Proceedings of the International Congress of Mathematicians Hyderabad, India, 2010

Modules for elementary abelian p-groups

David J. Benson

Abstract. Let E ≅ (ℤ/p)^{r} (r ≥ 2) be an elementary abelian p-group and let k be an algebraically closed field of characteristic p. A finite dimensional kE-module M is said to have constant Jordan type if the restriction of M to every cyclic shifted subgroup of kE has the same Jordan canonical form. I shall begin by discussing theorems and conjectures which restrict the possible Jordan canonical form. Then I shall indicate methods of producing algebraic vector bundles on projective space from modules of constant Jordan type. I shall describe realisability and non-realisability theorems for such vector bundles, in terms of Chern classes and Frobenius twists. Finally, I shall discuss the closely related question: can a module of small dimension have interesting rank variety? The case p odd behaves throughout these discussions somewhat differently to the case p = 2.

Dramatis Personæ: B=Benson, C=Carlson, F=Friedlander, P=Pevtsova, S=Suslin.

1. Introduction

Many questions in modular representation theory of finite groups reduce to questions about elementary abelian subgroups. A prototype for such a reduction is Chouinard’s Theorem [11], which states that a module is projective if and only if its restriction to every elementary abelian subgroup is projective. Quillen [19, 20] described the spectrum of the cohomology ring in terms of the elementary abelian subgroups, and this was generalised to the theory of varieties for modules by Carlson [7, 8], Alperin and Evens [1], Avrunin and Scott [2]. The classification of localising subcategories of the stable module category also reduces to elementary abelian subgroups, see Benson, Iyengar and Krause [5, 6]. These theorems and others motivate the study of modules for an elementary abelian p-group

E = ⟨g_{1}, …, g_{r}⟩ ≅ (ℤ/p)^{r}

over an algebraically closed field k of characteristic p. These are generally unclassifiable, so we often restrict our attention to particular classes of modules that we might hope to understand better.

We shall only be interested in finite dimensional kE-modules in this talk. Dade’s Lemma [12] states that a finite dimensional kE-module M is projective (or equivalently, free) if and only if its restriction to every cyclic shifted subgroup is free. A cyclic shifted subgroup is a subgroup of the group algebra kE of order p generated by an element of the form 1 + X_{α} where

X_{α} = λ_{1}X_{1} + ⋯ + λ_{r}X_{r},  (1.1)

X_{i} = g_{i} − 1 ∈ J(kE) and α = (λ_{1}, …, λ_{r}) ∈ 𝔸^{r}(k) \ {0}. This motivates Carlson’s definition of the rank variety of M:

Definition 1.2. The rank variety of a kE-module M is defined to be the closed homogeneous subset of 𝔸^{r} = 𝔸^{r}(k) given by

V_{E}^{r}(M) = {α ∈ 𝔸^{r} \ {0} | M↓_{⟨1+X_{α}⟩} is not free} ∪ {0}.

Example 1.3. Let r = 4, let p = 2, and let M be the four dimensional kE-module defined by

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

Then

X_{α} ↦ ( \begin{matrix}0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ λ_{1} & λ_{2} & 0 & 0 \\ λ_{3} & λ_{4} & 0 & 0 \end{matrix} )

and V_{E}^{r}(M) is the variety defined by the vanishing of the minor λ_{1}λ_{4} − λ_{2}λ_{3}.

We write k[Y_{1}, …, Y_{r}] for the coordinate ring of affine space k[𝔸^{r}]. With this notation, in Example 1.3, V_{E}^{r}(M) is the irreducible quadric with equation given by Y_{1}Y_{4} − Y_{2}Y_{3} = 0. This, when projectivised, gives the Segre embedding of ℙ^{1} × ℙ^{1} in ℙ^{3}.

2. Jordan type

Let k, E and M be as described in the introduction. The elements X_{1}, …, X_{r} are generators for J(kE), and the elements X_{α} of equation (1.1) form a set of coset representatives of J^{2}(kE) in J(kE). Since X_{α}^{p} = 0, the action of X_{α} on M breaks up into Jordan blocks of length between 1 and p with eigenvalue 0. We write

[p]^{a_{p}} … [1]^{a_{1}}

for the Jordan type.

Warning 2.1. If x, y ∈ J(kE), x − y ∈ J^{2}(kE), it can happen that x and y have different Jordan types on M.

Definition 2.2. Nilpotent Jordan types are partially ordered: if A and B are nilpotent square matrices of the same size, A ≥ B if and only if for all s > 0 we have rank(A^{s}) ≥ rank(B^{s}). If we think of Jordan types as represented by partitions, then this is the well known dominance ordering.

We say that x ∈ J(kE) \ J^{2}(kE) has maximal Jordan type if it is maximal with respect to this partial order.

Maximal Jordan type was examined in depth by Friedlander, Pevtsova and Suslin [14], and the following theorem relating it to generic Jordan type forms the beginning of their investigation.

Modules for elementary abelian p-groups 3

Theorem 2.3 (FPS). If x, y ∈ J(kE) and x − y ∈ J^{2}(kE) then x has maximal Jordan type if and only if y does, so that the elements of maximal Jordan type determine a well defined subset of J(kE)/J^{2}(kE). This is a dense open subset. This Jordan type is the same as that of the element X_{α} for a generic point α defined over a large enough transcendental extension of k.

Because of this theorem, we talk of the generic Jordan type of M, which is the Jordan type of a generic element X_{α} of M.

3. Modules of constant Jordan type

Definition 3.1 (CFP [10]). We say that M has constant Jordan type [p]^{a_{p}} … [1]^{a_{1}} if every element of J(kE) \ J^{2}(kE) has the same Jordan canonical form on M.

Rather than working in the module category, we often wish to work in the stable module category where we ignore projective summands of a module. So if we forget the term [p]^{a_{p}} in the Jordan type of a module, as it can be recovered from the remaining terms and the dimension, we talk of a module M of stable constant Jordan type [p − 1]^{a_{p−1}} … [1]^{a_{1}}. Dade’s lemma shows that a module M has empty stable constant Jordan type if and only if M is projective.

It is obvious that a direct sum of modules of constant Jordan type is again such. It is not quite so obvious, but it follows from Theorem 2.3, that a direct summand of a module of constant Jordan type is again such. After all, if there’s a closed subset where one summand has smaller Jordan type, the other would have to have larger Jordan type on the same closed subset, in order for the sum to be constant.

Warning 3.2. If M and N are kE-modules, it is not necessarily true that if α ∈ 𝔸^{r} \ {0} then M ⊗_{k} N (with diagonal group action) restricted to X_{α} is isomorphic to M↓_{X_{α}} ⊗_{k} N↓_{X_{α}}.

Nonetheless, we have the following theorem.

Theorem 3.3 (CFP [10]). If M and N have constant Jordan type then so do M^{∗} and M ⊗_{k} N.

The fundamental question then arises:

Question 3.4. Let us assume that r ≥ 2, so that the situation is not trivial. Then what stable constant Jordan types can occur?

4. Endotrivial modules

What modules have stable constant Jordan type [1]? Suppose that M is such a module. Then M ⊗_{k} M^{∗} ≅ Hom_{k}(M, M) also has stable constant Jordan type [1]. The canonical maps k → M ⊗_{k} M^{∗} → k, defined using the inclusion of the identity map and the trace map on matrices, compose to give dim M times the identity map. But dim M ≡ 1 (mod p), so M ⊗_{k} M^{∗} decomposes as a direct sum of a trivial module and another summand; this other summand is projective, by Dade’s lemma.

Definition 4.1. A module M is said to be endotrivial if M ⊗_{k} M^{∗} is a direct sum of a trivial module and a projective module.

Theorem 4.2 (Dade [12]). If M is an endotrivial module for an elementary abelian p-group then M ≅ Ω^{n}(k) ⊕ (projective) (n ∈ ℤ).

The notation here is as follows. If n ≥ 0 then Ω^{n}(k) is the nth kernel in a minimal projective resolution of k while Ω^{−n}(k) is the nth cokernel in a minimal injective resolution of k. The module Ω^{n}(k) has stable constant Jordan type [1] if p = 2 or n is even, while if both p and n are odd it has stable constant Jordan type [p − 1].

So we have seen that the indecomposable modules of stable constant Jordan type [1] are precisely the modules Ω^{n}(k), where n is even if p is odd.

We can deal with indecomposable modules of stable constant Jordan type [p−1] (p odd) in the same way. If M is such a module, then again M ⊗_{k} M^{∗} has stable constant Jordan type [1], and we deduce using the same arguments that M is isomorphic to Ω^{n}(k) with n odd.

So what about modules of stable constant Jordan type [a] with 1 < a < p − 1? It was conjectured by CFP [10] that there are no modules of stable constant Jordan type [2] if p ≥ 5. I proved this conjecture while visiting MSRI in 2008 [4].

Theorem 4.3 (B, MSRI 2008). If r ≥ 2 then there are no kE-modules of stable constant Jordan type [a] with 1 < a < p − 1.

The technique of proof was to take exterior and symmetric powers, and use Dade’s lemma to get incompatible congruences on dim_{k} M.

There are two conjectures, each of which imply the theorem above. The first was formulated by Rickard at MSRI in 2008, based on a computer printout of data from a large number of modules of constant Jordan type.

Conjecture 4.4 (Rickard, MSRI 2008). Suppose that r ≥ 2 and M is a module of constant Jordan type. If there are no Jordan blocks of length i then the total number of Jordan blocks of length greater than i is divisible by p.

The second was formulated by Suslin and recorded in CFP [10].

Conjecture 4.5 (S). Suppose that r ≥ 2 and M is a module of constant Jordan type. If for some i with 2 ≤ i ≤ p − 1, the module M has a Jordan block of length i, then it also has a Jordan block either of length i − 1 or of length i + 1.

The smallest cases which remain unresolved, and which would follow from either of these conjectures, are the existence of modules of stable constant Jordan type [3][1] and [2]^{2} for p ≥ 5. But each conjecture disallows types allowed by the other, and we have no reasonable conjecture in general as to exactly what types occur.

5. Vector bundles on projective space

We use the phrase vector bundle on ℙ^{r−1} in the algebraic sense, so that it equivalent to the phrase locally free sheaf of O-modules, where O is the structure sheaf of ℙ^{r−1}.

Remarks 5.1. The only line bundles on ℙ^{r−1} are O(n) for n ∈ ℤ.

r = 2: every vector bundle on ℙ^{1} is a direct sum of line bundles; the decomposition is essentially unique (Grothendieck).

r ≥ 3: It is moderately easy to construct indecomposable vector bundles on ℙ^{r−1} of every rank ≥ r − 2.

The only known indecomposable vector bundles with rank bigger than 1 and less than r − 2 are:

On ℙ^{4}, the Horrocks–Mumford bundle [16] F_{HM} of rank 2 with 15, 000 symmetries (the group is 5^{1+2}SL(2, 5)),

On ℙ^{5}, Horrocks’ Parent bundle [15] of rank 3,

On ℙ^{5} in characteristic 2, the Tango bundle [21] of rank 2,

Some further rank 2 bundles on ℙ^{4} and rank 3 bundles on ℙ^{5} constructed by Kumar [17], and by Kumar, Peterson and Rao [18], in positive characteristic only…

… and bundles obtained from these by twisting and pulling back through self-maps of projective space.

In particular, it remains unknown whether there are indecomposable vector bundles of rank two on ℙ^{6} in any characteristic and on ℙ^{5} in characteristic other than 2.

We construct vector bundles F_{i}(M) (1 ≤ i ≤ p) from a module M of constant Jordan type as follows. We let

ℙ^{r−1} = Proj k[Y_{1}, …, Y_{r}]

where the Y_{i} are the functions on 𝔸^{r} defined by Y_{i}(X_{j}) = δ_{ij} (Kronecker delta). Given a kE-module M, we set M̃ = M ⊗_{k} O, a trivial bundle over ℙ^{r−1} whose rank is equal to the dimension of M.

Definition 5.2 (FP [13]). We define a map of vector bundles

θ: M̃(j) → M̃(j + 1) (j ∈ ℤ)

by the formula

θ(m ⊗ f) = ∑_{i} X_{i}m ⊗ Y_{i}f.

Definition 5.3 (BP, MSRI 2008).

F_{i}(M) = \frac{Ker θ ∩ Im θ^{i−1}}{Ker θ ∩ Im θ^{i}}

as a subquotient of M̃, for 1 ≤ i ≤ p.

Remark 5.4. This definition has the feature that F_{i}(M) is a vector bundle for 1 ≤ i ≤ p if and only if M has constant Jordan type. The rank of F_{i}(M) is equal to the number of Jordan blocks of length i on a cyclic shifted subgroup.

Example 5.5. Let E = (ℤ/p)^{2} = ⟨g_{1}, g_{2}⟩, kE = k[X_{1}, X_{2}]/(X_{1}^{p}, X_{2}^{p}), X_{i} = g_{i} − 1. Let M be the kE-module given by

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

Then

θ = (\begin{matrix}0 & 0 & 0 \\ Y_{1} & 0 & 0 \\ Y_{2} & 0 & 0 \end{matrix})

has kernel of rank two and image of rank one; F_{1}(M) and F_{2}(M) are both rank one bundles.

Example 5.6. More generally, for E of any rank, the module M = kE/J^{2}(kE) has constant Jordan type [2][1]^{r−1}. We have F_{1}(M) ≅ T (−1) and F_{2}(M) ≅ O(−1), where T is the tangent bundle.

Example 5.7. The kE-module M = Soc^{2}(kE) also has constant Jordan type [2][1]^{r−1}, but F_{1}(M) ≅ Ω(1), F_{2}(M) ≅ O where Ω = T^{∗} denotes the cotangent bundle.

Example 5.8 (B, MSRI 2008). If p ≥ 7 and r = 5 then there exists a kE-module M of dimension 30p^{5} of stable constant Jordan type [p − 1]^{30}[2]^{2}[1]^{26} such that F_{2}(M) ≅ F_{HM}(−2), a twist of the Horrocks–Mumford bundle. See the preprint [3] for details.

Example 5.9 (B, Summer 2008). Let E = (ℤ/2)^{6}, and M be the module for which θ is the following 30 × 30 matrix:

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

Then M has constant Jordan type [2]^{14}[1]^{2} and F_{1}(M) is the Tango bundle of rank two on ℙ^{5}.

6. Properties of F_{i}(M)

The following properties of the vector bundles F_{i}(M) were found by BP, MSRI 2008.

• F_{p−i}(ΩM) ≅ F_{i}(M)(−p + i) (1 ≤ i ≤ p − 1))

• F_{i}(M^{∗}) ≅ F_{i}(M)^{∨}(−i + 1) (1 ≤ i ≤ p)

• F_{1}(M ⊗_{k} N) ≅ ⨁_{i=1}^{p−1} F_{i}(M) ⊗_{O} F_{i}(N)(i − 1)

• The short exact sequence 0 → ΩM → P_{M} → M → 0 induces a complex

0 → F_{p}(ΩM) → F_{p}(P_{M}) → F_{p}(M) → 0

which is not exact, but has homology only in the middle, where it is

⨁_{i=1}^{p−1} F_{i}(M)(−p + i).

Realisability of vector bundles by modules of constant Jordan type is addressed in the following theorem.

Theorem 6.1 (BP, MSRI 2008). Given a vector bundle F of rank s on ℙ^{r−1}, there exists a kE-module of stable constant Jordan type [1]^{s} such that 1. if p = 2 then F_{1}(M) ≅ F,

2. if p is odd then F_{1}(M) ≅ F^{∗}(F)

where F: ℙ^{r−1} → ℙ^{r−1} is the Frobenius map.

The method of proof of the theorem is to take a resolution of the vector bundle by sums of twists of the structure sheaf. The maps in the resolution are polynomials in Y_{1}, …, Y_{r}. If p = 2, a polynomial of degree d is realised by a map from Ω^{d}k to k. Then there is a construction in the stable module category of kE which produces a single module from these maps.

If p is odd then the pth power of the polynomial is realised by a map from Ω^{2d}k to k, and so we only get to realise bundles whose resolutions involve only pth powers of polynomials. This is where the Frobenius twist comes in.

When p is odd, there is an obstruction to improving the theorem to F_{1}(M) ≅ F coming from Chern classes.

7. Chern classes

The Chow group A^{∗}(ℙ^{r−1}) is isomorphic to ℤ[h](h^{r}). Given a vector bundle F on ℙ^{r−1}, there is a Chern polynomial

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

whose coefficients c_{i}(F) are the Chern numbers of F.

Theorem 7.1 (B, Summer 2008). Suppose that r ≥ 2, and let M be a kE-module of stable constant Jordan type [1]^{s}. Then p | c_{i}(F_{1}(M)) for 1 ≤ i ≤ p − 2.

If p = 2 this gives no information, but for p odd it gives a genuine restriction on the vector bundles that can occur this way.

Example 7.2. The rank two Horrocks–Mumford bundle F_{HM} on ℙ^{4} has Chern numbers c_{1}(F_{HM}(i)) = 2i + 5 and c_{2}(F_{HM}(i)) = i^{2} + 5i + 10. So no twist of F_{HM} can occur as F_{1}(M) for a module of stable constant Jordan type [1]^{2}. This explains why, in Example 5.8, it was necessary to have Jordan blocks of lengths other than one and p.

8. Small modules with interesting varieties

We now move away from constant Jordan type, and look at some more speculative questions. Looking back at Example 1.3, can we mimic this construction if p is odd? The most obvious attempts fail, so we are left with the general question: are there small modules with interesting varieties, when p is odd? Can we find good bounds and good constructions to address this question?

A construction of Carlson produces modules with any desired closed homogeneous subvariety of 𝔸^{r} as its rank variety, but the dimension of the module produced this way is large. For example, if we do this with the ruled quadric Y_{1}Y_{4} − Y_{2}Y_{3} = 0 for (ℤ/2)^{4}, then we are required to interpret this as an element of H^{2}(E, k) ≅ Ext_{kE}^{1}(Ωk, k) and take the corresponding extension of k by Ωk, of dimension 16. Carlson’s method in general is to realise hypersurfaces in this way and then tensor modules for a general variety written as an intersection of hypersurfaces.

A modified version of Carlson’s method produces Example 1.3. Namely, instead of using a single element of cohomology, we use a 2 × 2 matrix of elements ( \begin{matrix}Y_{1} & Y_{2} \\ Y_{3} & Y_{4} \end{matrix} ) and interpret it as an element of Ext_{kE}^{1}(k ⊕ k, k ⊕ k). The corresponding four dimensional module gives the 4 × 4 matrices of the example.

When p is odd, the single element method is even worse. The generators in degree one of cohomology are nilpotent, and cannot be used in Carlson’s construction. One is forced to use the generators of degree two, so Y_{1}Y_{4} − Y_{2}Y_{3} corresponds to an element of H^{4}(E, k) ≅ Ext_{kE}^{1}(Ω^{3}k, k). The dimension of the module obtained in this way is 13p^{4}.

If we go with the 2×2 matrix method for p odd, we still end up with an element of Ext_{kE}^{1}(Ωk ⊕ Ωk, k ⊕ k), giving a module of dimension 2p^{4}.

But we can do better than this. If a row of the matrix does not use all of the variables, we can use a relative syzygy of k instead of the absolute syzygy. The condition for making this work is that the variety of the subgroup used for the relative syzygy should be contained in the variety defined by the polynomials appearing in the row. So for the 2×2 matrix we’re considering, we can use relative syzygies for two subgroups of order p^{2} to obtain a module of dimension 2p^{2} with the required variety. Here is a diagram for this module when p = 3:

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

In this diagram the actions of the four elements X_{i} = g_{i} − 1 (1 ≤ i ≤ 4) are represented by the single, double, wavy and dotted edges respectively. The leftmost and rightmost vertices are identified to make a module of dimension 18.

This module of dimension 2p^{2} is still far from the lower bound given by Bézout’s theorem, as described in Carlson [9]:

Theorem 8.1 (C, 1993). Let E be an elementary abelian p-group of rank r. If M is a kE-module whose rank variety has dimension m and degree d then dim_{k} M/p^{r−m} is an integer at least as big as d.

In our case, the Bézout bound is 2p, but it is not hard to prove that there is no module of dimension 2p with the required variety for p odd. I suspect that the smallest module with the ruled quadric as an irreducible component of its variety is the above one of dimension 2p^{2}.

Question 8.2. Let E be an elementary abelian p-group of rank r. Given a closed homogeneous subvariety V ⊆ 𝔸^{r}, what is the smallest dimension of a kE-module M with V_{E}(M) = V?

9. Modules for (ℤ/p)^{2}

For an elementary abelian group of rank two, Question 8.2 is not interesting, because the projective line does not have interesting subvarieties. But when we mix conditions on the variety with conditions on the Jordan type, we do get some very interesting questions. I shall describe just one theorem in this direction.

Let E = (ℤ/p)^{2} with p odd. For an indecomposable kE-module the possible varieties are the whole of 𝔸^{2} and a single line through the origin. In the latter case, namely the case of periodic modules, we can look at the Jordan type of the module on the cyclic shifted subgroup corresponding to that line. The following theorem gives rather surprising information about the dimensions of such modules.

Theorem 9.1 (B, 2010). If an indecomposable periodic kE-module M has Jordan blocks of length p on all cyclic shifted subgroups then the dimension of M is at least p^{\frac{3}{2}} . Furthermore, there exists such a module M of dimension at most p^{\frac{3}{2}} + \sqrt{2} p^{\frac{5}{4}} .

The constant \sqrt{2} in the theorem is probably not best possible. There must be further theorems of a similar type waiting to be discovered, but at the moment this seems to be the only one known.

References

[1] J. L. Alperin and L. Evens, Varieties and elementary abelian subgroups, J. Pure & Applied Algebra 26 (1982), 221–227.

[2] G. S. Avrunin and L. L. Scott, Quillen stratification for modules, Invent. Math. 66 (1982), 277–286.

[3] D. J. Benson, Modules of constant Jordan type and the Horrocks–Mumford bundle, preprint, 2008.

[4] , Modules of constant Jordan type with one non-projective block, Algebras and Representation Theory 13 (2010).

[5] D. J. Benson, S. B. Iyengar, and H. Krause, Local cohomology and support for triangulated categories, Ann. Scient. Éc. Norm. Sup. (4) 41 (2008), 1–47.

[6] , Stratifying modular representations of finite groups, Preprint, 2008.

[7] J. F. Carlson, The complexity and varieties of modules, Integral representations and their applications, Oberwolfach, 1980, Lecture Notes in Mathematics, vol. 882, Springer-Verlag, Berlin/New York, 1981, pp. 415–422.

[8] , The varieties and cohomology ring of a module, J. Algebra 85 (1983), 104– 143.

[9] , Varieties and modules of small dimension, Arch. Math. (Basel) 60 (1993), 425–430.

[10] J. F. Carlson, E. M. Friedlander, and J. Pevtsova, Modules of constant Jordan type, J. Reine & Angew. Math. 614 (2008), 191–234.

[11] L. Chouinard, Projectivity and relative projectivity over group rings, J. Pure & Applied Algebra 7 (1976), 278–302.

[12] E. C. Dade, Endo-permutation modules over p-groups, II, Ann. of Math. 108 (1978), 317–346.

[13] E. M. Friedlander and J. Pevtsova, Constructions for infinitesimal group schemes, Preprint, 2008.

[14] E. M. Frielander, J. Pevtsova, and A. Suslin, Generic and maximal Jordan types, Invent. Math. 168 (2007), 485–522.

[15] G. Horrocks, Examples of rank three vector bundles on five dimensional projective space, J. London Math. Soc. 18 (1978), 15–27.

[16] G. Horrocks and D. Mumford, A rank 2 vector bundle on ℙ^{4} with 15, 000 symmetries, Topology 12 (1973), 63–81.

[17] N. M. Kumar, Construction of rank two vector bundles on ℙ^{4} in positive characteristic, Invent. Math. 130 (1997), 277–286.

[18] N. M. Kumar, C. Peterson, and A. P. Rao, Construction of low rank vector bundles on ℙ^{4} and ℙ^{5}, J. Alg. Geometry 11 (2002), 203–217.

[19] D. G. Quillen, The spectrum of an equivariant cohomology ring, I, Ann. of Math. 94 (1971), 549–572.

[20] , The spectrum of an equivariant cohomology ring, II, Ann. of Math. 94 (1971), 573–602.

[21] H. Tango, On morphisms from projective space ℙ^{n} to the Grassmann variety Gr(n, d), J. Math. Kyoto Univ. 16 (1976), 201–207.

David Benson, Aberdeen E-mail: bensondj at maths dot abdn dot ac dot uk