THE AVRUNIN SCOTT THEOREM AND A CLASS OF LOCAL ALGEBRAS

KARIN ERDMANN AND MILES HOLLOWAY

1. Introduction

The theory of varieties for kG-modules (with G a finite group and k an algebraically closed field of characteristic p > 0 dividing |G|) has had a significant impact on modular representation theory and has led to the development of similar theories in other areas; p-Lie algebras (Friedlander and Parshall [9], Jantzen [10] et al), Steenrod algebras (Palmieri [11]) and (well behaved) finite dimensional self-injective k-algebras (Snashall and Solberg, [12], see also [8]).

One of the key results in the theory for groups (due to Quillen and Carlson) is that the support variety of a kG-module can be constructed from the support varieties for the module after restricting to elementary abelian p-subgroups E ≤ G. Another key result is the Avrunin-Scott theorem ([1]), which precisely relates the support variety of a kE-module to the rank variety of the module. This is important because the rank variety for a kE-module, as well as being of independent interest, has the virtue of being computationally explicit.

In this paper we present a new proof of the Avrunin-Scott theorem (in subsection 5.1). To do so we provide a more representation theoretic description of the support and rank varieties (we call these stable map descriptions). We then turn to a certain class of finite dimensional algebras, Λ_{m}, which possess a rank variety theory (see [7]). Moreover these algebras are well behaved and so the support variety theory ([12],[8]) is applicable. We consider the analogue of the Avrunin-Scott theorem for these algebras and, with our more representation theoretic perspective, we prove the corresponding analogue of the Avrunin-Scott theorem (Theorem 9.1).

The paper is organised as follows. In §2 we provide some background on the varieties considered. In §3 we recall the definition of the support variety for a kG-module, and in §3.5 give our stable map description of these varieties. In §4 we review the definition of the rank variety for a kE-module and provide a stable map description of these varieties. In §5 we state the Avrunin-Scott theorem and discuss some of the issues involved in existing proofs, before giving our alternative proof in §5.1. The rest of the paper is concerned with the class of algebras Λ_{m}, which are defined in §6. §7 briefly recalls some of the definitions and results of the support variety theory for general self-injective k-algebras, while §8 does the same for the rank variety theory for the algebras Λ_{m}. Finally, in §9, motivated by the arguments used in our proof of the Avrunin-Scott theorem, we prove the analogue for the algebras Λ_{m}.

Throughout the paper k will always denote an algebraically closed field and where char(k) becomes important we’ll specify it explicitly. G will denote an arbitrary finite group, with p dividing |G|, and E will always be an elementary abelian p-group of rank n, where char(k) = p. All algebras considered will be finite-dimensional k-algebras and given such an algebra Λ we’ll only consider finite dimensional Λ-modules. The category of left (resp. right) Λ-modules will be denoted by Λ -mod (resp. mod- Λ) and the full subcategory of projective Λ-modules will be denoted by Λ -proj (resp. proj- Λ). We adopt the following notation for the dualities D− := Hom_{Λ}(−, k) : Λ -mod → mod- Λ and −^{∨} := Hom_{Λ}(−, Λ) : Λ -proj → proj- Λ. For symmetric algebras, such as group algebras, it’s well know that D− ≅ −^{∨} and we simply denote this common duality by −^{∗}. Recall also that for a symmetric algebra Λ, if M is a Λ bimodule which is projective as a left or right Λ module then M^{∗} ⊗_{Λ} − is both left and right adjoint to M ⊗_{Λ} −. Given a finite-dimensional k-algebra Λ, Λ^{e} := Λ ⊗_{k} Λ^{op} denotes the enveloping algebra and we will freely view Λ-bimodules as equivalent to left Λ^{e}-modules. Given a graded ring R = R^{∗}, the degree of a homogeneous element z ∈ R will be written deg(z).

Acknowledgement. The second author is thankful for the support of an EPSRC postdoctoral fellowship during which part of this work was carried out and is also very grateful to Fiona Barron for financial support during the remainder of this work.

2. Background on the varieties considered

To describe the support varieties for kG-modules, one starts with the cohomology ring, R = R^{∗} := H^{∗}(G, k) = Ext_{kG}^{∗}(k, k). Thanks to the Evens-Golod-Venkov theorem this is a finitely generated graded commutative k-algebra, with degree zero part R^{0} = k. The ambient variety V_{G} is defined to be the maximal ideal spectrum of R, that is V_{G} := MaxSpec- R, then one associates a proper homogeneous ideal I(M) to a given kG-module M and defines the variety to be the closed subvariety V (I(M)). But V_{G} has a distinguished point, viz the maximal ideal R^{+} = ⊕_{i≥1}R^{i}. Moreover as I(M) is a homogeneous ideal (so I(M) ⊆ R^{+}), V (I(M)) is a homogeneous affine variety that contains the distinguished point. A presentation of R as R = k[x_{1}, …, x_{n}]/J (with x_{i} homogeneous elements, deg(x_{i}) = n_{i} > 0 and J a homogeneous ideal) yields an embedding of V_{G} as a closed set in 𝔸^{n}. Moreover R^{+} then corresponds to the origin in 𝔸^{n} under this embedding. Typically, because the varieties considered are homogeneous, one can view them as collections of lines through the origin. However one has to be careful, taking into account the grading, in defining the lines (see §5.4 in [3]). To be more precise, given λ ∈ k, there is a ring homomorphism m_{λ} : R → R which simply multiplies an element of degree r by λ^{r}. The induced map m_{λ}^{∗} : MaxSpec- R → MaxSpec- R, is called dilation by λ and given a point α ∈ MaxSpec- R, the set {λα|λ ∈ k} forms a homogeneous subvariety of V_{G} that is a line in V_{G}. For example if R = k[x, y, z], with deg(x) = 1, deg(y) = 2, deg(z) = 3, then the line through (1, 2, 2) ∈ MaxSpec- R would be defined by the homogeneous ideal (2x^{2} − y, 2x^{3} − z). Because the varieties considered are invariant under dilation, it might be more efficient to consider the projective variety

\overline{V}_{G} = Proj R := \{p ⊊ R^{+}, p a homogeneous prime ideal\}.

We will mostly (in keeping with the existing literature) view the varieties considered as homogeneous affine varieties, however occasionally it may be more convenient to consider them as projective varieties.

3. The support varieties for group algebras

In this section we recall the definition and some of the properties of support varieties for group algebras. We then re-interpret some of them to provide what we call a stable map description of the support variety for a module.

3.1. Definitions. Recall that for M ∈ kG -mod, we have the ring homomorphism

H^{∗}(G, k) ⟶^{−⊗_{k}M} Ext_{kG}^{∗}(M, M).

The Evens-Golod-Venkov theorem tells us that H^{∗}(G, k) is an affine k-algebra. If I(M) is the ideal that is the kernel of the map − ⊗_{k} M, then the support variety, V_{G}(M), of M is defined to be the variety of the ideal I(M). That is, with V_{G} := MaxSpec- H^{∗}(G, k), then

V_{G}(M) := V (I(M)) = {m ∈ V_{G}|I(M) ⊆ m}.

This is a homogeneous affine variety.

Given two kG-modules, M and M′, we denote by I(M, M′) the ideal that is the kernel of the action of H^{∗}(G, k) on Ext_{kG}^{∗}(M, M′), and V_{G}(M, M′) the corresponding variety. This action is given by cup product, which admits a factorisation either as − ⊗_{k} M followed by Yoneda composition or − ⊗_{k} M′ followed by Yoneda composition. It follows that

I(M′, M) ⊇ I(M) + (M′), and hence
V_{G}(M, M′) ⊆ V_{G}(M) ∩ V_{G}(M′)

Some further relatively elementary properties of these support varieties are:

(i) V_{G}(M) = V_{G}(M^{∗}) = V_{G}(ΩM).

(ii) V_{G}(M ⊕ M′) = V_{G}(M) ∪ V_{G}(M′).

(iii) V_{G}(M) = ⋃_{S} V_{G}(S, M).

(iii) V_{G}(M) = {0} ⟺ M is projective.

(iv) V_{G}(M) ∩ V_{G}(M′) = {0} ⇒ Ext_{kG}^{n}(M, M′) = 0, for n > 0.

In (iii) S runs over the composition factors of M. In addition we have the crucial tensor intersection rule

(v) V_{G}(M ⊗_{k} M′) = V_{G}(M) ∩ V_{G}(M′).

3.2. The L_{ζ} modules. Here we recall the construction of an important class of modules, L_{ζ} (ζ ∈ H^{∗}(G, k) a homogeneous element). Given ζ ∈ H^{n}(G, k) we can represent ζ as a map ζ : Ω^{n}k → k, and L_{ζ} is simply the kernel of this map. In fact, with P an injective hull for Ω^{n}k, and M_{ζ} := Ω^{−1}L_{ζ} we have the following diagram

[figure: 0; 0; L_{ζ}; L_{ζ}; 0; Ω^{n}k; ζ; P; Ω^{n−1}k; 0; 0; k; M_{ζ}; Ω^{n−1}k; 0; 0; 0]

What this diagram is really saying is that we have a triangle, in kG -\underline{mod},

Ω^{n}k ⟶^{ζ} k → M_{ζ} → Ω^{−1}(Ω^{n}k)

Rotating the triangle we have

(3.1)  k → M_{ζ} → Ω^{n−1}k ⟶^{−Ω^{−1}(ζ)} Ω^{−1}k

We will need the following important fact about these modules, which can be proved without recourse to the Avrunin-Scott theorem, for example essentially the same proof as that used in [8] will do (but compare with [3]).

Theorem 3.1.

V_{G}(M_{ζ}) = V_{G}(L_{ζ}) = V (⟨ζ⟩).

Where V (⟨ζ⟩) denotes the hypersurface determined by the element ζ.

3.3. Elementary abelian p-groups. Recall E is an elementary abelian p-group of rank n. The cohomology ring of E (see Corollary 3.5.7 of [2]) is given by.

Theorem 3.2.

H^{∗}(E, k) ≅ \begin{cases}k[y_{1}, …, y_{n}] & with deg(y_{i}) = 1, for p = 2, \\ k[z_{1}, …, z_{n}] ⊗ Λ(y_{1}, …, y_{n}) & with deg(y_{i}) = 1, deg(z_{i}) = 2 for p > 2. \end{cases}

Here Λ(y_{1}, …, y_{n}) is the exterior algebra. In the case p > 2 a Noether normalisation for H^{∗}(E, k) is given by the polynomial subalgebra k[z_{1}, …, z_{n}]. Clearly in this case a choice of k-basis for the vector subspace of H^{2}(E, k) spanned by the {z_{i}} determines an identification V_{E} ≅ 𝔸^{n}. This vector subspace is the image of the Bockstein map β : H^{1}(E, k) → H^{2}(E, k) (see §3.4). The case p = 2 is simpler with H^{∗}(E, k) already a Noether normalisation and choice of system of parameters amounting to a choice of k-basis for H^{1}(E, k) = Ext_{kE}^{1}(k, k).

3.4. The Bockstein. The Bockstein homomorphism is a degree 1 graded map

β : H^{∗}(G, 𝔽_{p}) → H^{∗+1}(G, 𝔽_{p})

It is defined to be the connecting homomorphism in the long exact sequence associated to the short exact sequence

0 → 𝔽_{p} → ℤ/p^{2} → 𝔽_{p} → 0.

Given an extension field k/𝔽_{p}, we have H^{∗}(G, k) ≅ k ⊗_{𝔽_{p}} H^{∗}(G, 𝔽_{p}) The Bockstein map is extended semilinearly, through the Frobenius map on k, to a map β : H^{∗}(G, k) → H^{∗+1}(G, k). That is, given ∑_{i} λ_{i}x_{i} ∈ H^{r}(G, k), with x_{i} ∈ H^{r}(G, 𝔽_{p}), λ_{i} ∈ k, then

β(∑_{i} λ_{i}x_{i}) = ∑_{i} λ_{i}^{p}β(x_{i}).

For the case G = E, we can take, for p > 2, the generators {z_{i}} of the Noether normalisation as z_{i} = β(y_{i}). If p = 2, then β(y_{i}) = y_{i}^{2}.

3.5. The stable map description. For M ∈ kG -mod we know that V_{G}(M) is a homogeneous affine variety. Given a point α ∈ V_{G}, we denote by ℓ_{α} the line in V_{G} through α (see §2 for the meaning of the lines). In the case G = E then V_{E} ≅ 𝔸^{n} and λ ∈ k^{n} determines a point in V_{E}. For E we seek a representation theoretic condition to check whether ℓ_{λ} ⊂ V_{E}(M). In order to do this we will need to construct modules whose variety is exactly ℓ_{λ}. That this is possible follows from the more general fact.

Theorem 3.3. Given any closed homogeneous subvariety, V ⊆ V_{G} there exists a module M ∈ kG -mod such that V_{G}(M) = V.

Note for V = ℓ_{α} every indecomposable non-projective submodule of the module given also has variety ℓ_{α}. The following result describes modules whose varieties are lines.

Theorem 3.4. If M ∈ kG -mod is indecomposable and has variety a line, then M is a periodic module and the period must divide the degree of one of the generators of a Noether normalisation of H^{∗}(G, k).

The upshot of these theorems is that we are guaranteed an indecomposable module T′_{α} whose variety is the line ℓ_{α} (for α ∈ V_{G}) and which is periodic, say of period r. By taking T_{α} := T′_{α} ⊕ ⋯ ⊕ Ω^{r−1}T′_{α}, we can ensure that ΩT_{α} ≅ T_{α}, as well as V_{G}(T_{α}) = ℓ_{α}.

There is also a converse result, due to Carlson, which says

Theorem 3.5. If M ∈ kG -mod is indecomposable and periodic, then V_{G}(M) is a single line through the origin in V_{G}.

We now consider G = E and use the modules T_{λ} (for λ ∈ k^{n}, also note in this case r ≤ 2) to provide the following stable map description of varieties.

Theorem 3.6. Let M ∈ kE -mod and λ ∈ 𝔸^{n}. If T_{λ} ∈ kE -mod is such that V_{E}(T_{λ}) = ℓ_{λ} and ΩT_{λ} ≅ T_{λ}, then

ℓ_{λ} ⊆ V_{E}(M) ⟺ \underline{Hom}_{kE}(T_{λ}, M) ≠ 0.

Proof. As E is a p-group, we have V_{E}(N) = V_{E}(N, k) for any N ∈ kE -mod. By the tensor intersection property, V_{E}(T_{λ} ⊗ M^{∗}) = V_{E}(T_{λ}) ∩ V_{E}(M^{∗}) = V_{E}(T_{λ}) ∩ V_{E}(M). But V_{E}(T_{λ}) = ℓ_{λ} and so

ℓ_{λ} ∈ V_{E}(M) ⟺ V_{E}(T_{λ} ⊗ M^{∗}) ≠ {0}.

Moreover V_{E}(T_{λ} ⊗M^{∗}) = V_{E}(T_{λ} ⊗M^{∗}, k), but Ext_{kE}^{∗}(T_{λ} ⊗M^{∗}, k) = Ext_{kE}^{∗}(T_{λ}, M). Using ΩT_{λ} ≅ T_{λ}, it’s clear that Ext_{kE}^{i}(T_{λ}, M) ≅ \underline{Hom}_{kE}(T_{λ}, M) for i > 0. In other words, V_{E}(T_{λ} ⊗ M^{∗}) ≠ \{0\} if and only if \underline{Hom}_{kE}(T_{λ}, M) ≠ 0. □

4. The rank variety for elementary abelian p-group algebras

In this section we recall Carlson’s definition of the rank variety for an elementary abelian p-group. We then re-interpret the definition to provide a stable map description of these varieties analogous to that given in Theorem 3.6.

4.1. Definitions. Given E, an elementary abelian p-group of rank n, the rank variety assigns a homogeneous affine variety (or projective variety), denoted by V_{E}^{r}(M) to M ∈ kE -mod. In order to define this variety we will need to fix a coordinate system for the underlying affine space V_{E}^{r} := 𝔸^{n}. This amounts to choosing a k-basis for the ndimensional vector space J(kE)/J^{2}(kE). If E is presented as E := ⟨g_{1}, …, g_{n}|g_{i}^{p}, [g_{i}, g_{j}]⟩ then an obvious basis for J(kE)/J^{2}(kE) is given by {x_{i} := g_{i} − 1}, for 1 ≤ i ≤ n. Note also that then

kE = k⟨x_{1}, …, x_{n}⟩ ≅ k[X_{1}, …, X_{n}]/(X_{i}^{p}).

Before recalling the definition of the rank variety, note that a choice of k-basis for J(kE)/J^{2}(kE) is equivalent, via duality, to a choice of basis for the vector space

Hom_{k}(J(kE)/J^{2}(kE), k) = Hom_{kE}(J(kE)/J^{2}(kE), k).

But we have a natural isomorphism

Hom_{kE}(J(kE)/J^{2}(kE), k) ≅ Ext_{kE}^{1}(k, k).

To summarise, a choice of identification of V^{r} with 𝔸^{n} is equivalent to fixing a k-basis for the vector space Ext_{kE}^{1}(k, k).

The definition of the rank variety rests upon the notion of a shifted cyclic subgroup. Given 0 ≠ λ = (λ_{1}, …, λ_{n}) ∈ k^{n} define

u_{λ} := ∑_{i=1}^{n} λ_{i}x_{i} ∈ kE

and note that u_{λ}^{p} = 0, so that ⟨1 + u_{λ}⟩ is a cyclic p-group inside kE; a so called shifted subgroup.

Definition 4.1. Given M ∈ kE -mod, the rank variety of M is defined by

V_{E}^{r}(M) := {0} ∪ {0 ≠ λ ∈ V_{E}^{r}| rank(u_{λ}) < ((p − 1)/p) dim M},

where u_{λ} : M → M is multiplication by u_{λ}.

This is clearly a homogeneous affine variety. The rank condition is equivalent to the condition that M ↓_{⟨1+u_{λ}⟩} is not a free k⟨1+u_{λ}⟩-module. This follows easily by considering the structure of indecomposable modules for a cyclic p-group. The same considerations also show that the rank condition is equivalent to the condition:

(4.1)  nullity(u_{λ}) − dim M + nullity(u_{λ}^{p−1}) > 0.

Some of the properties of the cyclic modules kEu_{λ} are given by

Proposition 4.2. For 0 ≠ λ ∈ k^{n}, the cyclic module kEu_{λ} is an indecomposable periodic module, of period 2 if p > 2 and period 1 if p = 2. In particular

Ω(kEu_{λ}) ≅ kEu_{λ}^{p−1}.

Moreover we have

V_{E}^{r}(kEu_{λ}) = ℓ_{λ}.

Proof. The first statement is easy to see. Indeed we may choose a basis of J(kE)/J^{2}(kE) starting with u_{λ}. In other words we may assume u_{λ} = x_{1}, so that dim(kEx_{1}) = p^{n−1}(p−1). The first statement is now obvious. It’s also clear that both statements are trivial if n = 1, so we can assume n ≥ 2. To prove the second statement we have to show that if 0 ≠ μ = (μ_{1}, …, μ_{n}) ∈ k^{n}, then

nullity(u_{μ}) ≤ p^{n−2}(p − 1) if μ_{i} ≠ 0, for some i > 1,
nullity(u_{μ}) > p^{n−2}(p − 1) if μ = (1, 0, …, 0).

If some μ_{i} ≠ 0 (for i > 1) then again we may take this as our second basis vector in a basis of J(kE)/J^{2}(kE). So we may assume x_{2} = u_{μ} and we need to examine ker(u_{μ}). But it’s clear that

{∏_{i=3}^{n} x_{i}^{s_{i}}x_{2}^{p−1}x_{1}^{r}|1 ≤ r ≤ p − 1, and 0 ≤ s_{i} ≤ p − 1 for i > 2}

constitutes a basis and so nullity(u_{μ}) = p^{n−2}(p − 1), that is μ ∉ V_{E}^{r}(kEx_{1}), if some μ_{i} ≠ 0 (for i > 1). The only possibility left is μ_{i} = 0 for i > 1, that is u_{μ} = μ_{1}x_{1}, but it’s clear that then

{∏_{i=2}^{n} x_{i}^{s_{i}}x_{1}^{p−1}|0 ≤ s_{i} ≤ p − 1 for i > 1}

constitutes a basis for ker(u_{μ}), and hence nullity(u_{μ}) = p^{n−1} > p^{n−2}(p−1), so V_{E}^{r}(kEu_{λ}) = ℓ_{λ}. □

4.2. The stable map description. The elements u_{λ} ∈ kE, for 0 ≠ λ ∈ k^{n}, can be used to give the following, stable homomorphism, description of the rank variety. The proof of this is essentially contained in ([7], Lemma 3.7), but for completeness we include a proof here.

Theorem 4.3. Given M ∈ -mod kE and 0 ≠ λ ∈ k^{n} then

ℓ_{λ} ⊆ V_{E}^{r}(M) ⟺ \underline{Hom}_{kE}(kEu_{λ} ⊕ kEu_{λ}^{p−1}, M) ≠ 0.

Proof. Applying Hom_{kE}(−, M) to the exact sequence

kE ⟶^{u_{λ}} kE → kEu_{λ} → 0

and using the identification Hom_{kE}(kE, M) ≅ M, we obtain

Hom_{kE}(kEu_{λ}, M) ≅ ker(u_{λ} : M → M).

Similarly Hom_{kE}(kEu_{λ}^{p−1}, M) ≅ ker(u_{λ}^{p−1} : M → M). We now apply Hom_{kE}(−, M) to the short exact sequence

0 → kEu_{λ}^{p−1} → kE ⟶^{u_{λ}} kEu_{λ} → 0

giving the exact sequence

0 → Hom_{kE}(kEu_{λ}^{p−1}, M) → Hom_{kE}(kE, M) → Hom_{kE}(kEu_{λ}, M) → \underline{Hom}_{kE}(kEu_{λ}, M) → 0.

Using the identifications above and taking the alternating sum of the dimensions we arrive at

nullity(u_{λ}) − dim M + nullity(u_{λ}^{p−1}) = dim \underline{Hom}_{kE}(kEu_{λ}, M).

Similarly applying Hom_{kE}(−, M) to the short exact sequence

0 → kEu_{λ} → kE ⟶^{u_{λ}^{p−1}} kEu_{λ}^{p−1} → 0,

we have

nullity(u_{λ}) − dim M + nullity(u_{λ}^{p−1}) = dim \underline{Hom}_{kE}(kEu_{λ}^{p−1}, M).

These results together with the remark before equation (4.1) are exactly what we need.

□

5. The Avrunin-Scott Theorem

Here we recall the statement of the Avrunin-Scott theorem (Theorem 5.1) and discuss some of the issues involved in existing proofs. We then present an alternative proof from a more representation theoretic viewpoint using the stable map description of the varieties described earlier.

Given M ∈ kE -mod, we have two different varieties associated to M; the support variety V_{E}(M) and the rank variety V_{E}^{r}(M). The Avrunin-Scott theorem says that these two varieties agree. To be more precise there exists a morphism of the underlying affine varieties V_{E}^{r} and V_{E} that maps the subvariety V_{E}^{r}(M) onto V_{E}(M). Before stating the Avrunin-Scott theorem, we need to define this morphism.

From §3.3 we know that V_{E} ≅ 𝔸^{n}, and a particular identification amounts to, via the Bockstein map, a choice of k-basis of the vector space Ext_{kE}^{1}(k, k). From §4 we have V_{E}^{r} ≅ 𝔸^{n}, with a particular identification amounting to a choice of k-basis of Ext_{kE}^{1}(k, k). Clearly we should make the same choice of basis for Ext_{kE}^{1}(k, k) in both cases. Once this is done we can define the morphism of varieties F : V_{E}^{r} → V_{E} by

F : V_{E}^{r}(= 𝔸^{n}) →V_{E}(= 𝔸^{n})
F ((λ_{1}, …, λ_{n})) = \begin{cases}(λ_{1}, …, λ_{n}) & if p = 2 \\ (λ_{1}^{p}, …, λ_{n}^{p}) & for p > 2. \end{cases}

For p > 2 this is essentially the Bockstein map β : H^{1}(E, k) → H^{2}(E, k) whose image W is the subspace of Ext_{kE}^{2}(k, k) spanned by the generators of a Noether normalisation. Moreover in the case p > 2, the map F is a morphism of varieties and a bijection of sets, but clearly there is no inverse morphism to F.

We can now state

Theorem 5.1 (Avrunin-Scott). Given M ∈ kE -mod then

F (V_{E}^{r}(M)) = V_{E}(M).

Before giving an alternative proof of this, it’s worth dwelling on some aspects of the existing proofs (see [3]).

One of the difficulties (as outlined in [3]) in proving Theorem 5.1, is that, for H_{α} := ⟨1 + u_{α}⟩ a cyclic shifted subgroup of kE, the following diagram does not necessarily commute

\xymatrix{Ext_{kE}^{∗}(k, k) \ar[d]^{_{res_{E,Hα}}} \ar[r]^{−⊗M} & Ext_{kE}^{∗}(M, M) \ar[d]^{_{res_{E,Hα}}} \\ Ext_{kH_{α}}^{∗}(k, k) \ar[r]^{−⊗M} & Ext_{kH_{α}}^{∗}(M, M)}

Let’s assume p > 2 and suppose that the above diagram did commute and then see how an argument proving Theorem 5.1 would proceed. Let J(M) = I(M) ∩ k[x_{1}, … x_{n}] be the ideal defining the variety V_{E}(M), that is J(M) is the annihilator, in the Noether normalisation subring of H^{∗}(E, k), of 1_{M} ∈ Hom(M, M) the identity map. Let ζ ∈ J(M) be a homogeneous element, say ζ = f(x_{1}, … x_{n}), then to establish

F (V_{E}^{r}(M)) ⊆ V_{E}(M)

we would want to see that f(α_{1}^{p}, …, α_{n}^{p}) = 0 for α = (α_{1}, …, α_{n}) ∈ V_{E}^{r}(M). With the above diagram commuting we would have

res_{E,H_{α}}(ζ)1_{M} = res_{E,H_{α}}(ζ1_{M}) = 0.

A result of Carlson’s ([4],Proposition 2.20) says

res_{E,H_{α}}(ζ) = f(α_{1}^{p}, …, α_{n}^{p})γ

where γ is the canonical generator of Ext_{kH_{α}}^{∗}(k, k) in degree 2 deg(f). By construction α ∈ V_{E}^{r}(M) means M ↓_{H_{α}} is not free and so γ1_{M} ≠ 0. Hence f(α_{1}^{p}, …, α_{n}^{p}) = 0 as desired.

To obtain the reverse inclusion

V_{E}(M) ⊆ F (V_{E}^{r}(M)),

consider the corresponding inclusion of ideals. One would want to know that for ζ homogeneous and arbitrary subject to res_{E,H_{α}}(ζ) = 0 for all α ∈ V_{E}^{r}(M) (in other words ζ is obtained from a polynomial defining V_{E}^{r}(M) by raising coefficients to the power p), then ζ ∈ I(V_{E}(M)) = \sqrt{J(M)}. Still assuming the commutativity of the diagram we have

0 = res_{E,H_{α}}(ζ)1_{M} = res_{E,H_{α}}(ζ1_{M}).

Another theorem of Carlson’s ([5], Theorem 3.1) says that θ ∈ Ext_{kE}^{t}(M, M) (t ≥ 0) is nilpotent if and only if res_{E,H_{α}}(θ) is nilpotent for all H_{α} such that α ∈ V_{E}^{r}(M). Hence ζ1_{M} is nilpotent, i.e. ζ ∈ \sqrt{J(M)}, and we have the reverse inclusion.

In any event the above diagram does not necessarily commute, essentially because in restricting to a shifted subgroup we’re changing the group action on the image of the functor − ⊗_{k} M. In [3] this problem is somewhat circumvented by noting that V_{E}(M) = V_{E}(k, M) (as E is a p-group) and that the action of H^{∗}(E, k) on V_{E}(k, M) employs Yoneda composition i.e. not the Hopf structure needed for tensor products over k. However one still has to check that the morphism F (equivalently the Bockstein) is unaffected by changing to a shifted subgroup and this is what is done in [3].

5.1. Alternative proof of Theorem 5.1. To prove Theorem 5.1 we will use the stable map description of the varieties given in §3.5 and §4.2. For the support variety, the only properties needed for the modules T_{λ} in the statement of Theorem 3.6 were

ΩT_{λ} ≅ T_{λ}, and V_{E}(T_{λ}) = ℓ_{λ}.

The main fact to establish in proving Theorem 5.1 is the following

Proposition 5.2. For 0 ≠ λ ∈ k^{n},

V_{E}(kEu_{λ}) = ℓ_{F (λ)}.

With this proposition we have

Proof of Theorem 5.1. As kEu_{λ} is indecomposable and periodic, the remark above together with Proposition 5.2 says we may take T_{F (λ)} = kEu_{λ} ⊕ kEu_{λ}^{p−1} if p > 2, and simply take T_{F (λ)} = T_{λ} = kEu_{λ} if p = 2.

The stable map description of the support variety, Theorem 3.6, says

ℓ_{F (λ)} ⊆ V_{E}(M) ⟺ \underline{Hom}_{kE}(T_{F (λ)}, M) ≠ 0.

The stable map description of the rank variety, Theorem 4.3, says

ℓ_{λ} ⊆ V_{E}^{r}(M) ⟺ \underline{Hom}_{kE}(kEu_{λ} ⊕ kEu_{λ}^{p−1}, M) ≠ 0.

(obviously \underline{Hom}_{kE}(kEu_{λ}, M) ≠ 0 suffices for p = 2).

Taken together they clearly finish the proof. □

Before proving Proposition 5.2, we will need to understand how the elements; z_{i} ∈ Ext_{kE}^{2}(k, k), for p > 2, and y_{i} ∈ Ext_{kE}^{1}(k, k), for p = 2, described in Theorem 3.2 act when viewed as maps in the stable category. Let kE = k⟨x_{1}, …, x_{n}⟩ be as described in §4. Under the identification in the definition of the morphism F described above, the y_{i} ∈ Ext_{kE}^{1}(k, k) appearing in §3.3 are the dual basis vectors, y_{i} = x_{i}^{∨} and, for p > 2, z_{i} = β(y_{i}). With this notation we have the following

Lemma 5.3. For p > 2 the maps Ω^{−1}z_{i} ∈ \underline{Hom}_{kE}(Ωk, Ω^{−1}k) (for 1 ≤ i ≤ n) can be factored as Ω^{−1}z_{i} = πη_{i}, where

π : kE → kE/ soc kE ≅ Ω^{−1}k

is the natural quotient map and η_{i} : Ωk → kE is the linear map defined by η_{i}(rad Ωk) = 0 and

η_{i}(x_{j}) = \begin{cases}0 & for i ≠ j \\ c_{i} := ∏_{\{s|s≠i\}} x_{s}^{p−1} & if i = j. \end{cases}

If p = 2, then the maps Ω^{−1}y_{i} ∈ \underline{Hom}_{kE}(k, Ω^{−1}k) admit a factorization Ω^{−1}y_{i} = πη_{i}, where π is as before and η_{i} : k → kE is given by

η_{i}(1) = c_{i} := ∏_{{s|s≠i}} x_{s}.

Proof. We will assume p > 2 as the proof in the case p = 2 is similar and somewhat simpler. The cohomology of kE can be obtained by noting that

kE = k[X_{1}, …, X_{n}]/(X_{i}^{p}) = k[X_{1}]/(X_{1}^{p}) ⊗_{k} ⋯ k[X_{n}]/(X_{n}^{p}).

Elements in the cohomology of k[X_{i}]/(X_{i}^{p}) are then tensored together to give, by the Künneth theorem, elements in the cohomology of kE. In giving this presentation of kE in terms of x_{i} we have chosen a basis for Ext_{kE}^{1}(k, k), the corresponding elements z_{i} ∈ Ext_{kE}^{2}(k, k) as described in §3.3 are then easily seen to be obtained, by the above process, from maps z′_{i} in the following diagram

\xymatrix{0 \ar[r] & kX_{i}^{p−1} \ar[r] \ar[d]^{z′_{i}} & k[X_{i}]/(X_{i}^{p}) \ar[r]^{X_{i}} & k[X_{i}]/(X_{i}^{p}) \\  & k}

where z′_{i} maps X_{i}^{p−1} to 1. If we carry out this tensoring, we see that z_{i} comes from the diagram

\xymatrix{0 \ar[r] & Ω^{2}(k) \ar[r] \ar[d]^{z_{i}} & ⨁^{n} kE \ar[r]^{m} & kE \\  & k}

where m = (^{X_{1}} ^{···} ^{X_{n}} )^{tr} and z_{i} is the map that sends X_{i}^{p−1}kE ⊂ Ω_{kE}^{2}(k) to k, induced by z′_{i} (note all terms involving X_{j} for j ≠ i are sent to 0.)

To calculate Ω^{−1}(z_{i}) we must calculate the map α_{i} in the diagram

\xymatrix{0 \ar[r] & Ω^{2}(k) \ar[r] \ar[d]^{z_{i}} & ⨁^{n} kE \ar[r] \ar[d]^{α_{i}} & im(m) = Ω(k) \ar[r] \ar[d]^{Ω^{−1}(z_{i})} & 0 \\ 0 \ar[r] & k \ar[r] & kE \ar[r]^{π} & kE/ soc(kE) \ar[r] & 0}

A diagram chase shows that α_{i} maps the i^{th} summand to kE by sending 1 to c_{i} = ∏_{{s|s≠i}} x_{s}^{p−1}, and all other summands to 0. Hence we have the factorisation of Ω^{−1}(z_{i}) as desired. □

Given 0 ≠ μ = (μ_{1}, …, μ_{n}) ∈ k^{n}, take

ζ := \begin{cases}∑_{i=1}^{n} μ_{i}z_{i} ∈ Ext_{kE}^{2}(k, k) & if p > 2, \\ ∑_{i=1}^{n} μ_{i}y_{i} ∈ Ext_{kE}^{1}(k, k) & for p = 2. \end{cases}

Lemma 5.3 says Ω^{−1}ζ = πη where η = ∑_{i=1}^{n} μ_{i}η_{i}. The corresponding module M_{ζ} constructed in §3.2 has, by Theorem 3.1, support variety V_{E}(M_{ζ}) = V (⟨ζ⟩) which is the hypersurface determined by the element ζ. But clearly under the identification V_{E} = 𝔸^{n}, this hypersurface is the hyperplane through the origin perpendicular to the line ℓ_{μ} and we’ll denote this hyperplane by ^{⊥}ℓ_{μ}. So we have

V_{E}(M_{ζ}) = ^{⊥}ℓ_{μ}.

The key fact we will need to establish Proposition 5.2 is the following.

Lemma 5.4. With the above notation, if 0 ≠ λ, μ ∈ k^{n} are such that F (λ) ∈ ^{⊥}ℓ_{μ} (i.e. ∑_{i=1}^{n} λ_{i}^{p}μ_{i} = 0 if p > 2, ∑_{i=1}^{n} λ_{i}μ_{i} = 0 if p = 2) then there is an monomorphism

kEu_{λ} → M_{ζ}

that yields a non-zero element in \underline{Hom}_{kE}(kEu_{λ}, M_{ζ}).

Proof. Assume p > 2 and recall the diagram defining M_{ζ},

[figure: 0; k; M_{ζ}; Ωk; −Ω^{−1}ζ; 0; 0; k; kE; π; Ω^{−1}k; 0]

It suffices to determine a map f in the following diagram

kEu_{λ} ⟶^{f} kE ⊕ Ωk ⟶^{(π,Ω^{−1}ζ)} Ω^{−1}k

so that the composite is zero. As we have a cyclic module we only have to determine the image of u_{λ}. Setting f(u_{λ}) = (η(u_{λ}), −u_{λ}) (recall Ω^{−1}ζ = πη), we will have such a map, provided the annihilator of u_{λ} kills f(u_{λ}). But the annihilator of u_{λ} is generated by u_{λ}^{p−1}. So we have to check that u_{λ}^{p−1}η(u_{λ}) = 0. By Lemma 5.4, η(u_{λ}) = ∑_{i=1}^{n} λ_{i}μ_{i}c_{i}, and calculation shows

u_{λ}^{p−1}c_{i} = λ_{i}^{p−1}c,

where c = ∏_{i=1}^{n} x_{i}^{p−1} generates the socle of kE. Hence we have

u_{λ}^{p−1}η(u_{λ}) = (∑_{i=1}^{n} λ_{i}^{p}μ_{i})c

and this is zero by our assumptions.

If p = 2, the corresponding diagram for M_{ζ} is

[figure: 0; k; M_{ζ}; k; −Ω^{−1}ζ; 0; 0; k; kE; π; Ω^{−1}k; 0]

As before it suffices to determine a map f in the following diagram

kEu_{λ} ⟶^{f} kE ⊕ k ⟶^{(π,Ω^{−1}ζ)} Ω^{−1}k

so that the composite is zero, and that means we have to specify the image of u_{λ}. Setting f(u_{λ}) = (η(1), −1) will work (Ω^{−1}ζ = πη), again provided the annihilator of u_{λ} kills f(u_{λ}). But the annihilator of u_{λ} is generated by u_{λ} itself and clearly kills the trivial module k, so we have to check that u_{λ}η(1) = 0. By Lemma 5.4 η(1) = ∑_{i=1}^{n} μ_{i}c_{i}, and we have

u_{λ}(∑_{i=1}^{n} μ_{i}c_{i}) = ∑_{i,j} λ_{j}μ_{i}x_{j}c_{i} = (∑_{i=1}^{n} λ_{i}μ_{i})c

where c = ∏_{i=1}^{n} x_{i} generates the socle of kE. So we require ∑_{i=1}^{n} λ_{i}μ_{i} = 0, and this is our assumption.

In both constructions above it’s clear that f is a monomorphism and it’s easy to see that f doesn’t factor through a projective. □

Proof of Proposition 5.2. We know that kEu_{λ} is an indecomposable periodic (of period 2 if p > 2 and period 1 if p = 2) kE-module. It follows from Theorem 3.5 that the support variety is a line. That is

V_{E}(kEu_{λ}) = ℓ_{α},

for some 0 ≠ α ∈ k^{n}. As noted at the beginning of §5.1, we may therefore take T_{α} = kEu_{λ} ⊕ kEu_{λ}^{p−1} if p > 2, or T_{α} = kEu_{λ} if p = 2.

Now take 0 ≠ μ = (μ_{1}, …, μ_{n}) ∈ k^{n}, let

ζ = \begin{cases}∑_{i=1}^{n} μ_{i}z_{i} ∈ Ext_{kE}^{2}(k, k) & if p > 2, \\ ∑_{i=1}^{n} μ_{i}y_{i} ∈ Ext_{kE}^{1}(k, k) & for p = 2. \end{cases}

and consider the corresponding module M_{ζ}. Its support variety is a hyperplane, V_{E}(M_{ζ}) = ⊥ℓ_{μ}. From the stable map description, Theorem 3.6, we have

ℓ_{α} ⊆ V_{E}(M_{ζ}) ⟺ \underline{Hom}_{kE}(T_{α}, M_{ζ}) ≠ 0.

But we know from Lemma 5.4 that if we choose μ ∈ k^{n} so that F (λ) ∈ ^{⊥}ℓ_{μ}, then \underline{Hom}_{kE}(T_{α}, M_{ζ}) ≠ 0.

It follows from this that the line ℓ_{α} lies in every hyperplane that contains the line ℓ_{F (λ)}. The intersection of all hyperplanes containing a given line is precisely that line, hence we have ℓ_{α} = ℓ_{F (λ)}. In other words

V_{E}(kEu_{λ}) = ℓ_{F (λ)}.

□

6. The algebras Λ_{m}

The remainder of this paper will be concerned with examining an analogue of the Avrunin-Scott theorem for a class of finite-dimensional symmetric k-algebras, Λ_{m}, which are defined by Definition 6.1. For k an algebraically closed field,

Λ_{m} := k[X_{1}, …, X_{m}]/(X_{i}^{2}).

The point here is that we don’t specify the characteristic of the field k. If char(k) = 2 then, of course, Λ_{m}, is just the group algebra of an elementary abelian 2-group of rank m. For arbitrary characteristic Λ_{m} will not be a group algebra (or even obviously a Hopf algebra), so we will need a replacement for the support variety.

In the next section we recall the results from [12] and [8] that develop a suitable support variety for self-injective algebras, provided certain finiteness conditions are satisfied. These varieties are defined in terms of the Hochschild cohomology ring and so we record here the calculation of the Hochschild cohomology for the algebras Λ_{m}. It will be evident from these calculations that the algebras Λ_{m} satisfy the finiteness conditions needed for the support variety theory.

Theorem 6.2. If char(k) ≠ 2 then

HH^{∗}(Λ) ≅ (k[z_{1}, …, z_{m}] ⊗_{k} Λ(y_{1}, …, y_{m}))[w_{1}, …, w_{m}]/(w_{i}z_{i}, w_{i}y_{i}, w_{i}^{2})

where deg(z_{i}) = 2, deg(y_{i}) = 1 and deg(w_{i}) = 0. If char(k) = 2 then

HH^{∗}(Λ) ≅ k[z_{1}, …, z_{m}, w_{1}, …, w_{m}]/(w_{i}^{2}),

with deg(z_{i}) = 1 and deg(w_{i}) = 0.

Moreover

Ext_{Λ_{m}}^{∗}(k, k) = k[α_{1}, …, α_{m}]

where deg(α_{i}) = 1.

It’s also easy to see that Ext_{Λ_{m}}^{∗}(k, k) is finitely generated over the image of HH(Λ_{m}) under the action from the map ϕ = − ⊗_{Λ_{m}} k. In fact

Ext_{Λ_{m}}^{∗}(k, k) = im(ϕ) ⊕ ⨁_{i} im(ϕ)α_{i}.

7. Support varieties

In this section we start by considering an arbitrary finite dimensional self-injective kalgebra Λ, before specialising to the case Λ = Λ_{m} in subsection 7.2. In [12] and [8] a support variety for self-injective algebras is developed. This theory assumes certain finite generation conditions on the Hochschild cohomology of the algebra Λ. We briefly recall the definitions and finiteness conditions as well as some of the key results from [12] and [8] that are analogues of results in the group case. We then provide a stable map description of the support variety of Λ_{m} (Theorem 7.7) in analogy to Theorem 3.6

We first note that HH^{∗}(Λ) is a graded commutative k-algebra, and that given M, N ∈ Λ -mod we have an action of HH^{∗}(Λ) on Ext_{Λ}^{∗}(M, N) given by the map

HH^{∗}(Λ) ⟶^{−⊗_{Λ}M} Ext_{Λ}^{∗}(M, M)

followed by Yoneda composition. As shown in [12], this commutes, up to graded commutativity, with the map induced by − ⊗_{Λ} N followed by Yoneda composition. If M ∈ mod- Λ, then we consider the map HH^{∗}(Λ) ⟶^{M⊗_{Λ}−} Ext_{Λ}^{∗}(M, M) instead.

We recall the finite generation assumptions on Λ from [8]

Assumption 1 (Fg1). There exists a suitable graded subalgebra H = H^{∗} ⊆ HH^{∗}(Λ) such that

(i) H is a commutative affine ring, (ii) H^{0} = HH^{0}(Λ) = Z(Λ).

Clearly the crucial aspect of this assumption is on the existence of a suitable H. Generally this means the second assumption holding also.

Assumption 2 (Fg2). Ext_{Λ}^{∗}(Λ/𝔯, Λ/𝔯) is a finitely generated H-module.

Clearly, taken together assumptions Fg1 and Fg2 provide a replacement of the Evens-Golod-Venkov theorem for HH^{∗}(Λ), that is so crucial to the support variety for group algebras.

We will assume that Λ satisfies both Fg1 and Fg2. We now recall (see [12] and [8]) some definitions and basic properties. The underlying affine variety, V_{H} is defined to be

V_{H} := MaxSpec- H.

Given M, N ∈ Λ -mod we have seen that Ext_{Λ}^{∗}(M, N) becomes a finitely generated Hmodule. Denote the annihilator of this module by I(M, N) and define the support variety for the pair of modules to be the variety associated to the ideal I(M, N), that is

V_{H}(M, N) := {m ∈ V_{H}|I(M, N) ⊆ m}.

The support variety for M ∈ Λ -mod is then given by

V_{H}(M) := V_{H}(M, M).

Similarly for M ∈ mod- Λ. We also have V_{H} = V_{H}(Λ/𝔯). These varieties are homogeneous affine varieties.

These varieties satisfy analogues of the elementary properties (i)-(iv) in §3.1. We record only those that are slightly different.

(i)′ V_{H}(M) = V_{H}(DM) = V_{H}(ΩM).

(iii)′ V_{H}(M) = V_{H}(M, Λ/𝔯) = V_{H}(Λ/𝔯, M).

(iv)′ V_{H}(M) ∩ V_{H}(M′) = {0} ⇒ Ext_{Λ}^{n}(M, M′) = 0, for n >> 0.

However, as Λ does not necessarily possess a Hopf structure it doesn’t make sense for M ⊗_{k} N to be a Λ-module. Hence we do not have a tensor intersection rule, nevertheless there is a kind of replacement involving analogues of the L_{ζ} modules of §3.2.

7.1. The M_{ζ} bimodules. We need to recall the construction, in [8], of the Λ-bimodules M_{ζ} that are analogues of the modules M_{ζ} = Ω^{−1}(L_{ζ}) considered in §3.2. Given a homogeneous element ζ ∈ H^{n}, we may represent it as a map ζ : Ω_{Λ^{e}}^{n}(Λ) → Λ. The Λ^{e}-module M_{ζ} is then defined by the pushout diagram

[figure: 0; Ω_{Λ^{e}}^{n}(Λ); ζ; P^{n−1}; Ω_{Λ^{e}}^{n−1}(Λ); 0; 0; Λ; M_{ζ}; Ω_{Λ^{e}}^{n−1}(Λ); 0]

where P^{n−1} is a projective Λ-bimodule.

We now have the following result (see [8])

Theorem 7.1. For ζ ∈ H, a homogeneous element, and M ∈ Λ -mod, then

V_{H}(M_{ζ} ⊗_{Λ} M) = V_{H}(⟨ζ⟩) ∩ V_{H}(M).

Note if L_{ζ} := M_{ζ} ⊗_{Λ} Λ/𝔯 ∈ Λ -mod, then V_{H}(L_{ζ}) = V_{H}(⟨ζ⟩). The bimodules M_{ζ} are projective as left or right Λ-modules and can be used to construct a module with a given variety. In particular we have the analogue to Theorem 3.3.

Theorem 7.2. Given any closed homogeneous subvariety, V ⊆ V_{H} there exists a module M ∈ Λ -mod such that V_{H}(M) = V.

Proof. Let I(V) = ⟨ζ_{1}, …, ζ_{n}⟩, with ζ_{i} ∈ H^{∗} homogeneous elements. It then follows from Theorem 7.1 that with M := M_{ζ_{1}} ⊗_{Λ} ⋯ ⊗_{Λ} M_{ζ_{n}} ⊗_{Λ} Λ/𝔯, we have

V_{H}(M) = V (I(V)) ∩ V_{H}(Λ/𝔯) = V.
□

Finally before turning to Λ_{m} and a stable map description of its support varieties we recall the analogous result (see [8], Theorem 4.3 and Proposition 4.4) to Theorem 3.4

Theorem 7.3. If M ∈ Λ -mod is indecomposable and has variety a line, then M is a periodic module. Moreover the period must divide the degree of one of the homogeneous generators of H.

We also record here for completeness the analogue of (Carlson’s) Theorem 3.5 (see [8], Proposition 4.2)

Theorem 7.4. If M ∈ Λ -mod is an indecomposable periodic module then V_{H}(M) is a line.

7.2. Support varieties and stable map description for Λ_{m}. We now specialize to the case Λ = Λ_{m}, and provide a stable map description of support varieties, analogous to Theorem 3.6. For Λ_{m}, we have V_{HH(Λ_{m})} ≅ 𝔸^{m}, by Theorem 6.2, and hence lines ℓ_{λ} ⊆ V_{HH(Λ_{m})} are determined by 0 ≠ λ ∈ k^{m}.

Given 0 ≠ μ = (μ_{1}, …, μ_{n}) ∈ k^{n}, and ζ = ∑_{i=1}^{n} μ_{i}z_{i} ∈ HH^{2}(Λ_{m}), then the variety in 𝔸^{m} defined by the ideal ⟨ζ⟩ is simply ^{⊥}ℓ_{μ}, the hyperplane perpendicular to the line ℓ_{μ}. Clearly given some line ℓ_{λ} ⊆ 𝔸^{m} we can choose m − 1 hyperplanes ^{⊥}ℓ_{μ_{i}} whose intersection is the given line ℓ_{λ}. If ζ_{i} ∈ HH^{2}(Λ_{m}) are elements corresponding to μ_{i} ∈ k^{m} and if we set M\underline{ζ} := M_{ζ_{1}} ⊗_{Λ_{m}} ⋯ ⊗_{Λ_{m}} M_{ζ_{m−1}} then by repeated use of Theorem 7.1 we have

(7.1)  V_{HH(Λ_{m})}(M\underline{ζ} ⊗_{Λ_{m}} k) = ℓ_{λ}.

We record explicitly the results of Theorem 7.1 for this case, as this will be needed in section 9.

Lemma 7.5. For 0 ≠ λ ∈ 𝔸^{m}, and M\underline{ζ} chosen so that (7.1) holds then

ℓ_{λ} ⊆ V_{HH(Λ_{m})}(M) ⟺ V_{HH(Λ_{m})}(M\underline{ζ} ⊗_{Λ_{m}} M) ≠ 0.

We will also need to consider the dual bimodules, M_{ζ}^{∗}. In particular we have

Theorem 7.6. Let ζ ∈ HH^{n}(Λ_{m}) and M ∈ Λ_{m} -mod, then

V_{H}(M_{ζ}^{∗} ⊗_{Λ_{m}} M) = V_{H}(⟨ζ⟩) ∩ V_{H}(M).

Proof. Consider the dual diagram to the pushout diagram defining M_{ζ}. The top row of this dual diagram is

0 → (Ω_{Λ_{m}^{e}}^{n−1}(Λ_{m}))^{∗} → M_{ζ}^{∗} → Λ_{m}^{∗} → 0

Using the fact that (Ω^{k}(−))^{∗} ≅ Ω^{−k}(−^{∗}) together with Λ_{m}^{∗} ≅ Λ_{m} as bimodules (since Λ_{m} is symmetric) we can apply Ω^{n−1}(−) to this sequence and obtain the short exact sequence of bimodules

0 → Λ → Ω_{Λ_{m}^{e}}^{n−1}(M_{ζ}^{∗}) ⊕ proj → (Ω_{Λ_{m}^{e}}^{n−1}(Λ_{m})) → 0.

By considering the projective bimodule resolution of Λ_{m}, we must have some η ∈ HH^{n}(Λ_{m}) such that M_{η} = Ω_{Λ_{m}^{e}}^{n−1}(M_{ζ}^{∗})⊕proj. It follows from Theorem 7.1 that we need only consider the case M = k, and in fact it suffices to show that

V_{HH(Λ_{m})}(⟨η⟩) = V_{HH(Λ_{m})}(M_{η} ⊗_{Λ_{m}} k) = V_{HH(Λ_{m})}(M_{ζ} ⊗_{Λ_{m}} k) = V_{HH(Λ_{m})}(⟨ζ⟩),

because Ω_{Λ}^{n−1}(M_{ζ}^{∗} ⊗_{Λ_{m}} k) ≅ Ω_{Λ_{m}^{e}}^{n−1}(M_{ζ}^{∗}) ⊗_{Λ_{m}} k ⊕ (proj) and using standard properties of the support varieties we have V_{HH(Λ_{m})}(M_{η} ⊗_{Λ_{m}} k) = V_{HH(Λ_{m})}(Ω_{Λ}^{n−1}(M_{ζ}^{∗} ⊗_{Λ_{m}} k)) = V_{HH(Λ_{m})}(M_{ζ}^{∗} ⊗_{Λ_{m}} k). We can now use the fact that for any M ∈ Λ_{m} -mod, V_{HH(Λ_{m})}(M) = V_{HH(Λ_{m})}(M, k) = V_{HH(Λ_{m})}(k, M), together with adjointness, to obtain

V_{HH(Λ_{m})}(M_{ζ}^{∗}⊗_{Λ_{m}}k) = V_{HH(Λ_{m})}(k, M_{ζ}^{∗}⊗_{Λ_{m}}k) = V_{HH(Λ_{m})}(M_{ζ}⊗_{Λ_{m}}k, k) = V_{HH(Λ_{m})}(M_{ζ}⊗_{Λ_{m}}k).

□

We now have the stable map description of the support varieties, analogous to Theorem 3.6.

Theorem 7.7. Let M ∈ Λ_{m} -mod, 0 ≠ λ ∈ k^{m} and M\underline{ζ} be chosen so that (7.1) holds. Denote by T_{λ} the non-projective summand of M\underline{ζ}. Then

ℓ_{λ} ⊆ V_{HH(Λ_{m})}(M) ⟺ \underline{Hom}_{Λ_{m}}(T_{λ} ⊕ ΩT_{λ}, M) ≠ 0.

Proof. Note first that by Theorem 7.3 the summands of T_{λ} are periodic of period at most 2. Hence T_{λ} is itself periodic of period at most 2.

We have by Theorem 7.6 that

V_{HH(Λ_{m})}(M ∗ ; \underline{ζ} ⊗_{Λ_{m}} M) = ℓ_{λ} ∩ V_{HH(Λ_{m})}(M).

But V_{HH(Λ_{m})}(M ∗ ; \underline{ζ} ⊗_{Λ_{m}} M) = V_{HH(Λ_{m})}(k, M ∗ ; \underline{ζ} ⊗_{Λ_{m}} M) = 0 if and only if Ext_{Λ}^{i}(k, M ∗ ; \underline{ζ} ⊗_{Λ_{m}} M) = 0 for i >> 0. But

Ext_{Λ_{m}}^{i}(k, M ∗ ; \underline{ζ} ⊗_{Λ_{m}} M) = Ext_{Λ_{m}}^{i}(M\underline{ζ} ⊗_{Λ_{m}} k, M) = Ext_{Λ_{m}}^{i}(T_{λ}, M) = \underline{Hom}_{Λ_{m}}(Ω^{ϵ}T_{λ}, M).

For ϵ = i( mod 2). In other words ℓ_{λ} ⊆ V_{HH(Λ_{m})}(M) if and only if Ext_{Λ_{m}}^{i}(T_{λ}, M) ≠ 0 for i >> 0, which is equivalent to \underline{Hom}_{Λ_{m}}(T_{λ} ⊕ ΩT_{λ}, M) ≠ 0. □

Remark. This result is weaker than the corresponding Theorem 3.6 because we haven’t replaced T_{λ} by any module which has support variety a line ℓ_{λ}.

8. Rank varieties

In this section we briefly recall the rank variety for Λ_{m} modules as well as a stable map description that were introduced in [7].

If ρ : C_{m} → GL(W) is an irreducible representation of the m-generated Clifford algebra, C_{m}, and M is a Λ_{m}-module where the action of X_{i} on M is denoted by y_{i}, then for 0 ≠ λ ∈ k^{n} the following map, σ(λ), is considered in [7] where

σ_{M}(λ) := ∑_{j=1}^{m} λ_{j}ρ(e_{j}) ⊗_{k} y_{j} : W ⊗_{k} M → W ⊗_{k} M.

Here e_{j} are the standard generators of the Clifford algebra.

In [7] the rank variety was defined as a projective variety, but in order to be consistent with the other varieties we’re considering we’ll equivalently view it as a homogeneous affine variety. With this in mind the rank variety, V^{r}(M), of the module M is defined by

V^{r}(M) := {0} ∪ {0 ≠ λ ∈ k^{n}| rank(σ_{M}(λ)) < (1/2) dim(W) dim(M)}.

Clearly V^{r}(M) is a union of lines in the underlying affine m-space 𝔸^{m} which we’ll denote by V^{r}. As noted in [7] (section 4) there is an action of the group ∏^{m} ℤ/2ℤ, of all coordinate sign changes, on V^{r} and V^{r}(M) is invariant under this action; rephrased in the notation of section 9, if μ, λ ∈ V^{r} are such that λ ∈ V^{r}(M) and F (λ) = F (μ) then μ ∈ V^{r}(M).

If M = Λ_{m} then the module V (λ) is defined to be the image of the map σ : W ⊗_{k} Λ_{m} → W ⊗_{k} Λ_{m} defined above. It is then shown in [7] that the rank variety admits the following stable map interpretation.

Theorem 8.1. For 0 ≠ λ ∈ k^{m} and M ∈ -mod Λ_{m} then

ℓ_{λ} ⊆ V^{r}(M) ⟺ \underline{Hom}_{Λ_{m}}(V (λ), M) ≠ 0.

Moreover the modules V (λ) are periodic and in fact Ω(V (λ)) ≅ V (λ). We will also need the important fact that the rank variety detects projectivity (see Theorem 4.3 in [7]).

Theorem 8.2. For M ∈ Λ_{m} -mod, then

V^{r}(M) = {0} ⟺ M ∈ Λ_{m} -proj .

We finish this subsection with an easy observation that will prove useful. Given ζ ∈ HH(Λ_{m}) some homogeneous element, of degree n say, and M ∈ Λ_{m} -mod, apply − ⊗_{Λ_{m}} M to the short exact sequence that is the second line in the pushout diagram defining M_{ζ} in subsection 7.1. Since all modules in that diagram are projective as right Λ_{m} modules, we have the short exact sequence

0 → M → M_{ζ} ⊗_{Λ_{m}} M → Ω_{Λ_{m}^{e}}^{n−1}(Λ_{m}) ⊗_{Λ_{m}} M → 0.

We can now use the fact that Ω_{Λ_{m}^{e}}^{n−1}(Λ_{m}) ⊗_{Λ_{m}} M ≅ Ω^{n−1}(M) ⊕ proj together with some basic properties of rank varieties (see [7]) to see Lemma 8.3. With the above notation, we have

V^{r}(M_{ζ} ⊗_{Λ_{m}} M) ⊆ V^{r}(M).

9. Analogue of the Avrunin Scott theorem

With both the support and rank variety theory for the algebras Λ_{m} looking similar to the group case, we might hope that there’s also a close connection between the two as that in the group case embodied by the Avrunin-Scott theorem.

As before we first need to consider what should be the map (or morphism) between the underlying affine varieties. In the identification of V^{r} with 𝔸^{m} of §8 there was no need to make a choice of basis for rad(Λ_{m})/ rad^{2}(Λ_{m}) as it already had a distinguished basis ( viz the X_{i}). Similarly for the Hochschild cohomology ring in §6, the generators z_{i} of the Noether normalisation are again distinguished generators. So, in effect, we have already chosen compatible identifications of V^{r} and V_{HH(Λ_{m})} with 𝔸^{m}.

We now define the morphism F by

F : V^{r}(= 𝔸^{m}) →V_{HH(Λ_{m})}(= 𝔸^{m})
F ((λ_{1}, …, λ_{n})) = \begin{cases}(λ_{1}, …, λ_{n}) & if char(k) = 2, \\ (λ_{1}^{2}, …, λ_{n}^{2}) & for char(k) ≠ 2. \end{cases}

We can now state the analogue of the Avrunin-Scott theorem.

Theorem 9.1. Given M ∈ Λ_{m} -mod then

F (V^{r}(M)) = V_{HH(Λ_{m})}(M).

As in the group case, the main task will be to establish

Proposition 9.2. For 0 ≠ λ ∈ k^{n},

V_{HH(Λ_{m})}(V (λ)) = ℓ_{F (λ)}.

With this proposition we have

Proof of Theorem 9.1. The stable map description of the rank variety, Theorem 8.1, gives

ℓ_{λ} ⊆ V^{r}(M) ⟺ \underline{Hom}_{Λ_{m}}(V (λ), M) ≠ 0

Since V (λ) is periodic of period 1, we have

\underline{Hom}_{Λ_{m}}(V (λ), M) ≠ 0 ⟺ Ext_{Λ_{m}}^{i}(V (λ), M) ≠ 0 (for i > 0).

But property (iv)′ of the support varieties listed in section 7 says

Ext_{Λ_{m}}^{n}(V (λ), M) ≠ 0 (for n >> 0) ⇒ V_{HH(Λ_{m})}(V (λ)) ∩ V_{HH(Λ_{m})}(M) ≠ {0}.

By Proposition 9.2, this is equivalent to ℓ_{F (λ)} ⊆ V_{HH(Λ_{m})}(M), so that we have an inclusion

F (V^{r}(M)) ⊆ V_{HH(Λ_{m})}(M).

To obtain the reverse inclusion, let ℓ_{μ} ⊆ V_{HH(Λ_{m})}(M) and take M\underline{ζ}, as in subsection 7.2, such that V_{HH(Λ_{m})}(M\underline{ζ} ⊗_{Λ_{m}} k) = ℓ_{μ}. By Lemma 7.5, V_{HH(Λ_{m})}(M\underline{ζ} ⊗_{Λ_{m}} M) = ℓ_{μ}.

Consider now F (V^{r}(M\underline{ζ}⊗_{Λ_{m}}M)), we have seen that this is contained in V_{HH(Λ_{m})}(M\underline{ζ}⊗_{Λ_{m}} V (λ)). Suppose F (V^{r}(M\underline{ζ}⊗_{Λ_{m}}M)) = \{0\}, so that V^{r}(M\underline{ζ}⊗_{Λ_{m}}V (λ)) = \{0\} then this would mean M\underline{ζ} ⊗_{Λ_{m}} M is projective, by Theorem 8.2. In particular V_{HH(Λ_{m})}(M\underline{ζ} ⊗_{Λ_{m}} M) = \{0\}. Since this is not the case, we must have V^{r}(M\underline{ζ} ⊗_{Λ_{m}} M) = \{ℓ_{λ}|F (λ) = μ\}. To finish, we simply use Lemma 8.3 to obtain {ℓ_{λ}|F (λ) = μ} ⊆ V^{r}(M). □

The proof of Proposition 9.2 will be, with suitable modifications, a variation on that given for Proposition 5.2 in the group case. We begin by giving the suitable modifications of Lemma 5.3 (as Lemma 9.3) and Lemma 5.4 (as Lemma 9.5).

Given z_{i} ∈ HH^{2}(Λ_{m}), one of the generators of the Noether normalisation, we have z_{i} ⊗_{Λ_{m}} k ∈ Ext_{Λ_{m}}^{2}(k, k) and it’s these elements that allow us to write out the analogues of Lemmas 5.3 and 5.4.

Lemma 9.3. The maps Ω^{−1}(z_{i} ⊗_{Λ_{m}} k) ∈ \underline{Hom}_{Λ_{m}}(Ωk, Ω^{−1}k) (for 1 ≤ i ≤ m) can be factored as Ω^{−1}(z_{i} ⊗_{Λ_{m}} k) = πη_{i}, where

π : Λ_{m} → Λ_{m}/ soc Λ_{m} ≅ Ω^{−1}k

is the natural quotient map and η_{i} : Ωk → Λ_{m} is the linear map defined by η_{i}(rad Ωk) = 0 and

η_{i}(x_{j}) = \begin{cases}0 & for i ≠ j \\ c_{i} := ∏_{\{s|s≠i\}} x_{s} & if i = j. \end{cases}

Proof. Exactly the same as that for Lemma 5.3. □

Given 0 ≠ μ = (μ_{1}, …, μ_{n}) ∈ k^{n}, and ζ = ∑_{i=1}^{n} μ_{i}z_{i} ∈ HH^{2}(Λ_{m}) then Lemma 9.3 says Ω^{−1}(ζ ⊗_{Λ_{m}} k) = πη where η = ∑_{i=1}^{m} μ_{i}η_{i}. We recall that the corresponding module M_{ζ} ⊗_{Λ_{m}} k has variety a hyperplane,

V_{HH(Λ_{m})}(M_{ζ} ⊗_{Λ_{m}} k) = ^{⊥}ℓ_{μ}.

With this notation we have the following

Lemma 9.4. For 0 ≠ μ, λ ∈ k^{m} let T_{λ} ∈ Λ_{m} -mod be such that V_{HH(Λ_{m})}(T_{λ}) = ℓ_{λ} then

\underline{Hom}_{Λ_{m}}(T_{λ}, M_{ζ} ⊗_{Λ_{m}} k) ≠ 0 ⟺ ℓ_{λ} ⊆ ^{⊥}ℓ_{μ}.

Proof. We have by adjunction,

\underline{Hom}_{Λ_{m}}(T_{λ}, M_{ζ} ⊗_{Λ_{m}} k) ≠ 0 ⟺ \underline{Hom}_{Λ_{m}}(M_{ζ}^{∗} ⊗_{Λ_{m}} T_{λ}, k) ≠ 0.

But

\underline{Hom}_{Λ_{m}}(M_{ζ}^{∗} ⊗_{Λ_{m}} T_{λ}, k) ≠ 0 ⟺ V_{HH(Λ_{m})}(M_{ζ}^{∗} ⊗_{Λ_{m}} T_{λ}) ≠ \{0\},

and from Theorem 7.6 we have

V_{HH(Λ_{m})}(M_{ζ}^{∗} ⊗_{Λ_{m}} T_{λ}) = ⟨ζ⟩ ∩ V_{HH(Λ_{m})}(T_{λ}) = ^{⊥}ℓ_{μ} ∩ ℓ_{λ}.

The result follows. □

Continuing with the above notation (i.e. ζ ∈ HH^{2}(Λ_{m}) such that V_{HH(Λ_{m})}(M_{ζ} ⊗_{Λ_{m}} k) = ⊥ℓ_{μ}), we have the following key fact that, in conjunction with the previous Lemma, will allow us to prove Proposition 9.2 Lemma 9.5. If 0 ≠ λ, μ ∈ k^{m} are such that F (λ) ∈ ^{⊥}ℓ_{μ} (i.e. ∑_{i=1}^{n} λ_{i}^{2}μ_{i} = 0) then there is an monomorphism

V (λ) → M_{ζ} ⊗_{Λ_{m}} k

that yields a non-zero element in \underline{Hom}_{Λ_{m}}(V (λ), M_{ζ} ⊗_{Λ_{m}} k).

Proof. It would suffice to construct a map from V (λ) to a finite direct sum ⊕{M_{ζ} ⊗_{Λ} Λ/r} having the stated properties. Using the notation from [7], V (λ) = (W ⊗ Λ)σ_{λ} (recall W is an irreducible representation of the Clifford algebra C_{n}). W ⊗ M_{ζ} ⊗_{Λ} Λ/r as an Λ-module is just dim W copies of M_{ζ} ⊗_{Λ} Λ/r. Write ζ̃ = Ω^{−1}ζ ⊗_{Λ} Λ/r = πη. Then to construct our map it suffices to find a map f in the following diagram

V (λ) ⟶^{f} W ⊗ (Λ ⊕ Ωk) ⟶^{(W ⊗π,W ⊗ζ̃)} W ⊗ Ω^{−1}k

so that the composite is zero. As V (λ) is a cyclic module, it suffices to determine f(σ_{λ}). We set f(σ_{λ}) = (W ⊗ η(σ_{λ}), −W ⊗ σ_{λ}). As before this will work provided σ_{λ}f(σ_{λ}) = 0. This reduces to requiring σ_{λ}η(σ_{λ}) = 0. Using the notation from [7], we have

σ_{λ}η(σ_{λ}) = σ_{λ} ∑ λ_{j}b_{j} ⊗ μ_{j}c_{j} = (∑ λ_{i}b_{i} ⊗ X_{i})(∑ λ_{j}b_{j} ⊗ μ_{j}c_{j}) = (∑ λ_{i}^{2}μ_{i})c

which is zero by assumption. Again it’s relatively simple to see that the map f doesn’t factor through a projective module and hence we have our non-zero element as required.

□

We are now almost ready to prove Proposition 9.2. However there is one minor technicality concerning the decomposability of the modules V (λ) that we must first address. The proof of the following Lemma is straightforward using results from [7].

Lemma 9.6. For 0 ≠ λ ∈ k^{m} and t := |{i : λ_{i} = 0}| then

V (λ) ≅ \begin{cases}⊕^{2^{⌊t/2⌋}}V, if m is odd, \\ ⊕^{2^{⌊t+1/2⌋}}V, if m is even. \end{cases}

where V is an indecomposable Λ_{m}-module.

Proof of Proposition 9.2. Thanks to Lemma 9.6 we know that V (λ) decomposes into a direct sum of copies of V, an indecomposable periodic module, of period 1. By Theorem 7.4 the support variety of V and hence that of V (λ) must be a line. That is, V_{HH(Λ_{m})}(V (λ)) = ℓ_{α} say, for some 0 ≠ α ∈ k^{m}.

Now choose 0 ≠ μ ∈ k^{m} and take ζ ∈ HH^{2}(Λ_{m}) such that V_{HH(Λ_{m})}(M_{ζ} ⊗_{Λ_{m}} k) = ^{⊥}ℓ_{μ}. Since V_{HH(Λ_{m})}(V (λ)) = ℓ_{α}, Lemma 9.4 says

ℓ_{α} ⊆ ^{⊥}ℓ_{μ} ⟺ \underline{Hom}_{Λ_{m}}(V (λ), M_{ζ} ⊗_{Λ_{m}} k) ≠ 0.

But we know from Lemma 9.5 that if we choose μ ∈ k^{m} so that F (λ) ∈ ^{⊥}ℓ_{μ}, then \underline{Hom}_{Λ_{m}}(V (λ), M_{ζ} ⊗_{Λ_{m}} k) ≠ 0.

The proof is now exactly the same as that in Proposition 5.2 and we conclude

V_{HH(Λ_{m})}(V (λ)) = ℓ_{F (λ)}.
□

References

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

[2] D.J. Benson, Representations and cohomology vol. I, Cambridge studies in advanced mathematics, 30 (1991) CUP

[3] D.J. Benson, Representations and cohomology vol. II, Cambridge studies in advanced mathematics, 31 (1991) CUP

[4] J. Carlson, The varieties and the cohomology ring of a module, J. of Algebra 85 (1983) 104-143.

[5] J. Carlson, The cohomology ring of a module, Journal of Pure and Applied Algebra 36 (1985) 105-121.

[6] E.C Dade, Endo-Permutation Modules II, Ann. Math 108 (1978) 317-346.

[7] K. Erdmann, M. Holloway, Rank varieties and projectivity for a class of local algebras, Math. Z. 247(3) (2004) 441-460.

[8] K. Erdmann, M. Holloway, N. Snashall, Ø. Solberg, R. Taillefer, Support varieties for selfinjective algebras, K theory (07/09/2004) (http://www.kluweronline.com/issn/0920-3036).

[9] E. Friedlander and B. Parshall, Support varieties for restricted Lie algebras, Invent. Math 86 (1986) 553-562.

[10] J.C. Jantzen, Kohomologie von p-Lie algebren und nilpotente Elemente, Abh. Math. Sem. Univ. Hamburg 56 (1986) 191-219.

[11] J.H. Palmieri, Quillen stratification for the Steenrod algebra, Ann. Math 149 (1999) 421-449.

[12] N.Snashall and Ø.Solberg, Support varieties and Hochschild cohomology rings, Proc. London Math. Soc 88(3) (2004) 705-732.

Karin Erdmann, Mathematical Institute, 24–29 St. Giles, Oxford OX1 3LB, England

   E-mail address: erdmann@maths.ox.ac.uk

Miles Holloway, Mathematical Institute, 24–29 St. Giles, Oxford OX1 3LB, England

   E-mail address: holloway@maths.ox.ac.uk