VARIETIES FOR COHOMOLOGY WITH TWISTED COEFFICIENTS

JON F. CARLSON

1. Introduction

Let G be a finite group and let k be a field of characteristic p > 0. If M is a kG-module then the support variety V_{G}(M) of M is the variety of the annihilator of the cohomology Ext_{kG}^{∗}(M, M) in the maximal ideal spectrum of H^{∗}(G, k). The variety measures a lot of the homological behavior of a module. For example the complexity of M, or polynomial growth rate of a projective resolution of M is the dimension of V_{G}(M). The module M is projective if and only if its variety is zero. If M and N are two kG-module then V_{G}(M ⊗_{k} N) = V_{G}(M)∩V_{G}(N). Some exposition of the theory can be found in the books ([2],[11], [9]).

In this paper we look at the same sort of homological behavior for Ext_{kG}^{∗}(M, N) where M and N are two finitely generated modules in the same kG-block. Certainly the original definition of V_{G}(M) in [1] was as the variety of the intersection of the annihilators of Ext_{kG}^{∗}(M, N) for all finitely generated modules N. Our aim here is much more specific. It is made possible by a recent solution [4] of a conjecture on the vanishing of cohomology that appeared in [7]. To understand the result we need some notation. For H a subgroup of G, let res_{G,H} : H^{∗}(G, k) ⟶ H^{∗}(H, k) be the restriction map. Let res_{G,H}^{∗} : V_{H}(k) ⟶ V_{G}(k) be the corresponding map on varieties.

Definition 1.1. [7]. The nucleus N ⊆ V_{G}(k) is the union

N = ⋃ res_{G,E}^{∗}(V_{E}(k))

where E runs through the collection of all elementary abelian p-subgroups such that C_{G}(E) is not p-nilpotent.

The Theorem of Benson in [4] says that if M is a finitely generated module in the principal block B_{0}(G), then H^{∗}(G, M) = 0 only if V_{G}(M) ⊆ N. But it does not answer the question of what might be the variety of H^{∗}(G, M). By this variety we mean the closed subset W_{G}(M) of all maximal ideal in the maximal ideal spectrum V_{G}(k), of H^{∗}(G, k), that contain the annihilator in H^{∗}(G, k) of H^{∗}(G, M).

One of our main theorems is to show that the difference between V_{G}(M) and W_{G}(M) is a subset of the nucleus N. This sheds some light on the variety of Ext_{kG}^{∗}(M, N) ≅ H^{∗}(G, Hom_{k}(M, N)). However there are some real difficulties with the block theory in the case that M and N are not in B_{0}(G). For B a nonprincipal block, there is a B-nucleus, N_{B}, such that the difference between V_{G}(M) ∩ V_{G}(N) and the variety of the annihilator of Ext_{kG}^{∗}(M, N) is a subset of N_{B} whenever M and N are in B. However there are still some problems in defining the nucleus of a block. Some of them are illustrated in the examples of the last section. This paper is not the definitive explanation of the vanishing of cohomology Ext_{kG}^{∗}(M, N), but rather it is intended to be a first attempt at explaining some the phenomena in the light of recent advances.

∗ partially supported by grants from NSF and EPSRC.

The organization of the paper is roughly as follows. Section 2 is mostly preliminaries concerned with the thick subcategories of the stable module category. Of particular importance is the role of the thick subcategory generated by the trivial module k. In Section 3 we look at the variety of H^{∗}(G, M) for M in the principal block. For other blocks we need full force of Benson’s result on induction of periodic modules. In Section 4, we give an independent proof of this theorem that does not depend on the technology of infinitely generated modules [14]. Rather it is based on transfer maps [10] and standard induction-restriction arguments. Section 5 consider the nucleus for nonprincipal blocks and the final section is composed of the promised examples.

2. Notation and Preliminaries

Throughout the paper, G is a finite group and k is an algebraically closed field of characteristic p > 0. Except when noted, the modules in the paper are finitely generated. The cohomology ring H^{∗}(G, k) = Ext_{kG}^{∗}(k, k) is a finitely generated k-algebra and its maximal ideal spectrum is denoted by V_{G}(k). If M is a kG-module, then let J (M) ⊆ H^{∗}(G, k) be the annihilator of Ext_{kG}^{∗}(M, M) = H^{∗}(G, Hom_{k}(M, M)) and let V_{G}(M) ⊆ V_{G}(k) be the subvariety consisting of all maximal ideals that contain J (M).

For a kG-module M, let Ω(M) be the kernel of a projective cover P ⟶^{ϵ} M. That is, P is a projective module of minimal dimension so that ϵ is a surjection. Inductively we let Ω^{n}(M) = Ω(Ω^{n−1}(M)) for n > 1. Similarly Ω^{−1}(M) is the cokernel of the inclusion of M into its injective hull: M ⟶ Q. Note that Q is also projective since kG is a Frobenius algebra. Again Ω^{−n}(M) = Ω^{−1}(Ω^{−n+1}(M)) for n > 1.

The most natural setting for our results is the stable category stmod(kG) of kGmodules modulo projectives. Let mod(kG) be the category of all finitely generated kG-modules and kG-module homomorphisms. The stable category has the same objects as mod(kG) but the morphisms are given by the formula

\underline{Hom}_{kG}(M, N) = Hom_{stmod(kG)}(M, N) = Hom_{kG}(M, N)/PHom_{kG}(M, N).

Here PHom(−, −) is the set of homomorphisms which factor through projective modules. The usual cohomology functors are natural in the stable category in the sense that

Ext_{kG}^{n}(M, N) ≅ \underline{Hom}_{kG}(Ω^{n}(M), N) ≅ \underline{Hom}_{kG}(Ω^{a+n}(M), Ω^{a}(N))

for M and N any kG-modules, n > 0 and any a ∈ ℤ.

An important feature of the stable category is that it is triangulated. A triangle

L ⟶^{α} M ⟶^{β} N ⟶^{γ} Ω^{−1}(L)

is stmod(kG) correponds roughly to the exact sequences

0 ⟶ L ⟶^{α′} M ⊕ proj ⟶^{β′} N ⟶ 0

or

0 ⟶ M ⟶^{β′′} N ⊕ proj ⟶^{γ′′} Ω^{−1}(L) ⟶ 0

where cls(α′) = α, cls(β′) = cls(β′′) = β and cls(γ′′) = γ. The operation Ω^{−1} is called the translation functor on the stable category. See [13] or [9] for details on the triangulations.

A thick subcategory of stmod(kG) is a triangulated subcategory that is also closed under the taking of direct summands. So if C is a thick subcategory of stmod(kG) and M and N are in C, then any direct summand of any extension of M by N is also in C. For M in stmod(kG) let Gen(M) denote the subcategory of stmod(kG) generated by M. The following is implicit in [6] though not clearly stated.

Proposition 2.1. The following are equivalent for any kG-modules M and X.

(i) M ∈ Gen(X). (ii) There exists a kG-module M′ such that M ⊕ M′ has a filtration:

{0} = L_{0} ⊆ L_{1} ⊆ ⋯ ⊆ L_{n} = M ⊕ M′

where for each i = 1, …, n, L_{i}/L_{i−1} ≅ Ω^{n_{i}}(X) for some n_{i}.

Proof. We know that Gen(X) contains Ω^{n}(X) for all n, because it is closed under the translation functor and its inverse. Additionally, it is closed under extensions and direct summand. So any kG-module M that satisfies condition (ii) of the Proposition must be in Gen(X). That is, let C is the full subcategory of stmod(kG) of all objects that satisfy condition (2); then C ⊂ Gen(X).

On the other hand, X ∈ C and C, as given above, is a thick triangulated subcategory of stmod(kG). Hence Gen(X) ⊆ C and we are done.

The following was used extensively in [6], though not clearly stated. For notation let K = Gen(k) be the thick subcategory generated by the trivial kG-module.

Corollary 2.2. Suppose that C is any thick subcategory of stmod(kG) and that M ∈ C. Let W ∈ K. Then M ⊗ W ∈ C.

Here ⊗ means ⊗_{k} with diagonal G-action. By the Proposition there is kG-module W′ such that W ⊕ W′ has a filtration where the factors have the form Ω^{n}(k) for some values of n. Then (M ⊗ W) ⊕ (M ⊗ W′) has a filtration where the factors have the form M ⊗ Ω^{n}(k) ≅ Ω^{n}(M) ⊕ proj for some values of n. These modules are all in C.

3. Varieties for cohomology in the principal block

In this section we look at varieties for H^{∗}(G, M) where M is a nonprojective module in the principal block B_{0} = B_{0}(G). That is, we let I_{G}(M) denote the annihilator in H^{∗}(G, k) of H^{∗}(G, M). This is a homogeneous ideal, and it has an associated variety W_{G}(M) ⊆ V_{G}(M) consiting of all maximal ideals that contain I_{G}(M).

Let Ĥ^{∗}(G, M) and Êxt_{kG}^{∗}(M, N) be the Tate cohomology groups. So Ĥ^{n}(G, M) = Êxt_{kG}^{n}(k, M) = H^{n}(G, M), if n > 0, and in all cases Ĥ^{n}(G, M) ≅ \underline{Hom}_{kG}(Ω^{n}(k), M). Note that the annihilator of Ĥ^{∗}(G, M) has the same radical as that of H^{∗}(G, M). Hence the two H^{∗}(G, k)-modules define the same variety.

Lemma 3.1. [7] Suppose that M ∈ K = Gen(k). If M is not projective then Ĥ^{∗}(G, M) ≠ 0 and moreover Ĥ^{n}(G, M) ≠ {0} for an infinite number of positive values of n.

Proof. By Proposition 2.1 there is a kG-module M′ such that M ⊕ M′ has a filtration by modules of the form Ω^{m}(k) for some values of m. Now because M is not projective Ext_{kG}^{n}(M, M) ≠ {0} for an infinite number of values of n. Hence Ext_{kG}^{n}(M ⊗M′, M) ≠ {0} for an infinite number of values of n. Then by repeated use of the long exact sequence on cohomology we can deduce from the filtration on M ⊕ M′ that

Êxt_{kG}^{n}(Ω^{m}(k), M) ≅ Êxt^{n+m}(k, M) = Ĥ^{n+m}(G, M)

is not zero for an infinite number of positive values of n.

Proposition 3.2. Suppose that M is a kG-module and that N ∈ K = Gen(k). Then

\sqrt{I_{G}(M ⊗ N)} ⊇ I_{G}(M)

or equivalently W_{G}(M ⊗ N) ⊆ W_{G}(M).

Proof. Let N′ be a kG-module such that N ⊕ N′ has a filtration

{0} = L_{0} ⊆ L_{1} ⊆ ⋯ ⊆ L_{m} = N ⊕ N′

with L_{i}/L_{i−1} ≅ Ω^{n_{i}}(k) for each i, as in Proposition 2.1. Then (M ⊗ N) ⊕ (M ⊗ N′) has a filtration where the factors have the form (M ⊗ L_{i})/(M ⊗ L_{i−1}) ≅ Ω^{n_{i}}(M) ⊕ proj. Now H^{∗}(G, M ⊗ N) is a direct summand of H^{∗}(G, M ⊗ (N ⊗ N′)) as an H^{∗}(G, k)-module. Hence I_{G}(M ⊗ (N ⊕ N′)) ⊆ I_{G}(M ⊗ N). Therefore it is sufficient to show that

I_{G}(M) ⊆ \sqrt{I_{G}(M ⊗ (N ⊕ N′))}

For any i = 1, …, m and γ ∈ H^{n}(G, k) we have a commutative diagram

… ⟶ H^{r}(G, M ⊗ L_{i−1}) ⟶ H^{r}(G, M ⊗ L_{i}) ⟶ H^{r}(G, Ω^{n_{i}}(M)) ⟶ …
↓^{γ}  ↓^{γ}  ↓^{γ}
… ⟶ H^{r+n}(G, M ⊗ L_{i−1}) ⟶ H^{r+n}(G, M ⊗ L_{i}) ⟶ H^{r+n}(G, Ω^{n_{i}}(M)) ⟶ …

Notice that H^{s}(G, Ω^{n_{i}}(M)) ≅ Ĥ^{s−n_{i}}(G, M). So if γ annihilates Ĥ^{r−n_{i}}(G, M) then

γ · H^{r}(G, M ⊗ L_{i}) ⊆ Image(H^{r+n}(G, M ⊗ L_{i−1})).

The consequence of this is that if we choose sufficiently many elements γ_{1}, …, γ_{t} ∈ I_{G}(M) then

γ_{1}, …, γ_{t}H^{∗}(G, M ⊗ L_{i}) ⊆ Image(H^{∗}(G, M ⊗ L_{0})) = {0}.

We should be cautious here because it need not be the case that γ_{i}Ĥ^{s−n_{i}}(G, M) = 0 even when γH^{∗}(G, M) = 0. That is, s − n_{i} may not be positive. However, the sequence n_{i}, …, n_{m} is fixed and hence any product of sufficiently many elements of I_{G}(M) will annihilate H^{s−n_{i}}(G, M) for any s ≥ 0. Therefore we have proved the proposition.

Theorem 3.3. Let M be an indecomposable module in the principal block of kG. Then V_{G}(M) ∖ W_{G}(M) ⊆ N where N is the nucleus.

Proof. The first thing we should notice is that W_{G}(M) ⊆ V_{G}(M) or equivalently that J_{G}(M) ⊆ I_{G}(M). That is, H^{∗}(G, M) ≅ Ext_{kG}^{∗}(k, M) is a finitely generated left module over Ext_{kG}^{∗}(M, M). So if ζ ∈ J_{G}(M) then ζ · Id_{M} = 0 and ζ · H^{∗}(G, M) = {0}.

Suppose that α ∈ V_{G}(M) and α ∉ N. Then there exists a kG-module N with the properties that N ∈ K and V_{G}(N) = ⟨α⟩, the line through α. So M ⊗ N is a module in the principal block and V_{G}(M ⊗ N) = V_{G}(M) ∩ V_{G}(N) = ⟨α⟩. Because α ∈ N it must be that H^{∗}(G, M ⊗ N) ≠ {0} in infinitely many degrees by Benson’s Theorem [4]. Actually we know that M ⊗ N ∈ K by [6]. But most importantly we have that W_{G}(M ⊗ N) ≠ {0}. It follows that W_{G}(M ⊗ N) = ⟨α⟩ since it is a homogeneous variety and is contained in ⟨α⟩. Since W_{G}(M ⊗ N) ⊆ W_{G}(M) we know that α ∈ W_{G}(M) and we are done.

In the above proof, if M is in K then M ⊗N is in K and H^{∗}(G, M ⊗N) ≠ 0 in infinitely many degrees by Lemma 3.1. so in this case if α ∈ V_{G}(M) then α ∈ W_{G}(M) regardless of whether α is in the complement of the nucleus. Hence we have proved the following.

Corollary 3.4. Suppose that M ∈ K. Then W_{G}(M) = V_{G}(M).

4. Induction and periodic modules

The purpose of this section is to present an elementary proof of Benson’s Theorem [4] that an indecomposable periodic kG-module is stably induced from the diagonalizer of a minimal elementary abelian p-subgroup on which its variety is supported. The theorem will be needed in a fundamental way in the next section. Benson’s original proof used Rickard’s indempotent modules [14] and the theorem of varieties for infinitely generated modules [5]. Our proof deals only with finitely generated modules. We begin with some notation and terminology.

Let E be an elementary abelian p-subgroups of G. The diagonalizer D_{G}(E) is the set of all elements y ∈ G such that there exists a ∈ ℤ/pℤ with yxy^{−1} = x^{a} for all x ∈ E. Hence it is the set of all elements of N_{G}(E) whose conjugation action on E is by scalar multiplication. Clearly C_{G}(E) ⊆ D_{G}(E) ⊆ N_{G}(E) and D_{G}(E)/C_{G}(E) is a cyclic group whose order divides p − 1.

We say that subvariety V ⊆ V_{G}(k) is supported on a subgroup H ⊆ G if V ⊂ res_{G,H}^{∗}(V_{H}(k)). We say that V is minimally supported on H if V ⊈ res_{G,F}^{∗} (V_{F} (k)) for any subgroup F ⊆ H. A consequence of Quillen’s Theorem on the components of the variety is that every point or irreducible closed subvariety of V_{G}(k) is minimally supported on an elementary abelian p-subgroup of G. In the case of a line in V_{G}(k) we can characterize the condition as follows.

Proposition 4.1. Suppose that ℓ ⊆ V_{G}(k) is a line and that ℓ is minimally supported on an elementary abelian p-subgroup E of G. Let D = D_{G}(E) and let I = Tr_{D}^{G}(H^{∗}(D, k)) be the image of the transfer map. Then ℓ ∩ V_{G}(I) = {0} .

Proof. The quickest way to see the result is to use the main theorem of [3]. Let C be the set of all subgroups H of G with the property that the Sylow p-subgroup of the centralizer C_{G}(H) is not conjugate to a subgroup of D. Then by Theorem 1.1 of [3]

\sqrt{I} = ⋂_{H∈C} \sqrt{R_{H}}

where R_{H} is the kernel of the restriction to H. Note that V_{G}(R_{H}) = res_{G,H}^{∗}(V_{H}(k)) and hence we have that V_{G}(I) = ⋃_{H⊆C} res_{G,H}^{∗}(V_{H}(k)). Because C_{G}(E) ⊆ D, E ⊈ H for any

H ∈ C. Consequently res_{G,E}^{∗}(V_{E}(k)) ⊈ V_{G}(I). Then because ℓ is minimally supported on E, then ℓ ⊈ V_{G}(I) and we are done.

Proposition 4.2. [10] Suppose that M is a finitely generated indecomposable kG-module whose variety V_{G}(M) = ℓ is a line in V_{G}(k). Suppose that ℓ is minimally supported on an elementary abelian p-subgroup E. Then M is relatively D-projective for D = D_{G}(E).

Proof. By the last proposition there is an element γ ∈ H^{m}(D, k) such that ℓ∩V_{G}(Tr_{D}^{G}(γ)) = {0}. Let ζ = Tr_{H}^{D}(γ). Then we have an exact sequence

0 ⟶ L_{ζ} ⟶ Ω^{m}(k) ⟶^{ζ̂} k ⟶ 0

where ζ̂ is a cocycle representing ζ. Now we know that V_{G}(L_{ζ}) = V_{G}(ζ) and hence V_{G}(M) ∩ V_{G}(L_{ζ}) = {0} (see Proposition 5.9.1 of [2]). So M ⊗ L_{ζ} is projective and ζ ⊗ Id_{M} : Ω^{m}(k) ⊗ M ⟶ M has a right inverse σ. But because ζ is a transfer from D, ζ ⊗ Id_{M} factors through a relatively D-projective modules. Hence so does (ζ ⊗ Id_{M}) ∘ σ = Id_{M} and we can conclude that M is relatively D-projective.

We are now prepared to prove the Theorem of Benson.

Theorem 4.3. (Theorem 1.5 of [4]). Suppose that M is an indecomposable kG-module such that V_{G}(M) = ℓ is a line which is minimally supported on an elementary abelian p-subgroup E of G. Let D = D_{G}(E). Then there is a kD-module L and a projective module P such that L^{↑G} ≅ M ⊕ P.

Proof. By the last proposition there is an indecomposable direct summand U of M_{D} such that M is a direct summand of U^{↑G}. We claim that

U^{↑G} ↓_{E}≅ ∑_{x∈N_{G}(E)/D}(x ⊗ U_{E}) ⊕ (proj).

This assertion follows from the Mackey Subgroup Theorem. For let N = N_{G}(E). Then

U^{↑G} ↓_{E}≅ ∑_{x∈N/D} x ⊗ U_{E} ⊕ ∑_{\begin{matrix}ExD \\ x∉N \end{matrix}}((x ⊗ U)_{E∩xDx^{−1}})^{↑E}

If E ∩ xDx^{−1} ≠ E, then ((x ⊗ U)_{E∩xDx^{−1}})^{↑G} is projective as a kE-module because the variety of U is minimally supported on E.

So suppose that E = E ∩ xDx^{−1} but x ∉ N. Note that if F is any subgroup of D with E ⊈ F then U_{F} is a projective kF -module. Then U ↓_{x^{−1}Ex∩D} is projective and by conjugation (x ⊗ U)_{E∩xDx^{−1}} is also projective. So

U^{↑G} ↓_{E}= ∑_{x∈N/D}(x ⊗ U)_{E} = U^{↑N} ↓_{E} ⊕(proj).

To finish the proof we notice that if ℓ̂ = V_{E}(U_{E}) then V_{E}(x ⊗ U_{E}) = xℓ̂ ≠ ℓ̂ for any x ∈ N, x ∉ D. So suppose that U^{↑G} ≅ M ⊕ M′. Then M′ ↓_{E} has no component with variety ℓ̂ by the Krull-Schmidt Theorem. On the other hand we must have that V_{G}(M′) ⊆ res_{G,E}^{∗}(ℓ̂) = ℓ. This can be seen for example by noticing that if we tensor U^{↑G} with any module whose variety avoids ℓ then the tensor product is a projective module. Hence we must have that V_{G}(M′) = {0} and M′ is projective.

Remark 1. The above proof is written with the assumption that M is a finitely generated module. With careful modification it can be made to work for infinitely generated modules. Of course the modification must employ the extended notions of varieties for infinitely generated modules as developed in [5].

5. Varieties for modules in nonprincipal blocks

Suppose that B is a block for kG. We want to investigate the homological behavior of Ext_{kG}^{∗}(M, N) where M and N are finitely generated modules in B. In this regard, let I_{G}(M, N) be the annihilator in H^{∗}(G, k) of Ext_{kG}^{∗}(M, N) and let W_{G}(M, N) = V_{G}(I_{G}(M, N)) be the corresponding closed subvariety of V_{G}(k). Notice that Ext_{kG}^{∗}(M, N) = Ext_{kG}^{∗}(k, M^{∗} ⊗ N) = H^{∗}(G, M^{∗} ⊗ N). So W_{G}(M, N) = W_{G}(M^{∗} ⊗ N) depends only on the component (direct summand) of M^{∗} ⊗ N that is in the principal block. Indeed if M and N are in the prinicipal block then we can prove the following.

Proposition 5.1. Suppose that M and N are in B_{0}. Then

V_{G}(M) ∩ V_{G}(N) ∖ W_{G}(M, N) ⊆ N

Proof. First V_{G}(M^{∗} ⊗ N) = V_{G}(M) ∩ V_{G}(N) and so W_{G}(M, N) ⊆ V_{G}(M) ∩ V_{G}(N) as in the proof of Theorem 3.3. Let α ∈ V_{G}(M) ∩ V_{G}(N) be an element such that α ∉ N. Let L be a kG-module in K such that V_{G}(L) = ⟨α⟩, the line through α. Then V_{G}(M ⊗ L) = V_{G}(N ⊗ L) = ⟨α⟩ and because α ∉ N, M ⊗ L and N ⊗ L are in K. So W_{G}(M ⊗ L, N ⊗ L) = W_{G}(M^{∗} ⊗ N ⊗ L^{∗} ⊗ L) = ⟨α⟩ and this line is in W_{G}(M^{∗} ⊗ N) by Proposition 3.2.

The examples of the next section show that for nonprincipal blocks, things are not nearly so clean and simple. To prove a comparable theorem for an arbitrary block B we require the notion of the B-nucleus. First we need a definition of covering of blocks. The reader should be warned that the next definition is not standard.

Definition 5.2. Suppose that H ⊆ G is a subgroup, that B is a block of kG and b is a block of kH. We say that B cover b if there is a kG-module M in B such that M ↓_{H} has a nonprojective component in b.

Now for the definition of the nucleus. Let A be the set of all elementary abelian p-subgroups of G.

Definition 5.3. Suppose that B is a block of kG and let D be the defect group of B. Let A_{B} ⊆ A be the set of all elementary abelian p-subgroups E of G which satisfy either (1) C_{G}(E) is not p-nilpotent, or

(2) C_{G}(E) is p-nilpotent but B-covers more than one block of C_{G}(E). Then the B-nucleus is the set

N_{B} = ( ⋃_{E∈A_{B}} res_{G,E}^{∗}(V_{E}(k))) ∩ res_{G,D}^{∗}(V_{D}(k)).

Because every module in B is relatively D-projective it makes sense that the N_{B} should be a subset of res_{G,D}^{∗}(V_{D}(k)).

Our main theorem is the following

Theorem 5.4. Let B be a block of kG and suppose that M and N are finitely generated modules in B. Then

V_{G}(M) ∩ V_{G}(N) ∖ W_{G}(M, N) ⊆ N_{B}.

Most of the analysis in the proof is standard (e.g. see [8]) and some of it could be shortened by quoting results from the literature. However for the sake of completeness we sketch the whole proof.

The proof goes in several steps. As in some previous proofs we begin by choosing α ∈ V_{G}(M) ∩ V_{G}(N), α ∉ N_{B}. Our objective is to show that α ∈ W_{G}(M, N). Let E be an elementary abelian p-subgoups on which α is minimally supported. Then E ∉ A_{B} and so C_{G}(E) is p-nilpotent and B covers only one block b of kC_{G}(E). Let L ∈ K be a module with V_{G}(L) = ⟨α⟩, the line through α. Now if Êxt_{kG}^{∗}(L ⊗ M, L ⊗ N) ≠ {0} then Êxt_{kG}^{∗}(k, L^{∗}⊗L⊗M^{∗}⊗N) ≠ 0 and α ∈ W_{G}(L^{∗}⊗L⊗M^{∗}⊗N) ⊆ W_{G}(M^{∗}⊗N) = W_{G}(M, N) as desired. So it is sufficient to consider the case that V_{G}(M) = ⟨α⟩ = V_{G}(N).

Step 1. We reduce to the diagonlizer D of E. By theorem 4.3, M ⊕ P ≅ M_{1}^{↑G} for some kD-module M_{1} and some projective module P. So by the Eckmann-Shapiro Lemma

Êxt_{kG}^{∗}(M, N) ≅ Êxt_{kD}^{∗}(M_{1}, N_{D})

Here α = res_{G,D}^{∗}(β) for some β ∈ V_{D}(k) and V_{G}(M_{1}) = ⟨β⟩. Also N_{D} must have a component N_{1} with V_{D}(N_{1}) = ⟨β⟩. So it is sufficient to show that β ∈ W_{D}(M_{1}, N_{1}).

Step 2. We reduce to the centralizer of E. Let C = C_{G}(E). Suppose that we can show that Êxt_{kC}^{∗}(M_{1}, N_{1}) ≠ {0}. Then Êxt_{kC}^{∗}(k, M_{1}^{∗} ⊗ N_{1}) ≠ 0. Hence (M_{1}^{∗} ⊗ N_{1})_{0}, the part of M_{1}^{∗} ⊗ N_{1} in the principal block of kC is not a projective module. Let H = O_{p′}(C) be the normal p-complement of C. Observe that (M_{1}^{∗} ⊗ N_{1})_{0} = (M_{1}^{∗} ⊗ N_{1})^{H} is the H-fixed points of M_{1}^{∗} ⊗ N_{1}. Because H is also equal to O_{p′}(D), then (M_{1}^{∗} ⊗ N_{1})^{H} is a kD-module in the principal kD-block. From the assumption we know that it is not projective. Moreover the variety of (M_{1}^{∗} ⊗ N_{1})^{H} must be ⟨β⟩ which is not in the kDnucleus. Therefore ⟨β⟩ ⊆ (W_{D} ((M_{1}^{∗} ⊗ N_{1})^{H})) ⊆ W_{D}(M_{1}^{∗} ⊗ N_{1}) = W_{D}(M_{1}, N_{1}). This completes Step 2.

Before continuing with the proof we should notice that (M_{1})_{C} and (N_{1})_{C} are in the same kC-block because of the assumption that B covers only one block of kC. Also V_{C}(M_{1}) = V_{C}(N_{1}) = (res_{D,C}^{∗})^{−1}(⟨β⟩) is a line in V_{C}(k) because D preserves the lines in res_{C,E}^{∗}(V_{E}(k)).

At this point we should recycle some notation in order to clean it up. We are down to considering a p-nilpotent group G = C_{G}(E) where E is an elementary abelian p-subgroup. B is a block of kG and M and N are kG-modules in B with V_{G}(M) = ⟨α⟩ = V_{G}(N), where α is minimally supported on E. It remains to show that α ∈ W_{G}(M, N).

Let H = O_{p′}(G). The block B when restricted to H is a union of kH-blocks all of which are in the same G-orbit. Let T be an irreducible module in one of the kH-blocks covered by B.

Step 3. We reduce to the stablizer J of T. Let J = {g ∈ G|gT ≅ T }. Then by the Clifford Theorem M ↓_{J} ≅ M_{1} ⊕ ⋯ ⊕ M_{t}, where M_{1} ↓_{H} is a direct sum of copies of T, and for i = 2, …, t, there is a g_{i} ∈ G, g_{i} ∉ J such that M_{i} = g_{i}M_{1}. In other words M = M_{1}^{↑G} and similarly N = N_{1}^{↑G} where N_{1} is also a direct sum of copies of T. Therefore

Ext_{kG}^{∗}(M, N) ≅ Ext_{kJ}^{∗}(M_{1}, N ↓_{J}) ≅ Ext_{kJ}^{∗}(M_{1}, N_{1})

Since res_{G,J}^{∗} : V_{J}(k) ⟶ V_{G}(k) is one-to-one on res_{J,E}^{∗}(V_{E}(k)) we have that V_{J}(M_{1}) = V_{J}(N_{1}) = ⟨γ⟩ for γ ∈ V_{J}(k) and res_{G,J}^{∗}(γ) = α. Hence if γ ∈ W_{J}(M_{1}, N_{1}) then α ∈ W_{G}(M, N), and we may assume that T is G-stable.

Step 4. We complete the proof. The first observation is that the kH-module T extends to a kG-module S. This fact can be deduced from Fong’s paper [12], but for convenience we sketch a quick proof using Fong’s idea. Let P = T^{↑G}. Then P is a projective module with dimension DimP = |G : H|DimT. Because p does not divide DimT and |G : H| is the order of the Sylow p-subgroup, we conclude that P is indecomposable. Note also that P ↓_{H} ≅ T^{|G:H|}. If S = P/RadP then kG contains DimS copies of P as direct summands. So kG ↓_{H} contains DimS|G : H| copies of T. So DimS = DimT and S ↓_{H} ≅ T.

If M is a kG-module then we claim that M ≅ S ⊗ Hom_{kH}(S, M). That is we have a surjective homomorphism S⊗Hom_{kH}(S, M) ⟶ M given by m⊗f ↦ f(m). However because the dimensions are the same this is an isomorphism. Note that M_{0} = Hom_{kH}(S, M) is a kG-module with trivial H-action. That is, it is in the principal block of kG. So we have that

Êxt_{kG}^{∗}(M, N) ≅ Êxt_{kG}(S ⊗ M_{0}, S ⊗ N_{0}) ≅ Êxt_{kG}^{∗}(k, (S ⊗ S^{∗}) ⊗ (M_{0}^{∗} ⊗ N_{0}))
≅ Êxt_{kG}^{∗}(k, M_{0}^{∗} ⊗ N_{0}) ≠ {0}

because M_{0} and N_{0} are in K. Hence α ∈ W_{G}(M, N). This completes the proof.

6. Some Examples

In this section we present three examples. The examples are meant to illustrate that the analysis of the previous sections does not tell the whole story on the vanishing of cohomology. In the first example the B-nucleus is all of V_{G}(k). In the third example the groups itself is p-nilpotent.

Example 1. Let p = 2 and let H = (ℤ/2)^{2} × ℤ/7 = ⟨x, y⟩ × ⟨w⟩. Let G = H ⋊ ℤ/3 where ℤ/3 is generated by v with x^{v} = v^{−1}xv = y, y^{v} = xy and w^{v} = w^{2}. Let ξ ∈ k be a primitive 7^{th} root of unity. Let M_{i} be the irreducible kH-module of dimension 1 with w · m = ξ^{i}m for m ∈ M_{i}. Notice that M_{1}^{↑G} = S is irreducible of dimension 3. Also S ↓_{H} ≅ M_{1} ⊕ M_{2} ⊕ M_{4}. Let b_{i} be the kH-block of M_{i} and let B be the block of kG with irreducible module S.

Now E = ⟨x, y⟩ is the unique elementary abelian p-subgroup. So V_{E}(k) = V_{H}(k) = k^{2} and V_{G}(k) is the orbit space under the action of v. It is an easy check that N = {0} while N_{B} = V_{G}(k). None of the points in V_{H}(k) are fixed by the action of ⟨v⟩ however there are two fixed lines spanned by eigenvectors.

Suppose that α ∈ V_{H}(k) and ⟨α⟩ is not one of the fixed lines. Let L be a kH-module in the principal block b_{0} such that V_{H}(L) = ⟨α⟩. Let N_{1} = M_{1} ⊗ L and N_{2} = M_{2} ⊗ L and X_{i} = N_{i}^{↑G} for i = 1, 2.

Proposition 6.1. W_{G}(X_{1}, X_{2}) = {0} Proof. Note that Ext_{kG}^{∗}(X_{1}, X_{2}) = Ext_{kH}^{∗}(N_{1}, N_{2}^{↑G} ↓_{H}). Now

N_{2}^{↑G} ↓_{H}= (M_{2} ⊗ L) ⊕ (M_{4} ⊗ vL) ⊕ (M_{1} ⊗ v^{2}L).

We have that

Ext_{kH}^{∗}(M_{1} ⊗ L, M_{2} ⊗ L) = {0} = Ext_{kH}^{∗}(M_{1} ⊗ L, M_{4} ⊗ vL)

because the modules are in different blocks. But also Êxt_{kH}^{∗}(M_{1} ⊗ L, M_{1} ⊗ v^{2}L) = {0} because the varieties intersect trivially.

On the other hand we have the following

Proposition 6.2. Suppose that X_{1} and X_{2} are two kG-modules with the property that either DimV_{G}(X_{1}) + DimV_{G}(X_{2}) ≥ 3 or V_{G}(X_{1}) ∩ V_{G}(X_{2}) contains the image of one of the ⟨v⟩-fixed lines of V_{H}(k). Then W_{G}(X_{1}, X_{2}) ≠ {0}.

Proof. We know that there are kH-modules Y_{1} and Y_{2} in b_{1} such that X_{1} = Y_{1}^{↑G}, X_{2} = Y_{2}^{↑G}. So we have as before that Êxt_{kG}^{∗}(X_{1}, X_{2}) = Êxt_{kH}^{∗}(Y_{1}, Y_{2}). So the question comes down to whether V_{G}(Y_{1}) ∩ V_{G}(Y_{2}) can be {0}. The answer is no for both cases of the hypothesis.

Example 2. This example is only a slight change from the previous example, and we use the same basic notation. This time let H = (ℤ/2)^{3} × ℤ/7 = ⟨x, y, z⟩ × ⟨w⟩, G = H ⋊ ℤ/3 and let x^{v} = xy, y^{v} = yz, z^{v} = z, w^{v} = w^{2} for v an element of order 3. Let B be the kG-block of S where S ↓_{H}= M_{1} ⊕ M_{2} ⊕ M_{4} as before.

Then V_{H}(k) = k^{3} is a free k⟨v⟩-module. Hence there are three fixed lines one of which is res_{H,Z}^{∗}(V_{Z}(k)) for Z = ⟨z⟩. This line is fixed pointwise and res_{G,Z}^{∗}(V_{Z}(k)) = N. Again N_{B} = V_{G}(k). By the same arguments as before, for any line ⟨γ⟩ ∈ V_{H}(k) which is not fixed by ⟨v⟩, we can find modules X_{1}, X_{2} in B such that V_{G}(X_{1}) = V_{G}(X_{2}) is the image of ⟨γ⟩ in V_{G}(k) and W_{G}(X_{1}, X_{2}) = {0}. Also if ℓ is one of the lines which is fixed by ⟨v⟩, then the image of ℓ in V_{G}(k) is in W_{G}(X_{1}, X_{2}) provided X_{1} and X_{2} are in B and the image of ℓ is in V_{G}(X_{1}) ∩ V_{G}(X_{2}). In particular we note that even though there is a nonprojective module M in B_{0} such that V_{G}(M) = N and H^{∗}(G, M) = 0, there are no modules X_{1} and X_{2} in B with N ⊆ V_{G}(X_{1}) ∩ V_{G}(X_{2}) and Êxt_{kG}^{∗}(X_{1}, X_{2}) = {0}. The reason for this is that modules in B must be induced from H.

We can also show, as in the last example, that if X_{1} and X_{2} are in B and if DimV_{G}(X_{1})+ DimV_{G}(X_{2}) ≥ 4 then W_{G}(X_{1}, X_{2}) ≠ {0}.

Example 3. Suppose that H ≅ (ℤ/2)^{2} × (ℤ/3)^{2} = ⟨a, b⟩ × ⟨y, z⟩ and p = 3. Let G = H ⋊ ℤ/3 where the subgroup of order 3 that is not in H is generated by x. Assume that a^{x} = b, b^{x} = ab, y^{x} = yz, z^{x} = z. H has 4 blocks, b_{0}, b_{1}, b_{2}, b_{3} with b_{0} being the principal block. Let M_{i} be the irreducible module in b_{i}. So kG has two blocks. B_{0} with irreducible module k = M_{0} and B with irreducible module S = M_{1}^{↑G}. This time N = {0} and N_{B} = res_{G,E}^{∗}(V_{E}(k)). E = ⟨y, z⟩ is the defect group of B.

We should notice that Z = ⟨z⟩ has p-nilpotent centralizer C_{G}(Z) = G. so if X_{1} and X_{2} are modules in B with res_{G,Z}^{∗}(V_{Z}(k)) contained in V_{G}(X_{1}) ∩ V_{G}(X_{2}) then res_{G,Z}^{∗}(V_{Z}(k)) ⊆ W_{G}(X_{1}, X_{2}). On the other hand if α ∈ N_{B}, α ∉ res_{G,Z}^{∗}(V_{Z}(k)) then α = res_{G,E}^{∗}(γ) and the line through γ is not fixed by the action of ⟨x⟩. So again we can find X_{1}, X_{2} in B with Êxt_{kG}^{∗}(X_{1}, X_{2}) = {0} and V_{G}(X_{1}) = V_{G}(X_{2}) = ⟨α⟩.

References

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

[2] D. J. Benson, “Representations and cohomology, I”, Cambridge University Press, 1991.

[3] D. J. Benson, The image of the transfer map, Arch. Math. 61 (1993), 7–11.

[4] D. J. Benson, Cohomology of modules in the principal block of a finite group, New York J. Math. 1 (1995), 196–205.

[5] D. J. Benson, J. F. Carlson and J. Rickard, Complexity and varieties for infinitely generated modules, II, Math. Proc. Cambridge Phil. Soc. 120 (1996), 597–615.

[6] D. J. Benson, J. F. Carlson and J. Rickard, Thick subcategories of the stable module category, Fund. Math. 153 (1997), 59–80.

[7] D. J. Benson, J. F. Carlson and G. R. Robinson, On the vanishing of group cohomology, J. Algebra. 131 (1990), 40–73.

[8] M. Broué, Equivalences of blocks of group algebras, Finite Dimensional Algebras and Related Topics (Ottawa, ON, 1992) NATO Adv. Sci. Inst., Ser. C Math. Phys. Sci. 424, Kluwer Acad. Pub., Dordrecht, 1994.

[9] J. F. Carlson, “Modules and group algebras”, ETH Lecture Notes, Birkhäuser Verlag, 1996.

[10] J. F. Carlson, C. Peng and W. W. Wheeler, Transfer maps and virtual projectivity, J. Algebra (to appear).

[11] L. Evens, “The Cohomology of Groups”, Oxford University Press, 1991.

[12] P. Fong, Solvable groups and modular representation theory, Trans. Amer. Math. Soc. 103 (1962), 484–494.

[13] D. Happel, “Triangulated categories in the Representation Theory of Finite Dimensional Algebras”, London Math. Soc. Lecture Notes No. 119, Cambridge University Press, 1988.

[14] J. Rickard, Idempotent modules in the stable category, J. London Math. Soc. 56 (1997), 149–170.

Department of Mathemtics, University of Georgia, Athens, GA 30602

  E-mail address: jfcsloth.math.uga.edu; http://www.math.uga.edu/∼jfc