(u_{1}) = v,
x_{5}(u_{2}) = v,
x_{6}(u_{3}) = v,
x_{i}(b) = 0 for all other basis elements b, 1 ≤ i ≤ 6,

where u_{i}^{2} = 0.

If L_{3} had Loewy length 2 then there would exist some sub-module giving us a filtration with two sub-quotients N_{1}, N_{2} where the kE_{3}-module action on the sub-quotients is trivial. Suppose such a filtration exists.

If we consider the basis element u_{1}u_{2}u_{3} then we know that the equivalence class represented by u_{1}u_{2}u_{3} must be non-zero in either N_{1} or N_{2}, so suppose without loss of generality that it is non-zero in N_{1}. Now if N_{1} contained any elements of degree 2, say u_{2}u_{3}, we would have that the induced action was non-trivial (e.g. x_{1}(u_{2}u_{3}) = u_{1}u_{2}u_{3}).

Therefore N_{2} contains all elements of degree 2. Similarly N_{1} must contain all elements of degree 1. As x_{4}(u_{1}) = v we have that v must lie in N_{2}.

This gives us the following two sub-modules:

N_{1} = {u_{1}, u_{2}, u_{3}, u_{1}u_{2}u_{3}},
N_{2} = {u_{1}u_{2}, u_{1}u_{3}, u_{2}u_{3}, v}.

The problem is that neither N_{1} or N_{2} is a sub-module of L_{3}. For N_{1} we have x_{1}(u_{2}) = u_{1}u_{2} ∉ N_{1}, for N_{2} we have x_{1}(u_{2}u_{3}) = u_{1}u_{2}u_{3} ∉ N_{2}.

Therefore this does not give us a filtration of length 2 of L_{n}, so no filtration of this length can exist. Therefore the result holds. Finally as we have shown that M_{3} and L_{3} have different Loewy lengths they must be non-isomorphic.

Having completed this example we now proceed to prove the result in general.

Theorem 3.2.15. Suppose that E_{n} = (ℤ/2ℤ)^{2n}, where n ∈ ℕ and n ≥ 3, k is an algebraically closed field of characteristic 2 and f_{n} = Z_{1}Z_{n+1} + Z_{2}Z_{n+2} + ⋯ + Z_{n}Z_{2n}. Let M_{n}, L_{n} be the modules with variety V (M_{n}) = V (L_{n}) = V (f_{n}) produced from our matrix factorisation construction and our L_{n} construction, respectively. Then M_{n} has Loewy length 2 and L_{n} has Loewy length greater than 2. In particular, M_{n} is not isomorphic to L_{n} for all n ≥ 3.

Proof. We split the proof into two halves, firstly showing that M_{n} has Loewy length 2, then showing that L_{n} has Loewy length greater than 2.

1. M_{n} has Loewy length 2: Firstly recall from Theorem 3.2.10 that the matrix A_{n} for the action of α_{1}x_{1} + ⋯ + α_{2n}x_{2n} is a 2m × 2m matrix (where m = 2^{n−1}) of the following form:

( \begin{matrix}0 & 0 \\ \underbrace{∗}_{m×m} & 0 \end{matrix} ) .

If we use b_{i} to denote the basis element corresponding to row i and column i of A_{n} then we can construct a filtration of M_{n} given by M_{n} ⊃ ⟨b_{1}, …, b_{m}⟩ ⊃ 0. The sub-quotients are then ⟨b_{m+1}, …, b_{2m}⟩ and ⟨b_{1}, …, b_{m}⟩. Considering the matrix A_{n} we see that the induced action of kE_{n} on each subquotient is trivial, therefore M_{n} has Loewy length 2.

2. L_{n} has Loewy length greater than 2: We begin by recalling the definition of L_{n}. By our earlier construction (See Definition 3.1.3) L_{n} is a 2^{n}-dimensional kE_{n}-module with basis

{u_{1}^{a_{1}}u_{2}^{a_{2}} ⋯ u_{n}^{a_{n}} : a_{1}, …, a_{n} ∈ {0, 1}, ∑_{i=1}^{n} a_{i} > 0} ∪ {v}.

The kE_{n}-module action on L_{n} is given by:

• If 1 ≤ i ≤ n:

x_{i}(u_{1}^{a_{1}}u_{2}^{a_{2}} ⋯ u_{n}^{a_{n}}) = u_{1}^{a_{1}}u_{2}^{a_{2}} ⋯ u_{i}^{a_{i+1}} ⋯ u_{n}^{a_{n}}, where u_{i}^{2} = 0,
x_{i}(v) = 0,

• If n + 1 ≤ i ≤ 2n:

x_{i}(u_{i−n}) = v,
x_{i}(b) = 0 for all other basis elements b.

Now suppose that L_{n} has Loewy length 2. Then there exists a sub-module of L_{n} having sub-quotients N_{1}, N_{2}, where the kE_{n}-module action on the sub-quotients is trivial.

If we consider the basis element u_{1} ⋯ u_{n} then we know that the equivalence class represented by u_{1} ⋯ u_{n} must be non-zero in either N_{1} or N_{2}, so suppose without loss of generality that it is non-zero in N_{1}. Then if N_{1} contained an element of degree n − 1 (e.g. u_{2} ⋯ u_{n}) the induced action would be non-trivial. Therefore N_{2} contains all elements of degree n−1. Similarly N_{1} must contain all elements of degree n − 2. Continuing this process we find that if n is even/odd then N_{1} contains all elements of even/odd degree and N_{2} all elements of odd/even degree, respectively. Note that v must lie in whichever N_{i} contains the elements of even degree as x_{n+1}(u_{1}) = v.

We now have a unique pair of possible sub-quotients N_{1}, N_{2}. The problem is that neither N_{1} or N_{2} is a sub-module of L_{n}. If we take the N_{i} containing elements of even degree then we find that x_{1}(u_{2}u_{3}) = u_{1}u_{2}u_{3} ∉ N_{i}. Similarly if we take the N_{i} containing elements of odd degree we find that x_{1}(u_{2}) = u_{1}u_{2} ∉ N_{i}. Therefore this does not give us a filtration of length 2 of L_{n}, so no filtration of this length can exist. Therefore the result holds.

Finally we have now shown that the modules M_{n} and L_{n} have different Loewy lengths for all n ≥ 3. Therefore M_{n} is not isomorphic to L_{n} for any n ≥ 3.

3.3 Modules of Loewy Length p

3.3.1 Introduction

We detail how the Loewy length of a module determines the possible varieties. Our first task is to show that the situation varies drastically depending on whether the characteristic of our field is odd or even. It may prove useful whilst reading to know that our destination is the following theorem:

Theorem 3.3.10. Let M be a kE-module of Loewy length p, where E is an elementary abelian p-group and k an algebraically closed field of positive characteristic p. If p is 2 then the variety of M may be any hypersurface (by which we mean, given any hypersurface H there exists an M such that the variety of M is given by H) but if p is odd then M must be a finite union of linear varieties.

Throughout this section E will be an elementary abelian p-group of rank r, i.e. E ≅ (ℤ/pℤ)^{r} ≅ ⟨g_{1}, …, g_{r}⟩. Recall that this gives us kE ≅ k[x_{1}, …, x_{r}]/(x_{1}^{p}, …, x_{r}^{p}), where x_{i} = g_{i} − 1.

Before we begin we will also note the following two Lemmas, which will simplify several proofs throughout this section. The first Lemma appears as Fact 5.17.4 of [8].

Lemma 3.3.1. If A_{1}, …, A_{n} is a collection of commuting m × m matrices (with m, n ∈ ℕ) then there exists a matrix S such that SA_{i}S^{−1} is lower triangular for all 1 ≤ i ≤ n.

Lemma 3.3.2. Let p be a prime and M a p × p matrix with non-zero entries on the lower off-diagonal and all other entries 0. Then M^{p−1} ≠ 0 and M^{p} = 0.

If the entries on the lower off-diagonal of M are denoted by m_{1}, …, m_{p−1} then M^{p−1} will have one non-zero entry, the entry in row p, column 1 being m_{1}m_{2} ⋯ m_{p−1}.

Proof. This result follows from the fact that, for l ∈ ℕ, the (i, j)th element of A^{l+1} will be the product of the (i, j + 1)th element of A and the (i + 1, j)th element of A^{l} (Note that the (p, j)th and (i, p)th elements will be 0 as M is lower-triangular).

We now begin with a description of the situation in characteristic 2.

3.3.2 Characteristic 2

Given any finite collection of hypersurfaces we can construct a kE-module of Loewy length 2 whose variety is the union of those hypersurfaces. Using the fact that a direct sum of modules corresponds to a union of their varieties (See Proposition 2.5.5) we are able to split the proof into two distinct halves. Firstly we deal with a single hypersurface then use the direct sum property to generalise to finite unions. Before proceeding to the general result we first provide an example to illustrate the method.

Example 3.3.3. In the interests of brevity we will consider a module whose variety is given by the product of two polynomials of low degree, f_{1} = Z_{1}Z_{2} + Z_{3}Z_{4} and f_{2} = Z_{5}Z_{6} + Z_{7}Z_{8}. The first step is to construct a module M_{1} with variety given by f_{1}. The low degree of f_{1} allows this to be quickly achieved by taking M_{1} to be the 4-dimensional kE-module with the action of α_{1}x_{1} + ⋯ + α_{4}x_{4} on M_{1} given by the following matrix:

( \begin{matrix}0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ α_{1} & α_{3} & 0 & 0 \\ α_{4} & α_{2} & 0 & 0 \end{matrix} ) .

We define M_{2} similarly with α_{1}, …, α_{4} replaced by α_{5}, …, α_{8}. The varieties of M_{1} and M_{2} are each given by computing the non-zero 2 × 2 minors, therefore given by f_{1} and f_{2}.

It is now fairly straightforward to construct a module with variety given by f_{1}f_{2}. We use the fact that the variety of a direct sum of modules corresponds to a union of the varieties. Let M be the kE-module such that the action of α_{1}x_{1} + ⋯ + α_{8}x_{8} on M is given by the following matrix (i.e. M = M_{1} ⊕ M_{2}):

( \begin{matrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ α_{1} & α_{3} & 0 & 0 & 0 & 0 & 0 & 0 \\ α_{4} & α_{2} & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & α_{5} & α_{7} & 0 & 0 \\ 0 & 0 & 0 & 0 & α_{8} & α_{6} & 0 & 0 \end{matrix} ) .

The variety of this module is determined by computing the non-zero 4 × 4 minors, giving variety corresponding to f_{1}f_{2}.

Having completed this example we now proceed to the more general case. The underlying ideas are essentially the same with the majority of the extra work being simply notational overhead.

Proposition 3.3.4. Given any homogeneous irreducible polynomial f ∈ k[Z_{1}, …, Z_{r}] there is a kE-module of Loewy Length 2 with corresponding variety.

Proof. We know from the theory of matrix factorisations (See Section 2.8) that there exists an m × m matrix (for some m ∈ ℕ) A over k[y_{1}, …, y_{r}] with linear entries whose determinant is given by some power of f.

Write A_{α} for the matrix given by replacing the entries y_{1}, …, y_{r} in A with α_{1}, …, α_{r} respectively (where α_{i} ∈ k). We now define a matrix B to be the 2m × 2m matrix obtained by placing A_{α} in the lower-left corner and setting all other entries to 0, i.e.,

( \begin{matrix}0 & 0 \\ A_{α} & 0 \end{matrix} ) .

Now define a kE-module M by letting B describe the action of α_{1}x_{1} + ⋯ + α_{r}x_{r}. The variety of M is then determined by computing the non-zero m × m minors, i.e. computing the determinant of A_{α}. So M has the desired variety. We now need to prove that M has Loewy length 2.

Returning to our matrix B, choose a basis b_{1}, …, b_{2m} of M by letting each b_{i} correspond to row i and column i in B. Now let M_{1} = ⟨b_{1}, …, b_{m}⟩. Then M ⊃ M_{1} ⊃ 0 gives a filtration of M and the subquotients N_{1}, N_{2} are given by N_{1} = ⟨b_{m+1}, …, b_{2m}⟩ and 0, respectively. So M has a filtration of length 2 for which the action of kE on each sub-quotient is trivial. Therefore the Loewy length of M can be at most 2. As B ≠ 0 we know it must be at least 2, so the result follows.

3.3.3 Odd Characteristic

Having dealt with characteristic 2 we now proceed to describe the situation in odd characteristic. The situation is slightly more complex as we are no longer able to realise arbitrary hypersurfaces. In fact we will see that the only possible varieties for modules of Loewy length p (in characteristic p, with p odd) are unions of linear hyperplanes.

It therefore seems natural to divide into two separate tasks. The first is to show that all finite unions of linear hyperplanes are realisable. Having done this we will show that no other varieties may occur.

Modules with Variety a Union of Linear Hyperplanes

We determine the result for a single hyperplane f, before detailing the more general case.

Lemma 3.3.5. Given any homogeneous linear polynomial f ∈ k[Z_{1}, …, Z_{r}] there exists a kE-module of Loewy length p with corresponding variety.

Proof. Let A = ( \begin{matrix}0 & 0 & & & & & 0 \\ f & 0 & & & & & 0 \\ 0 & f & & & & & 0 \\ 0 & 0 & · & & & & 0 \\ 0 & 0 & & · & & & 0 \\ & & & & · \\ 0 & 0 & & & & f & 0 \end{matrix} ) .

Now define M to be the kE-module with the action of α_{1}x_{1} + ⋯ + α_{r}x_{r} given by A_{α}, where A_{α} is the matrix given by taking A and replacing all occurrences of Z_{i} with the corresponding α_{i}. Note that this gives us a well-defined kE-module as having the non-zero entries on the lower off-diagonal and all being f ensures commutativity on restriction. Then M has the desired variety as the only non-zero \frac{p−1}{p} minor of A has determinant a power of f. We also need to show that M has Loewy length p.

Lemma 3.3.2 shows that A_{α}^{p−1} ≠ 0 but A_{α}^{p} = 0, so M has Loewy length at least p. As M has dimension p it must have Loewy length at most p, so the result holds.

Lemma 3.3.6. Suppose we have a finite collection of linear polynomials f_{1}, …, f_{n} ∈ k[y_{1}, …, y_{r}]. Define a collection of p × p matrices A_{i} by

A_{i} = ( \begin{matrix}0 & 0 & · & · & · & 0 & 0 \\ f_{i} & 0 & · & · & · & 0 & 0 \\ 0 & f_{i} & 0 & · & · & 0 & 0 \\ 0 & 0 & · & 0 & · & 0 & 0 \\ 0 & 0 & 0 & · & 0 & 0 & 0 \\ & & & & · \\ 0 & 0 & 0 & 0 & 0 & f_{i} & 0 \end{matrix} ) .

Now let A be the matrix with diagonal entries given by A_{i} and zeros everywhere else, i.e.,

A = ( \begin{matrix}A_{1} & & & & 0 \\ & · \\ & & · \\ & & & · \\ 0 & & & & A_{n} \end{matrix} ) .

Then A^{p−1} ≠ 0 but A^{p} = 0.

Proof. This follows from Lemma 3.3.2.

Proposition 3.3.7. Suppose f is a product of linear polynomials. Then there exists a kE-module M of Loewy length p with variety corresponding to f.

Proof. Follows from the combination of Lemma 3.3.5, Lemma 3.3.6 and the fact that V (M_{1} ⊕ M_{2}) = V (M_{1}) ∪ V (M_{2}).

3.3.4 No Other Modules

Our next task is to show that given an elementary abelian p-group E and an algebraically closed field k of characteristic p > 2 then any kE-module M of Loewy length p must have linear variety. We begin with a quick Lemma allowing us to put the matrix corresponding to the action of α_{1}x_{1} + ⋯ + α_{r}x_{r} on M into a useful form.

Lemma 3.3.8. Let pd be the dimension of M. Then there exists a basis such that the matrix corresponding to the action of x_{α} = α_{1}x_{1} + ⋯ + α_{r}x_{r} on M is given by:

A = ( \begin{matrix}0 & 0 & 0 & ⋯ & 0 \\ S_{(1,1)} & 0 & 0 & ⋯ & 0 \\ S_{(2,1)} & S_{(2,2)} & 0 & ⋯ & 0 \\ S_{(3,1)} & S_{(3,2)} & S_{(3,3)} & ⋯ & 0 \\ · \\ · \\ S_{((p−1),1)} & S_{((p−1),2)} & ⋯ & S_{((p−1),p)} & 0 \end{matrix} ) ,

where each of the S_{(i,j)} is a d × d matrix.

Proof. We prove this by first using Lemma 2.7.2 to give that A is lower-triangular and then showing that Im(T^{i}) = Rad^{i}(M), where T is the matrix described in Lemma 2.7.3, so that E acts trivially on Im(T^{i})/Im(T^{i+1}).

Firstly Im(T^{i}) ⊆ Rad^{i}(M) since T^{i} ∈ J^{i}(kE), where J(kE) denotes the Jacobson radical of kE.

Now recall that Lemma 2.7.6 gives us that Rad^{j}(M) ⊆ Soc^{p−j}(M). Using this and the fact that Ker(T^{p−i}) = Im(T^{i}) we obtain the following chain of inequalities:

Ker(T^{p−i}) = Im(T^{i}) ⊆ Rad^{i}(M) ⊆ Soc^{p−i}(M) ⊆ Ker(T^{p−i}),

therefore Im(T^{i}) = Rad^{i}(M).

Proposition 3.3.9. Suppose M is a kE-module of Loewy length p and dimension pd. Then the variety of M is linear.

Proof. Let A be the matrix corresponding to the action of x_{α} = α_{1}x_{1} + ⋯ + α_{r}x_{r} on M.

Then Lemma 3.3.8 gives us that A can be put into the following form:

A = ( \begin{matrix}0 & 0 & 0 & ⋯ & 0 \\ S_{(1,1)} & 0 & 0 & ⋯ & 0 \\ S_{(2,1)} & S_{(2,2)} & 0 & ⋯ & 0 \\ S_{(3,1)} & S_{(3,2)} & S_{(3,3)} & ⋯ & 0 \\ · \\ · \\ S_{((p−1),1)} & S_{((p−1),2)} & ⋯ & S_{((p−1),p)} & 0 \end{matrix} ) ,

where each of the S_{(i,j)} is a d × d matrix.

As M is a representation of (ℤ/pℤ)^{r} the matrix A must commute with the matrix T from Lemma 2.7.3. Note that we must be careful here as Lemma 2.7.3 also required us to make a choice of basis to put the matrix T in the appropriate form. If our current choice of basis gives us the matrix T′, corresponding to T under a change of basis, there exists some invertible matrix S such that T = S^{−1}T′S. This allows us to rewrite A as

A = ( \begin{matrix}0 & 0 & 0 & 0 & 0 & ⋯ & 0 \\ S_{1} & 0 & 0 & 0 & 0 & ⋯ & 0 \\ S_{2} & S_{1} & 0 & 0 & 0 & ⋯ & 0 \\ S_{3} & S_{2} & S_{1} & 0 & 0 & ⋯ & 0 \\ · & & & · \\ · & & & & · \\ S_{(p−1)} & S_{(p−2)} & · & · & · & S_{1} & 0 \end{matrix} ) ,

where the S_{i} are d × d matrices.

Now let A_{i}, A_{j} correspond to the restrictions M ↓_{⟨1+x_{i}⟩}, M ↓_{⟨1+x_{j}⟩} respectively. Then write

A_{i} = ( \begin{matrix}0 & 0 & 0 & 0 & 0 & ⋯ & 0 \\ S_{(i,1)} & 0 & 0 & 0 & 0 & ⋯ & 0 \\ S_{(i,2)} & S_{(i,1)} & 0 & 0 & 0 & ⋯ & 0 \\ S_{(i,3)} & S_{(i,2)} & S_{(i,1)} & 0 & 0 & ⋯ & 0 \\ · & & & · \\ · & & & & · \\ S_{(i,(p−1))} & S_{(i,(p−2))} & · & · & · & S_{(i,1)} & 0 \end{matrix} ) .

Next observe that A_{i} and A_{j} may only commute if S_{(i,1)}S_{(j,1)} = S_{(j,1)}S_{(i,1)} for all 1 ≤ i, j ≤ r. To see this note that the d × d sub-matrix beginning at row d + 1, column 1 of A_{i}A_{j} is given by S_{(i,1)}S_{(j,1)}. Therefore, as all of the A_{i} must commute, there must be a choice of basis such that the S_{(i,1)} are simultaneously lower triangular for all 1 ≤ i ≤ r (This follows from considering Lemma 3.3.1 together with the fact that the A_{i} and A_{j} must also commute not only with the matrix T but with powers of T).

Finally we now consider computing the variety of M. Firstly, as the characteristic of k is p we must determine the non-zero \frac{(p−1)d}{p} × \frac{(p−1)d}{p} minors of A. The dimensions of the S_{i} combined with the lower-triangularity mean that the only possible non-zero such minor of A_{i} has determinant det(S_{(i,1)})^{(p−1)}. Now we have that S_{1} is a lower triangular d × d matrix. Therefore the only possible non-zero d × d minor of S_{1} is obtained by taking the product of the entries on the diagonal. Therefore the variety of M must be linear.

3.3.5 Conclusion

Gluing together the above results we obtain the following theorem:

Theorem 3.3.10. Let M be a kE-module of Loewy length p, where E is an elementary abelian p-group and k an algebraically closed field of positive characteristic p. If p is 2 then the variety of M may be any hypersurface (by which we mean, given any hypersurface H there exists an M such that the variety of M is given by H) but if p is odd then M must be a finite union of linear varieties.

Further Research

There are several interesting directions for further research. These divide into three distinct categories, module constructions having particular varieties, minimal dimensions of modules with particular variety and comparison with other invariants of a module.

4.1 Module Constructions

We discussed several different constructions for modules with variety given by a homogeneous irreducible quadratic. The modules obtained from these constructions, whilst having the same dimensions, are not isomorphic in general. It would be interesting to explore how large a collection of modules exists having these varieties and dimensions and what other properties they have.

Our constructions also focused on working with k an algebraically closed field of characteristic 2. It would be interesting to see if either of our two main constructions (The L_{n} construction based upon Carlson’s L_{ζ} construction and the matrix factorisation construction) could be generalised to give modules with a particular variety in odd characteristic. The most promising approach would seem to be generalising the L_{n} construction but this would likely involve dealing with very large modules until a definite construction could be established.

4.2 Minimal Dimensions

Unfortunately we were unable find a proof for the minimal dimension of modules with particular variety. One area of particular interest in completing this proof would be to obtain a better understanding of indecomposable Maximal Cohen-Macaulay modules. In the case of modules with variety a quadratic it may prove enough to study the Maximal Cohen-Macaulay modules of powers of quadratics and compare them to those of irreducible quadratics. In the case of higher degree polynomials there seems to be little currently understood about the possible dimensions of corresponding Maximal Cohen-Macaulay modules. For more on the correspondence between Maximal Cohen-Macaulay modules and matrix factorisations, see [23] and [10].

An alternative would of course to be to find a counter-example demonstrating that the modules presented here do not have minimal dimension with particular varieties.

4.3 Other Invariants

We described a correspondence between the variety of a module and its Loewy length. There may exist several other interesting invariants for which relations to varieties for modules could be found. Also of note is that we only considered the case where a module’s Loewy length is equal to the characteristic of the field we are working over.

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