MODULES WITH SMALL LOEWY LENGTH

DAVE BENSON AND FERGUS REID

Abstract. A module of complexity c for E ≅ (ℤ/p)^{r} in characteristic p has Loewy length at least (p − 1)(r − c) + 1. We study the case of equality. If p is odd, the only rank varieties possible are finite unions of linear subspaces of dimension c, and every such rank variety occurs. If p = 2, the variety has to be equidimensional. If such a variety is a finite union of set theoretic complete intersections then it occurs for such a module, but otherwise the situation is unclear. Exterior algebras in any characteristic are also treated, and follow the same behaviour as the case p = 2 above.

1. Introduction

Let E ≅ (ℤ/p)^{r} be an elementary abelian p-group of rank r and let k be an algebraically closed field of characteristic p. If M is a finite dimensional kE-module then its rank variety V_{E}^{rk}(M) as defined by Carlson [2] is a closed homogeneous subset of affine r-space V_{E}^{rk} = 𝔸^{r}(k). The dimension of V_{E}^{rk}(M) is called the complexity of M, and this determines the rate of growth of the minimal resolution of M as a kE-module. If M has complexity c then its Loewy length is at least (p − 1)(r − c) + 1. It is the purpose of this paper to investigate what happens when equality holds. The case of Loewy length p and complexity r − 1 was investigated in the Ph.D. thesis of the second author [9], and this paper generalises those results to arbitrary codimension. The main theorem of this paper is as follows.

Theorem 1.1. Let E ≅ (ℤ/p)^{r}, k an algebraically closed field of characteristic p, and M be a finite dimensional kE-module of complexity c. Then M has Loewy length at least (p − 1)(r − c) + 1. If M has exactly this Loewy length then the following hold:

(1) If p is odd then V_{E}^{rk}(M) is a union of a finite number of linear subspaces of V_{E}^{rk} = 𝔸^{r}(k) of dimension c. Every such subvariety occurs for some M of Loewy length (p − 1)(r − c) + 1.

(2) If p = 2 then every irreducible component of V_{E}^{rk}(M) is equidimensional of dimension c. Given a closed homogeneous subvariety V ⊆ 𝔸^{r}(k) which is a finite union of set theoretic complete intersections of dimension c, there exists such a kE-module M of Loewy length r − c + 1 with V_{E}^{rk}(M) = V.

If p = 2, the group algebra kE is isomorphic to an exterior algebra on r generators. The case p = 2 of Theorem 1.1 generalises in an obvious way to modules in any characteristic for an exterior algebra as follows.

Theorem 1.2. Let Λ = Λ(X_{1}, …, X_{r}) be an exterior algebra on variables X_{1}, …, X_{r} over an algebraically closed field k, and M be a finite dimensional Λ-module of complexity c. Then M has Loewy length at least r − c + 1. If M has exactly this Loewy length then V_{Λ}^{rk}(M) is equidimensional of dimension c. Given a closed homogeneous subvariety V ⊆ 𝔸^{r}(k) which is a finite union of set theoretic complete intersections of dimension c, there exists such a Λ-module M of Loewy length r − c + 1 with V_{Λ}^{rk}(M) = V.

Remark 1.3. In the case p odd, Theorem 1.1 describes exactly which subvarieties can occur for modules with complexity c and Loewy length (p − 1)(r − c) + 1. For p = 2, and more generally for exterior algebras in any characteristic in Theorem 1.2 there is a gap, as described by the following series of observations.

The twisted cubic defined by the homogeneous radical ideal

(Y_{1}Y_{3} − Y_{2}^{2}, Y_{2}Y_{4} − Y_{3}^{2}, Y_{1}Y_{4} − Y_{2}Y_{3}) ⊆ k[Y_{1}, …, Y_{4}]

is not a complete intersection in the sense that this ideal is not a complete intersection; but as a variety it is a set theoretic complete intersection, because it can be expressed as an intersection (with multiplicity two) of the hypersurfaces

Y_{1}Y_{3} − Y_{2}^{2} = 0 and Y_{3}(Y_{2}Y_{4} − Y_{3}^{2}) − Y_{4}(Y_{1}Y_{4} − Y_{2}Y_{3}) = 0.

But not every equidimensional closed homogeneous subvariety of 𝔸^{r}(k) is a set theoretic complete intersection. For example, the ideal (Y_{1}, Y_{2}) ∩ (Y_{3}, Y_{4}) ⊆ k[Y_{1}, …, Y_{4}] describes the union of two 2-dimensional linear subspaces of 𝔸^{4} intersecting in the origin. It is shown in Example 9.17 of [8] that this has codimension two but arithmetic rank three, and in particular it is not a set theoretic complete intersection. Nonetheless, it is obviously a union of two complete intersections, and hence the theorem guarantees that there is a module of Loewy length three for (ℤ/2)^{4} in characteristic two (or an exterior algebra in four variables in any characteristic) with this variety.

There are also irreducible examples. Example 3.4.2 in Hartshorne [6], after homogenisation, is the three dimensional closed homogeneous subset of 𝔸^{5}(k) defined parametrically by coordinates

(s^{3}, s^{2}t, stu, su(u − s), u^{2}(u − s)).

The paper describes this over the complex numbers, but the argument given there shows that over any algebraically closed field k, this is not a set theoretic complete intersection. We do not know whether there is a module of Loewy length three for (ℤ/2)^{5} (or for an exterior algebra on five generators in other characteristics) with this variety.

We thank Srikanth Iyengar and Anurag Singh for referring us to the examples in [8] and Burt Totaro for pointing us to Hartshorne’s paper.

2. The inequality

We begin by recalling Carlson’s definition [2, 3] of rank variety. Let E = ⟨g_{1}, …, g_{r}⟩ ≅ (ℤ/p)^{r}, and k be an algebraically closed field of characteristic p, and let M be a finitely generated kE-module. We define X_{i} = g_{i} − 1 ∈ kE for 1 ≤ i ≤ r. We denote by V_{E}^{rk} the affine space 𝔸^{r}(k) of dimension r. If α = (λ_{1}, …, λ_{r}) ∈ V_{E}^{rk}, we set

(2.1)  X_{α} = λ_{1}X_{1} + ⋯ + λ_{r}X_{r}.

Then X_{α}^{p} = 0, so the rank rk(X_{α}, M) is at most equal to \frac{p−1}{p} dim_{k} M with equality if and only if all Jordan blocks of X_{α} on M have length exactly p. If α ≠ 0, this happens if and only if the restriction of M to the subalgebra k[X_{α}]/(X_{α}^{p}) ⊆ kE is projective. The rank variety of M is defined to be

V_{E}^{rk}(M) = \{α ∈ 𝔸^{r}(k) | rk(X_{α}, M) < \frac{p−1}{p} dim_{k}(M)\}.

This is a closed homogeneous subvariety of V_{E}^{rk} = 𝔸^{r}(k) whose dimension is equal to the complexity c of M.

In the case of an exterior algebra Λ = Λ(X_{1}, …, X_{r}) over an algebraically closed field k of any characteristic, the definition is similar. For α = (λ_{1}, …, λ_{r}) ∈ V_{Λ}^{rk} = 𝔸^{r}(k), we set X_{α} = λ_{1}X_{1} + ⋯ + λ_{r}X_{r}. Then X_{α}^{2} = 0, and the rank variety of a Λ-module M is defined to be

V_{Λ}^{rk}(M) = \{α ∈ 𝔸^{r}(k) | rk(X_{α}, M) < \frac{1}{2} dim_{k}(M)\}.

The theory of rank varieties for exterior algebras is a consequence of the work of Aramova, Avramov and Herzog [1]; see especially Theorem 3.1 of that paper.

The following summarises the basic properties of rank varieties that we shall be using in this paper.

Proposition 2.2. Let M and N be kE-modules.

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

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

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

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

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

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

Analogous properties hold for rank varieties of Λ-modules.

Example 2.3. Let W be a linear subspace of V_{E}^{rk} of dimension c. Choose a basis X_{β_{1}}, …, X_{β_{c}} for W, and let

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

This is a subalgebra flatly embedded in kE. Let M(W) be the kE-module kE ⊗_{kE′} k. Then dim_{k}(M(W)) = p^{r−c}, the Loewy length of M(W) is (p−1)(r −c)+1, and V_{E}^{rk}(M(W)) = W.

Let U be a linear subspace of V_{E}^{rk} of dimension r − c such that U ∩ W = {0}. Choose a basis α_{1}, …, α_{r−c} for U, and set

(2.4)  kE′′ = k[X_{α_{1}}, …, X_{α_{r−c}}]/(X_{α_{1}}^{p}, …, X_{α_{r−c}}^{p}).

Then M(W)↓_{kE′′} is a free module of rank one.

Similarly in the exterior algebra case, if W is a linear subspace of V_{Λ}^{rk} of dimension c with basis X_{β_{1}}, …, X_{β_{c}}, let

Λ′ = Λ(X_{β_{1}}, …, X_{β_{c}})

and let M(W) be the Λ-module Λ ⊗_{Λ′} k. Then dim_{k}(M(W)) = 2^{r−c}, the Loewy length of M(W) is r − c + 1, and V_{Λ}^{rk}(M(W)) = W.

Example 2.3 will appear in the proofs of several theorems in this paper. The following will be used in the proof of Theorem 5.1.

Proposition 2.5. Let X_{α_{1}}, …, X_{α_{c}} be a basis for a subspace U of V_{E}^{rk} of dimension r − c of V_{E}^{rk} and let kE′′ ⊆ kE be as defined in (2.4). Let M be a kE-module of dimension p^{r−c} such that M↓_{kE′′} is free of rank one. Then V_{E}^{rk}(M) is a linear subspace W of V_{E}^{rk} of dimension c complementary to U.

Proof. Since kE is commutative, its action on M is as kE′′-module homomorphisms. We have End_{kE′′}(kE′′) ≅ kE′′, with the isomorphism being given by multiplication by elements of kE′′. So there is a function f: {X_{1}, …, X_{r}} → kE′′ such that the action of X_{i} on M is the same as the action of f(X_{i}). The elements X_{i} − f(X_{i}) then act as zero on M. The images of these elements in J(kE)/J^{2}(kE) span a subspace W of dimension at least c intersecting U in the origin, and hence W has dimension exactly c and is a complement to U. It now follows from Lemma 6.4 of Carlson [3] that V_{E}^{rk}(M) = W. □

Lemma 2.6. Suppose that M is a module with Rad^{m}(M) = 0. Then for 0 ≤ i ≤ m we have Rad^{i}(M) ⊆ Soc^{m−i}(M). □

Lemma 2.7. If M_{1} and M_{2} are kE-modules of Loewy length ℓ_{1} and ℓ_{2} then M_{1} ⊗_{k} M_{2} has Loewy length at most ℓ_{1} + ℓ_{2} − 1.

Proof. Tensor products of semisimple kE-modules are semisimple. So defining

N_{i} = ∑_{i_{1}+i_{2}=i} J^{i_{1}}(kE)M_{1} ⊗_{k} J^{i_{2}}(kE)M_{2} ⊆ M_{1} ⊗_{k} M_{2},

it follows that N_{0} = M_{1} ⊗_{k} M_{2}, N_{i}/N_{i+1} is semisimple, and we have N_{ℓ_{1}+ℓ_{2}−1} = 0 because if i_{1} + i_{2} = ℓ_{1} + ℓ_{2} − 1 then either i_{1} ≥ ℓ_{1} or i_{2} ≥ ℓ_{2}. □

Definition 2.8. We say that a closed homogenous subvariety V of V_{E}^{rk} is equidimensional if all irreducible components of V have the same dimension. A (homogeneous) hypersurface is equidimensional of codimension one. Equivalently, it is defined by a single homogeneous polynomial equation.

Theorem 2.9. Let E ≅ (ℤ/p)^{r}, k an algebraically closed field of characteristic p and M a finite dimensional kE-module of complexity c. Then the Loewy length of M is at least (p − 1)(r − c) + 1. If equality occurs then V_{E}^{rk}(M) is equidimensional, and for 0 ≤ i ≤ (p − 1)(r − c) + 1 we have

Rad^{i}(M) = J^{i}(kE′)M,

where kE′′ = k[X_{α_{1}}, …, X_{α_{r−c}}]/(X_{α_{1}}^{p}, …, X_{α_{r−c}}^{p}) is a flatly embedded subalgebra such that M↓_{kE′′} is free.

Proof. Since V_{E}^{rk}(M) has dimension c, there exists a linear subspace U of V_{E}^{rk} of dimension r − c whose intersection with V_{E}^{rk}(M) is exactly the origin. Let α_{1}, …, α_{r−c} be a basis for U. Then

kE′′ = k[X_{α_{1}}, …, X_{α_{r−c}}]/(X_{α_{1}}^{p}, …, X_{α_{r−c}}^{p})

is a subalgebra flatly embedded in kE, and M↓_{kE′′} is a free module. So the radical length of M↓_{kE′′} is the radical length of kE′′, which is equal to (p − 1)(r − c) + 1. It follows that the radical length of M has to be at least this big.

Now suppose that equality occurs. Then letting m = (p − 1)(r − c) + 1, by Lemma 2.6 we have

J^{i}(kE′′)M ⊆ Rad^{i}(M) ⊆ Soc^{m−i}(M) ⊆ J^{i}(kE′′)M

and so Rad^{i}(M) = J^{i}(kE′′)M.

If V_{E}^{rk}(M) has an irreducible component V of dimension less than c then there exists a linear subspace W of dimension r−c+2 such that V ∩W has dimension one, and W intersects other irreducible components of V_{E}^{rk}(M) in {0}. Let M(W) be the module constructed in Example 2.3, with dimension p^{c−2} and Loewy length (p − 1)(c − 2) + 1. Then V_{E}^{rk}(W ⊗_{k} M) decomposes as a union of V ∩ W and another subvariety intersecting V ∩ W in {0}. So by a theorem of Carlson [4], W ⊗_{k} M has a direct summand M′ with V_{E}^{rk}(M′) = V ∩ W. The Loewy length of M′ is at most the Loewy length of W ⊗_{k} M, which by Lemma 2.7 is at most

((p − 1)(c − 2) + 1) + ((p − 1)(r − c) + 1) − 1 = (p − 1)(r − 2) + 1.

Since M′ has complexity one, it follows that it is a counterexample to the inequality in the first part of the theorem. This contradiction shows that V_{E}^{rk}(M) cannot have such an irreducible component. □

Next, we describe the corresponding theorem for exterior algebras. The analogue of Theorem 2.9 is the following, and the proof is the same. See for example Theorem 3.2 and Remark 3.4 (3) of [1].

Theorem 2.10. Let Λ = Λ(X_{1}, …, X_{r}) be an exterior algebra on variables X_{1}, …, X_{r} over an algebraically closed field k, and M a finite dimensional Λ-module of complexity c. Then the Loewy length of M is at least r−c+1. If equality occurs then V_{Λ}^{rk}(M) is equidimensional, and for 0 ≤ i ≤ r − c + 1 we have Rad^{i}(M) = J^{i}(Λ′′)M, where Λ′′ = Λ(X_{α_{1}}, …, X_{α_{r−c}}) is an exterior subalgebra of rank r − c flatly embedded in Λ in such a way that M↓_{Λ′′} is free. □

3. Hypersurfaces

In this section we recall from [9] the analysis of modules of complexity c = r − 1. This depends on the following result from the theory of matrix factorisations:

Theorem 3.1. Let f be a homogeneous polynomial of degree d in k[Y_{1}, …, Y_{r}]. Then there exist m > 0 and m × m matrices A_{1}, …, A_{d} ∈ Mat_{m}(k[Y_{1}, …, Y_{r}]) such that each A_{i} is a matrix whose entries are either degree one polynomials or zero, and A_{1} … A_{d} = f.I, where I is the m × m identity matrix.

Proof. This follows from Theorem 1.2 of Herzog, Ulrich and Backelin [7]. □

Remark 3.2. In the theorem, it follows that for 1 ≤ i ≤ d we also have

A_{i}A_{i+1} … A_{d}A_{1} … A_{i−1} = f.I.

Corollary 3.3. Let f be a homogeneous polynomial of degree d in k[Y_{1}, …, Y_{r}]. Then there exists m > 0 and a matrix A ∈ Mat_{m}(k[Y_{1}, …, Y_{r}]) whose entries are either degree one polynomials or zero, such that the hypersurface determined by det(A) is the same as that determined by f.

Proof. If f is irreducible, this follows from Theorem 3.1 by taking A = A_{1}, since

det(A_{1}) … det(A_{d}) = det(f.I) = f^{m}

and so det(A) is a power of f. For general f it follows by taking direct sums. □

Remark 3.4. In the corollary, one may insist that det(A) is a power of f, but the matrices may be larger, and there is no advantage to doing this.

Theorem 3.5. Let k be a field and Λ = Λ(X_{1}, …, X_{r}) be an exterior algebra over k. If M is a Λ-module of Loewy length two then V_{Λ}^{rk}(M) is either a hypersurface or the whole of V_{Λ}^{rk}. Conversely, given any homogeneous hypersurface V of V_{Λ}^{rk}, there exists a Λ-module M of Loewy length two with V_{Λ}^{rk}(M) = V.

Proof. The first statement follows from Theorem 2.9. For the converse, we’ll make use of Corollary 3.3. Interpret k[Y_{1}, …, Y_{r}] as the coordinate ring of 𝔸^{r}(k). Thus Y_{1}, …, Y_{r} are dual to X_{1}, …, X_{r}. For notation, if α = (α_{1}, …, α_{r}) ∈ 𝔸^{r}(k) and f ∈ k[Y_{1}, …, Y_{r}], we write f(α) to mean f(α_{1}, …, α_{r}). We can write V ⊆ V_{Λ}^{rk} as V_{Λ}^{rk}(f), the zeros of a homogeneous polynomial f(Y_{1}, …, Y_{r}) ∈ k[Y_{1}, …, Y_{r}]. By Corollary 3.3, there exists a matrix A ∈ Mat_{m}(k[Y_{1}, …, Y_{r}]) whose entries are either degree one polynomials or zero, such that the hypersurface determined by det(A) is equal to V. Let M be the module given by

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

Then V_{Λ}^{rk}(M) is the set of α such that this matrix has rank less than \frac{1}{2} dim_{k}(M). The only relevant minor is the bottom left square, and so the condition is that α is a zero of det(A)(α) = f(α). Thus V_{Λ}^{rk}(M) = V. □

Corollary 3.6. Let k be a field of characteristic two and let E be an elementary abelian 2-group. Given any homogeneous hypersurfaces V ⊆ V_{E}^{rk}, there exists a kE-module M of Loewy length two with V_{E}^{rk}(M) = V. □

Next, we turn to odd primes. The following completes the proof of Theorem 1.1 in the case c = r − 1. A proof is given in Theorem 5.1 for c arbitrary, and it does not use this special case, so logically it is not necessary to read this proof. On the other hand it is easier, and good preparation for reading the proof of Theorem 5.1.

Theorem 3.7. Let k be an algebraically closed field of characteristic p ≥ 3 and let E be an elementary abelian p-group. Let M be a kE-module of Loewy length p with V_{E}^{rk}(M) a proper subvariety of V_{E}^{rk}. Then V_{E}^{rk}(M) is a finite union of linear hypersurfaces in V_{E}^{rk}.

Proof. Choose β ∈ V_{E}^{rk} \ V_{E}^{rk}(M). Then the action of X_{β} on M is projective, and setting kE′ = k[X_{β}]/(X_{β}^{p}), it follows from Theorem 2.9 that for 0 ≤ i ≤ p we have Rad^{i}(M) = X_{β}^{i}.M. So M has a basis with respect to which the matrix for X_{β} is

(\begin{matrix}0 & 0 & ⋯ & 0 & 0 \\ I & 0 & & 0 & 0 \\ 0 & I & & 0 & 0 \\ ⋮ & & ... & & ⋮ \\ 0 & 0 & ⋯ & I & 0 \end{matrix}) .

Here, there are p blocks, and 0 and I represent zero and identity square matrices of size dim_{k}(M)/p. Since Rad^{i}(M) = X_{β}^{i}.M, the generators X_{1}, …, X_{r} act as matrices of the form

(\begin{matrix}0 & 0 & ⋯ & 0 & 0 \\ ∗ & 0 & & 0 & 0 \\ ∗ & ∗ & & 0 & 0 \\ ⋮ & & ... & & ⋮ \\ ∗ & ∗ & ⋯ & ∗ & 0 \end{matrix})

Furthermore, since the X_{j} commute with X_{β}, this forces the matrices representing the X_{j} to take the form

(\begin{matrix}0 & 0 & ⋯ & & 0 & 0 \\ A_{j} & 0 & & & 0 & 0 \\ B_{j} & A_{j} & & & 0 & 0 \\ ⋮ & ... & ... & & & ⋮ \\ & & ... & A_{j} & 0 & 0 \\ ∗ & & ⋯ & B_{j} & A_{j} & 0 \end{matrix})

Since p ≥ 3, the fact that these matrices commute with each other now forces A_{1}, …, A_{r} to commute with each other. Commuting matrices over an algebraically closed field may be simultaneously lower triangularised. So after a change of basis within the blocks above, we may assume that A_{1}, …, A_{r} are lower triangular.

We are now in a position to compute V_{E}^{rk}(M). There is only one minor of size \frac{p−1}{p} dim_{k}(M) that can be non-zero, namely the bottom left part of the matrices above, obtained by removing one block from the top of each column and one block from the right of each row. If α = (λ_{1}, …, λ_{r}), then this minor for X_{α} has determinant

det(λ_{1}A_{1} + ⋯ + λ_{r}A_{r})^{p−1}.

Since A_{1}, …, A_{r} are lower triangular, this is a product of linear forms in λ_{1}, …, λ_{r}. It follows that V_{E}^{rk}(M) is the union of linear hypersurfaces determined by this product. □

4. Complete intersections

In this section, we handle the case of exterior algebras for arbitrary complexity, thereby completing the proof of Theorem 1.2 and of part (2) of Theorem 1.1. Theorem 4.1. Let Λ = Λ(X_{1}, …, X_{r}) be an exterior algebra over an algebraically closed field k. If V ⊆ V_{Λ}^{rk} is a finite union of set theoretic complete intersections of dimension c, then there exists a Λ-module M of Loewy length r − c + 1 with V_{Λ}^{rk}(M) = V. Proof. Since the Loewy length of M_{1} ⊕ M_{2} is equal to the larger of the Loewy lengths of M_{1} and M_{2}, using Proposition 2.2 (3), it suffices to prove the theorem in the case where V is a complete intersection. Let f_{1}, …, f_{r−c} be a regular sequence of homogeneous elements in k[Y_{1}, …, Y_{r}] such that the variety of the ideal (f_{1}, …, f_{r−c}) is equal to V. For each f_{i}, Theorem 3.5 produces a module M_{i} of Loewy length two with V_{Λ}^{rk}(M_{i}) = V (f_{i}), the hypersurface defined by f_{i}. Set M = M_{1} ⊗_{k} ⋯ ⊗_{k} M_{r−c}. Using Proposition 2.2 (4), we have

V_{Λ}^{rk}(M) = V_{Λ}^{rk}(M_{1}) ∩ ⋯ ∩ V_{Λ}^{rk}(M_{r−c}) = V (f_{1}) ∩ ⋯ ∩ V (f_{r−c}) = V.

Furthermore, by Lemma 2.7 and induction on the codimension of V, the Loewy length of M is equal to r − c + 1. □ Remark 4.2. If k does not have characteristic two, it is necessary to consider Λ as a graded algebra with the X_{i} in degree one, in order for it to be a Hopf algebra so that we can tensor modules together. Fortunately, the modules M_{i} in the proof of the theorem may be regarded as graded Λ-modules. Corollary 4.3. Let k be an algebraically closed field of characteristic 2 and E = (ℤ/2)^{r}. If V ⊆ V_{E}^{rk} is a finite union of set theoretic complete intersections of dimension c, then there exists a kE-module M of Loewy length r − c + 1 with V_{E}^{rk}(M) = V. □

5. Odd primes

In this section, we complete the proof of Theorem 1.1 by proving that if p is odd and M has minimal Loewy length for its complexity, then its variety is a finite union of linear subspaces of the same dimension. The proof is a generalisation of the proof of Theorem 3.7, but the technical details are harder to follow.

Theorem 5.1. Let k be an algebraically closed field of characteristic p ≥ 3 and let E = (ℤ/p)^{r} be an elementary abelian p-group. Let M be a kE-module with complexity c and Loewy length (p − 1)(r − c) + 1. Then V_{E}^{rk}(M) is a finite union of linear subspaces of V_{E}^{rk} of dimension c. Every such subvariety occurs this way.

Proof. The fact that every such variety is realised by a module with the right Loewy length is easy to see by looking at direct sums of modules of the form M(W) described in Example 2.3. So we assume that M is a module with complexity c and Loewy length (p − 1)(r − c) + 1 and we examine its variety.

Let kE = k[X_{1}, …, X_{r}]/(X_{1}^{p}, …, X_{r}^{p}). As in the proof of Theorem 2.9, choose a linear subspace U of dimension r − c with U ∩ V_{E}^{rk}(M) = {0} and choose a basis α_{1}, …, α_{r−c} for U. Set

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

Then M↓_{kE′′} is a free module.

Set M_{0} = J(kE′)M, which by Theorem 2.9 is equal to Rad(M). For 1 ≤ i ≤ r − c, set

M_{i} = X_{α_{1}}M + ⋯ + X_{α_{i}}^{2}M + ⋯ + X_{α_{r−c}}M,
M′_{i} = X_{α_{1}}M + ⋯ + X_{α_{i}}^{3}M + ⋯ + X_{α_{r−c}}M.

For 1 ≤ i < j ≤ r − c, set

M_{ij} = X_{α_{1}}M + ⋯ + X_{α_{i}}^{2}M + ⋯ + X_{α_{j}}^{2}M + ⋯ + X_{α_{r−c}}M

(just two terms with squares). Then X_{α_{i}} induces isomorphisms

M/M_{0} → M_{0}/M_{i} → M_{i}/M′_{i}.

By Theorem 2.9, X_{1}, …, X_{r} send M into M_{0} and M_{0} into M_{i}, and hence also induce maps

M/M_{0} → M_{0}/M_{i} → M_{i}/M′_{i}.

Since they commute with X_{α_{i}}, if we identify these spaces using X_{α_{i}} then the two maps induced by each X_{ℓ} are the same. Since the X_{ℓ} commute with each other, we may choose a basis for M/M_{0} with respect to which these maps are represented by lower triangular matrices, which we denote A_{j,ℓ}. Furthermore, we have a commutative square of isomorphisms

\xymatrix{M/M_{0} \ar[r]^{X_{α_{j′}}} \ar[d]_{X_{α_{j}}} & M_{0}/M_{j′} \ar[d]^{X_{α_{j}}} \\ M_{0}/M_{j} \ar[r]_{X_{α_{j′}}} & (M_{j} + M_{j′})/M_{j,j′}}

The vertical maps induced by an X_{ℓ} commute with X_{α_{j′}} so they are both the same, and the horizontal maps induced by another X_{ℓ′} commute with X_{α_{j}} so they are both the same. Therefore the matrices A_{j,ℓ} and A_{j′,ℓ′} commute. It follows that we may choose a single basis for M/M_{0} so that all the matrices A_{j,ℓ} (1 ≤ j ≤ r − c, 1 ≤ ℓ ≤ r) are lower triangular. Write m_{1}, …, m_{n} for such a basis. So the image of m_{b} under A_{j,ℓ} is contained in the linear span of m_{b}, …, m_{n}. Then a basis for M is given by the elements

(5.2)  X_{α_{1}}^{a_{1}} … X_{α_{r−c}}^{a_{r−c}}.m_{b}

with 0 ≤ a_{i} ≤ p − 1 and 1 ≤ b ≤ n.

Next, we observe that the actions of X_{1}, …, X_{r} on M are as kE′′-module endomorphisms. We have End_{kE′′}(M) ≅ Mat_{n}(kE′′), and by Theorem 2.9, the actions of X_{1}, …, X_{r} are by elements of Mat_{n}(J(kE′′)). Furthermore, if x ∈ Mat_{n}(J^{2}(kE′′)) then we may write x = ∑_{i} y_{i}z_{i} with y_{i}, z_{i} ∈ Mat_{n}(J(kE′′)) and y_{i}^{p} = 0, z_{i}^{p} = 0 (to see this, it suffices to consider the case where x has only one non-zero entry, and where that entry is a product of elements of J(kE′′)). So by Proposition 2.2 of Friedlander and Pevtsova [5], if we adjust the action of X_{1}, …, X_{r} by elements of Mat_{n}(J^{2}(kE′′)) we do not change V_{E}^{rk}(M). We may make such an adjustment so that the action of X_{1}, …, X_{r} on M is determined by the equations

X_{ℓ}.m_{b} = ∑_{j} A_{j,ℓ}X_{α_{j}}.m_{b}.

These equations do determine the action on all basis elements (5.2) since X_{ℓ} commutes with X_{α_{1}}^{a_{1}} … X_{α_{r−c}}^{a_{r−c}}.

We make a filtration of M by submodules

M = N_{1} ⊇ ⋯ ⊇ N_{n+1} = {0}

where N_{i} is the span of the basis elements (5.2) of M with b ≥ i. The subquotient N_{i}/N_{i+1} (1 ≤ i ≤ n) has a basis consisting of the images of the basis elements (5.2) of M with b = i. It is an example of a module satisfying the hypothesis of the theorem, of dimension p^{r−c}, and where the matrices A_{j,ℓ} described above have only one entry. Proposition 2.5 now applies, and we deduce that V_{E}^{rk}(N_{i}/N_{i+1}) is a linear subspace W_{i} ⊆ V_{E}^{rk} of dimension c. It follows using Proposition 2.2 (5) that V_{E}^{rk}(M) ⊆ W_{1} ∪ ⋯ ∪ W_{n}. By Theorem 2.9, V_{E}^{rk}(M) is equidimensional of dimension c, so it follows that V_{E}^{rk}(M) is a union of some subset of {W_{1}, …, W_{n}}. □

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