QUILLEN STRATIFICATION FOR HOCHSCHILD COHOMOLOGY OF BLOCKS

JONATHAN PAKIANATHAN AND SARAH WITHERSPOON

with an appendix by STEPHEN F. SIEGEL

Abstract. We decompose the maximal ideal spectrum of the Hochschild cohomology ring of a block of a finite group into a disjoint union of subvarieties corresponding to elementary abelian p-subgroups of a defect group. These subvarieties are described in terms of group cohomological varieties and the Alperin-Broué correspondence on blocks. Our description leads in particular to a homeomorphism between the Hochschild variety of the principal block and the group cohomological variety. The proofs require a result of Stephen F. Siegel, given in the appendix, which states that nilpotency in Hochschild cohomology is detected on elementary abelian p-subgroups.

1. Introduction

Quillen’s two groundbreaking papers of 1971 [18] yield a description of the variety of a finite group G, that is, of the maximal ideal spectrum of the cohomology ring H^{∗}(G, k), as a disjoint union of subvarieties corresponding to elementary abelian psubgroups (where p is the characteristic of the commutative ring k). Avrunin and Scott vastly generalized this result to varieties of modules over the group ring kG, that is to the maximal ideal spectra of quotients of H^{∗}(G, k) associated to these modules [5]. In this paper we replace H^{∗}(G, k) by the Hochschild cohomology ring HH^{∗}(kG) of the group algebra kG, and find a suitable analog of the Quillen stratification, describing the maximal ideal spectrum of HH^{∗}(kG) as a disjoint union of varieties corresponding to elementary abelian p-subgroups. As this Hochschild cohomology decomposes according to the decomposition of kG into blocks (that is, indecomposable ideal direct summands), this description is given by a stratification of the Hochschild cohomology ring HH^{∗}(B) of each block B. This theory is parallel to that of Linckelmann [16], in which he gives a Quillen stratification for the maximal ideal spectrum of the block cohomology ring LH^{∗}(B) of a block B, where LH^{∗}(B) is a particular subring of the group cohomology ring of a defect group of B. However the techniques we develop to work with Hochschild cohomology are quite different from those of Linckelmann, and begin with a particular ring homomorphism (Theorem 2.5 below).

Date: August 5, 2003.

1991 Mathematics Subject Classification. Primary 20J06.

The second author was supported by National Security Agency Grant #MDS904-01-1-0067 and National Science Foundation Grant #DMS0245560.

The relationship between the Hochschild cohomology ring HH^{∗}(B) and Linckelmann’s block cohomology ring LH^{∗}(B) is not fully understood. In the case of the principal block, our results imply that their respective varieties are homeomorphic. The general case is more complicated, but is potentially where the main significance of these cohomology theories lies: Blocks other than the principal block are not augmented algebras themselves, and so one must use a cohomology theory such as Hochschild cohomology or Linckelmann’s block cohomology to study them. There are a number of recent papers that use Hochschild cohomology to study various types of finite dimensional algebras and their modules, for example see [10, 13, 24]. Theories of support varieties for modules have been built from Hochschild cohomology [21, 24] and from Linckelmann’s block cohomology [16]. It is hoped that the present paper will lead to a better understanding of the connections between these two cohomology theories for blocks of finite groups and their modules. This would augment our knowledge of both and give some insight into any connection between the Morita structure (related to Hochschild cohomology) and the local structure (related to block cohomology) of a block.

We will start by summarizing the main results of this paper in greater detail. From now on k will be an algebraically closed field of positive characteristic p dividing the order of the finite group G. The Hochschild cohomology ring HH^{∗}(kG) = Ext_{kG⊗(kG)^{op}}^{∗}(kG, kG) is known to be isomorphic to H^{∗}(G, kG) = Ext_{kG}^{∗}(k, kG), where G acts on kG via conjugation. This observation goes back to Eilenberg and Mac Lane, however the proof that the ring structures correspond appears in [22, §3]. The ring structure on H^{∗}(G, kG) is given by cup product followed by the algebra multiplication map kG ⊗_{k} kG → kG. Similarly, if B is a block of kG, then HH^{∗}(B) ≅ H^{∗}(G, B), where the action of G on B is via conjugation. In this paper, we will work almost exclusively with the rings H^{∗}(G, kG) and H^{∗}(G, B) in order to exploit group cohomological techniques.

It is known that HH^{∗}(kG) is a finitely generated, graded-commutative algebra. Thus the maximal ideal spectrum of HH^{∗}(kG) (equivalently, of its even degree subring when p ≠ 2) is an affine algebraic variety which we denote by X_{G}. The details, and the following theorems, appear in Section 2. If H is a subgroup of G, let C_{G}(H) denote the centralizer of H in G, that is C_{G}(H) = {g ∈ G | gh = hg for all h ∈ H}.

Theorems 2.5, 2.10. There is a ring homomorphism, with nilpotent kernel,

HH^{∗}(kG) → ∏_{E} (kC_{G}(E) ⊗_{k} H^{∗}(E, k))^{N_{G}(E)} ,

the product over a set of representatives E of conjugacy classes of elementary abelian p-subgroups E of G. Thus the Hochschild variety X_{G} of kG is a union of subvarieties X_{G,E}, the union taken over the same set.

The structure of the subvarieties X_{G,E} in the theorem is given roughly by the existence of a finite surjection Y_{G,E} → X_{G,E} where Y_{G,E} is a disjoint union of affine varieties of dimension equal to the rank of E, the union taken over the set of blocks of kC_{G}(E).

In order to refine this description, we consider the Hochschild variety of a block B of kG. Let kG ≅ B_{1}⊕⋯⊕B_{n} be the decomposition into blocks (that is, indecomposable ideal direct summands) B_{1}, …, B_{n}. As there are no nonzero kG-bimodule maps between distinct blocks, HH^{∗}(kG) ≅ HH^{∗}(B_{1}) ⊕ ⋯ ⊕ HH^{∗}(B_{n}) as algebras. The Hochschild variety X_{G} of kG is thus the disjoint union of the Hochschild varieties X_{B} of its blocks B, where X_{B} is the maximal ideal spectrum of HH^{∗}(B) ≅ H^{∗}(G, B). We again find that each X_{B} is a union of subvarieties X_{B,E}, where this time E ranges over a set of G-conjugacy representatives of elementary abelian p-subgroups of a defect group of B. There are finite surjective maps Y_{B,E} → X_{B,E} where Y_{B,E} is a disjoint union of affine varieties of dimension equal to the rank of E, the union taken over the set of blocks of kC_{G}(E) corresponding to B under the Alperin-Broué correspondence. In the union X_{B} = ∪_{E}X_{B,E}, we must determine how the pieces are glued together to form X_{B}. In contrast to group cohomology, X_{B,F} does not necessarily inject into X_{B,E} when F < E. Thus we find it necessary to introduce auxiliary varieties X_{B,E,F} with maps X_{B,F} ⟵^{Br^{∗}} X_{B,E,F} ↪ X_{B,E} that provide the gluing information for X_{B}. The map Br^{∗} is given by the Alperin-Broué correspondence and is neither injective nor surjective in general.

Let X_{B,E}^{+} = X_{B,E} − ⋃_{F <E} X_{B,E,F}, the union over all proper subgroups F of E, and define Y_{B,E}^{+} similarly. As before, there are finite surjective maps Y_{B,E}^{+} → X_{B,E}^{+}. Let W_{G}(E) = N_{G}(E)/C_{G}(E), the Weyl group of E. The following result depends on some technical work done in Section 3:

Theorems 4.2, 4.3. (Quillen stratification) Let B be a block of G and P a defect group of B. The Hochschild variety X_{B} of B is a disjoint union of subvarieties X_{B,E}^{+}, the union over a set of elementary abelian p-subgroups E of P, one from each Gconjugacy class of elementary abelian p-subgroups of G for which at least one member is contained in P. If E is an elementary abelian p-subgroup of P for which there is a unique block of kC_{G}(E) corresponding to B under the Alperin-Broué correspondence, then there is an inseparable isogeny Y_{B,E}^{+}/W_{G}(E) → X_{B,E}^{+}.

The varieties Y_{B,E}^{+} of the theorem are given explicitly in terms of group cohomological varieties and the Alperin-Broué correspondence. The last statement applies in particular to the principal block B_{0}, and in this special case we have the following corollary.

Corollary 4.5. The Hochschild variety X_{B_{0}} of the principal block B_{0} is homeomorphic to the maximal ideal spectrum of the group cohomology ring H^{∗}(G, k).

This proves a weak form of the conjecture that HH^{∗}(B_{0}) is isomorphic to H^{∗}(G, k), modulo their radicals [23, Conj. 1] (see also [22, §11, Question 2]). A more general question, motivated by the results of [17], involves Linckelmann’s block cohomology LH^{∗}(B) (see Definition 4.6):

Question 4.7. Let B be a block of kG. Are the maximal ideal spectra of HH^{∗}(B) and LH^{∗}(B) homeomorphic?

In the case of the principal block B_{0}, it is known that LH^{∗}(B_{0}) ≅ H^{∗}(G, k), and so Corollary 4.5 gives an affirmative answer to this question. Our paper [17] also gives a positive answer in the cases (1) B has a cyclic defect group and (2) G is a Frobenius group (p odd). In fact we proved the stronger result that HH^{∗}(B) and LH^{∗}(B) are isomorphic, modulo their radicals, in these cases. We hope that the results of the present paper will ultimately lead to an answer to this question in general. Such an answer could have important consequences in block theory.

The appendix, written by Stephen F. Siegel, contains a detection result that is needed for our Quillen stratification. Hochschild cohomology is not a functor on algebras. However, if the algebra is a group algebra, we may define restriction maps to subgroup algebras by keeping coefficients in the original group ring. Then nilpotent elements are detected by restrictions to elementary abelian p-subgroups, as stated in Proposition 2.1 in the next section. The proof is similar to the standard proof given for the analogous result in group cohomology, but there are some subtleties involved. This detection result is a special case of the following more general result in the appendix.

Lemma 5.1. Let A be an algebra on which G acts by automorphisms, and M an Amodule that is also a kG-module for which these two module structures are compatible. Let ζ ∈ H^{∗}(G, A) such that for each elementary abelian p-subgroup E of G, some power of res_{E}^{G}(ζ) ∈ H^{∗}(E, A) annihilates H^{∗}(E, M). Then some power of ζ annihilates H^{∗}(G, M).

The detection result we need is a consequence of Lemma 5.1 and is also a special case of a theorem of Suslin for finite group schemes [25]. To be complete and to allow for potential applications of Lemma 5.1 itself, we have decided to include the details of the proof in the appendix.

2. The Hochschild variety of G

We will begin with some results about the ring structure of HH^{∗}(kG) ≅ H^{∗}(G, kG) (where the action of G on kG is via conjugation) and their consequences for the structure of its maximal ideal spectrum. First we note that H^{∗}(G, kG) is gradedcommutative by a very general result of Gerstenhaber [12, §7, Cor. 1]. It is also finitely generated: There is an embedding of the group cohomology ring H^{∗}(G, k) into H^{∗}(G, kG) induced by the unit map k ↪ kG. Now H^{∗}(G, kG) is finitely generated as an H^{∗}(G, k)-module, and H^{∗}(G, k) is finitely generated [11, Thm. 7.4.1, Cor. 7.4.6]. The union of two such sets of generators yields a set of generators for H^{∗}(G, kG).

The following result on nilpotent elements in H^{∗}(G, kG) is also stated as Corollary 5.4, proved in the appendix. Alternatively, the proposition follows from a very general theorem of Suslin for finite group schemes [25]. If H < G, we will use the notation res_{H}^{G} and cor_{H}^{G} for the restriction and corestriction maps, respectively. (See [6] or [11] for details.) Proposition 2.1. Let ζ ∈ H^{∗}(G, kG). Then ζ is nilpotent if, and only if, res_{E}^{G}(ζ) is nilpotent for every elementary abelian p-subgroup E of G.

For any subgroup H of G, let \overline{H}^{∗}(H, kG) (respectively, \overline{H}^{∗}(H, k)) denote the quotient of the cohomology ring H^{∗}(H, kG) (respectively, H^{∗}(H, k)) by the ideal of “proper transfers” ∑_{K<H} Im(cor_{K}^{H}). We will be interested in the ring homomorphism

(2.2)  σ: H^{∗}(G, kG) → ∏_{E}(\overline{H}^{∗}(E, kG))^{N_{G}(E)},

the product over a set of representatives E of conjugacy classes of elementary abelian p-subgroups of G, given in the E-component by the composition \overline{res}_{E}^{G} of res_{E}^{G} with the quotient map. We will use the following result from [22]. The statement (i) in the proposition below is essentially the Universal Coefficients Theorem [8, VI, Thm. 3.3], whereas the proof of (ii) is more involved and uses a product formula for Hochschild cohomology [22, Thm. 5.1]. Proposition 2.3. Let H be a subgroup of G, acting on kG by conjugation. The following are isomorphisms of graded algebras:

(i) [22, Prop. 3.2] H^{∗}(H, kC_{G}(H)) ≅ kC_{G}(H) ⊗_{k} H^{∗}(H, k).

(ii) [22, Thm. 10.2] \overline{H}^{∗}(H, kG) ≅ kC_{G}(H) ⊗_{k} \overline{H}^{∗}(H, k).

For an elementary abelian p-group E, we have \overline{H}^{∗}(E, k) = H^{∗}(E, k) [11, Lem. 6.3.4], so part (ii) of the above proposition implies that there is a ring isomorphism

(2.4)  ψ_{E}: \overline{H}^{∗}(E, kG) ⟶^{∼} kC_{G}(E) ⊗_{k} H^{∗}(E, k).

Part (i) further implies that kC_{G}(E) ⊗_{k} H^{∗}(E, k) ≅ H^{∗}(E, kC_{G}(E)). We will sometimes consider ψ_{E} to be a map to H^{∗}(E, kC_{G}(E)) when it is convenient. Let ρ be the composition of σ with the direct product of these isomorphisms ψ_{E}. Another way to view ρ is as the product of restriction maps res_{E}^{G}, each followed by the map induced by the projection of kG = kC_{G}(E)⊕k(G−C_{G}(E)) onto the first summand, and then by the isomorphism of part (i) of the above proposition.

Theorem 2.5. There is a ring homomorphism, with nilpotent kernel,

ρ: H^{∗}(G, kG) → ∏_{E}(kC_{G}(E) ⊗_{k} H^{∗}(E, k))^{N_{G}(E)},

the product over a set of representatives E of conjugacy classes of elementary abelian p-subgroups of G.

Proof. As ρ is the composition of σ (2.2) with the isomorphisms ψ_{E} (2.4), it suffices to prove that Ker(σ) is nilpotent. This is equivalent to showing that Ker(σ) consists entirely of nilpotent elements, as H^{∗}(G, kG) is graded-commutative.

Let ζ ∈ Ker(σ). Then for each E in the product, res_{E}^{G}(ζ) ∈ ∑_{F <E} Im(cor_{F}^{E}). We will show that res_{E}^{G}(ζ) is nilpotent; this involves an argument similar to the proof of [26, Thm. 3.2, part 2], but we repeat it here for completeness. It will follow that ζ is nilpotent by Proposition 2.1, as each elementary abelian p-subgroup of G is conjugate to one of the subgroups E in the product, and res_{E}^{G}(ζ) is nilpotent if and only if its conjugates by all elements of G are nilpotent.

Write res_{E}^{G}(ζ) = ∑_{F <E} cor_{F}^{E}(ζ_{F}) for some elements ζ_{F} ∈ H^{∗}(F, kG). Then

(res_{E}^{G}(ζ))^{2} = res_{E}^{G}(ζ) · ∑_{F <E} cor_{F}^{E}(ζ_{F}) = ∑_{F <E} cor_{F}^{E}(res_{F}^{G}(ζ) · ζ_{F}).

As ζ ∈ Ker(σ), we also have res_{F}^{G}(ζ) = ∑_{L<F} cor_{L}^{F}(ζ_{L}) for some elements ζ_{L} ∈ H^{∗}(L, kG). Therefore

(res_{E}^{G}(ζ))^{2} = ∑_{F <E} cor_{F}^{E} (∑_{L<F} cor_{L}^{F}(ζ_{L}) · ζ_{F} )
= ∑_{F <E} ∑_{L<F} cor_{L}^{E}(ζ_{L} · res_{L}^{F}(ζ_{F})).

Similarly, we may calculate (res_{F}^{G}(ζ))^{3}, and so on. The sizes of the subgroups involved in the power (res_{E}^{G}(ζ))^{n} strictly decreases as n increases, and so (res_{E}^{G}(ζ))^{n} = 0 for some n. □

We may use the map ρ in the theorem to obtain a result regarding the maximal ideal spectrum of the Hochschild cohomology ring H^{}(G, kG), where

H^{}(G, kG) = \begin{cases}H^{2∗}(G, kG), if p is odd \\ H^{∗}(G, kG), if p = 2. \end{cases}

The ring H^{}(G, kG) is commutative, as H^{∗}(G, kG) is graded-commutative. The map ρ restricted to H^{}(G, kG) will also be denoted ρ.

For each elementary abelian p-subgroup E, the corresponding factor in the target of ρ consists of elements invariant under N_{G}(E), and so is contained in the invariants under C_{G}(E):

(kC_{G}(E) ⊗_{k} H^{}(E, k))^{N_{G}(E)} ⊆ (kC_{G}(E) ⊗_{k} H^{}(E, k))^{C_{G}(E)}.

However C_{G}(E) acts trivially on H^{}(E, k), so the latter ring is Z(kC_{G}(E))⊗_{k}H^{}(E, k), a commutative algebra. We will set

(2.6)  A^{}(E) = Z(kC_{G}(E)) ⊗_{k} H^{}(E, k),

and define the algebra map

(2.7)  ϕ_{E}: H^{}(G, kG) → A^{}(E)

by restriction of the map ψ_{E} ∘ \overline{res}_{E}^{G} to H^{}(G, kG), followed by inclusion into A^{}(E) (ψ_{E} is defined in (2.4) and \overline{res}_{E}^{G} in the text above (2.4)).

We now define several varieties. Definition 2.8. The Hochschild variety of G is X_{G} = max(H^{}(G, kG)), the maximal ideal spectrum of H^{}(G, kG). Let Y_{G,E} = max(A^{}(E)) and X_{G,E} = max(Im(ϕ_{E})). Note that ϕ_{E} induces a map ϕ_{E}^{∗}: Y_{G,E} → X_{G,E} ↪ X_{G}.

We first describe an approximate relationship between the varieties Y_{G,E} and X_{G,E}. Lemma 2.9. The algebra A^{}(E) is finitely generated over Im(ϕ_{E}), so that ϕ_{E}^{∗} : Y_{G,E} → X_{G,E} is a finite surjective map. In particular, dim(X_{G,E}) = dim(Y_{G,E}).

Proof. It suffices to show that kC_{G}(E)⊗_{k} H^{}(E, k) is finitely generated over the image of ϕ_{E}, as A^{}(E) = Z(kC_{G}(E)) ⊗_{k} H^{}(E, k) is a submodule of this, and H^{}(G, kG) is Noetherian.

Applying the isomorphism (2.4), we need to show that \overline{H}^{}(E, kG) is finitely generated as a module over res_{E}^{G}(H^{}(G, kG)), which will follow once we show that H^{}(E, kG) is finitely generated over res_{E}^{G}(H^{}(G, kG)). Now H^{}(E, kG) is finitely generated over H^{}(E, k) by [11, Thm. 7.4.1], and H^{}(E, k) is finitely generated over res_{E}^{G}(H^{}(G, k)) by [11, Cor. 7.4.7]. Therefore H^{}(E, kG) is finitely generated over res_{E}^{G}(H^{}(G, k)). (Take as a set of generators the set of pairwise products of the previous generators.) But res_{E}^{G}(H^{}(G, k)) ⊆ res_{E}^{G}(H^{}(G, kG)) as these restriction maps are defined compatibly, and so H^{}(E, kG) is finitely generated over res_{E}^{G}(H^{}(G, kG)). Thus ϕ_{E}^{∗} is a finite map [7, p. 171]. As Im(ϕ_{E}) is a subalgebra of A^{}(E), ϕ_{E}^{∗} : Y_{G,E} → X_{G,E} is surjective by the Going-up Theorem [4, Thm. 5.10]. It also follows from finiteness that dim(X_{G,E}) = dim(Y_{G,E}). □

As a consequence of the lemma, we may describe the relationship of the Hochschild variety X_{G} to the varieties X_{G,E} and Y_{G,E} as follows. Theorem 2.10. The Hochschild variety is a union,

X_{G} = ⋃_{E}ϕ_{E}^{∗}(Y_{G,E}) = ⋃_{E}X_{G,E},

taken over a set of representatives E of conjugacy classes of elementary abelian psubgroups of G.

Proof. By Theorem 2.5, the map

∏_{E} ϕ_{E}: H^{}(G, kG) → ∏_{E} A^{}(E)

has nilpotent kernel. Since the product is finite, Lemma 2.9 implies that the target of the map ∏_{E} ϕ_{E} is finitely generated as a module over its image. It follows from these two facts that

⋃_{E} ϕ_{E}^{∗}: ⋃_{E} Y_{G,E} → X_{G}

is a finite dominant map [7, p. 172]. Again by the Going-up Theorem [4, Thm. 5.10], ⋃_{E} ϕ_{E}^{∗} is surjective. □

3. The varieties X_{G,E} and Y_{G,E}

In this section, we work to understand better the varieties X_{G,E} and Y_{G,E} appearing in Theorem 2.10. First, Y_{G,E} clearly depends on the block structure of the group algebra kC_{G}(E), and accordingly we choose some notation. Definition 3.1. If E is an elementary abelian p-subgroup of G, denote by 𝔅_{E} the set of blocks (that is, block ideals) of kC_{G}(E). Thus kC_{G}(E) = ⊕_{b∈𝔅_{E}}b as an algebra. Denote the block idempotent associated to the block b by e_{b}, so that b = kC_{G}(E)e_{b}. For each block b ∈ 𝔅_{E}, let Y_{b} = k^{rk(E)}, the affine space of dimension equal to the rank of E.

The following result gives a simple picture of the varieties Y_{G,E}. Lemma 3.2. If E is an elementary abelian p-subgroup of G then

Y_{G,E} ≅ ⋃_{b∈𝔅_{E}} Y_{b},

the disjoint union of affine spaces of dimension rk(E), one for each block of kC_{G}(E).

Proof. By definition, Y_{G,E} = max(Z(kC_{G}(E)) ⊗_{k} H^{}(E, k)). If we write kC_{G}(E) = ⊕_{b∈𝔅_{E}}b then Z(kC_{G}(E)) = ⊕_{b∈𝔅_{E}}Z(b) where Z(b) is the center of the block b. Note that Z(b)/ rad(Z(b)) ≅ k, generated by the image of e_{b} [9, Prop. 56.16].

Now we may write A^{}(E) = ⊕_{b∈𝔅_{E}}(Z(b) ⊗_{k} H^{}(E, k)). Thus Y_{G,E} is the disjoint union of the maximal ideal spectra of the summands Z(b) ⊗_{k} H^{}(E, k). Since Z(b) ⊗_{k} H^{}(E, k) is isomorphic, modulo radicals, to Z(b)/ rad(Z(b)) ⊗_{k} H^{}(E, k), its maximal ideal spectrum is

max (Z(b)/ rad(Z(b)) ⊗_{k} H^{}(E, k)) = max(H^{}(E, k)) ≅ k^{rk(E)} = Y_{b}.
□

The picture of the varieties Y_{G,E} provided by the lemma gives us some understanding of the subvarieties X_{G,E} of X_{G}, as each ϕ_{E}^{∗} : Y_{G,E} → X_{G,E} is a finite surjective map. We will work to refine this picture, and to understand how the varieties X_{G,E} are glued together in the union of Theorem 2.10. For this, we will need the Brauer map. Let F < E be elementary abelian p-subgroups of G, and note that

F < E ≤ C_{G}(E) ≤ C_{G}(F).

The corresponding Brauer map is the linear function

br_{F}^{E} : Z(kC_{G}(F)) → Z(kC_{G}(E))

defined by br_{F}^{E}(x) = x if x ∈ C_{G}(E), and br_{F}^{E}(x) = 0 if x ∈ C_{G}(F) − C_{G}(E). This is in fact an algebra homomorphism by [1, (2.5)(3)] or [9, Lemma 58.2(v)], as Z(kC_{G}(F)) = (kC_{G}(F))^{C_{G}(F)} ⊂ (kC_{G}(F))^{E}. We similarly denote by br_{1}^{P} : kG → kC_{G}(P) the Brauer map defined analogously for any p-subgroup P of G.

Cohomologically, the Brauer map br_{F}^{E} is the following composition of maps, where we identify Z(kC_{G}(F)) with degree 0 Hochschild cohomology:

H^{0}(C_{G}(F), kC_{G}(F)) ⟶^{res_{E}^{C_{G}(F)}} H^{0}(E, kC_{G}(F)) → \overline{H}^{0}(E, kC_{G}(F)) ≅ H^{0}(E, kC_{G}(E)).

Indeed H^{0}(E, kC_{G}(E)) ≅ (kC_{G}(E))^{E}, and the image of the above composition of maps lies in kC_{G}(E)^{C_{G}(E)} = Z(kC_{G}(E)) since it factors through H^{0}(C_{G}(E), kC_{G}(E)). Since restriction and reducing modulo the ideal of proper transfers are algebra maps, we see that br_{F}^{E} : Z(kC_{G}(F)) → Z(kC_{G}(E)) is also an algebra map.

In case F = 1, we may thus view the Brauer maps as degree 0 components of the map σ defined in (2.2).

Let

(3.3)  A^{}(E, F) = Z(kC_{G}(E)) ⊗ H^{}(F, k),

and A^{}(E), A^{}(F) as defined in (2.6). Let

(3.4)  Res_{F}^{E} : A^{}(E) → A^{}(E, F) and Br_{F}^{E} : A^{}(F) → A^{}(E, F)

be defined by Res_{F}^{E} = Id ⊗ res_{F}^{E} and Br_{F}^{E} = br_{F}^{E} ⊗ Id. Let

ϕ_{E,F} : H^{}(G, kG) → A^{}(E, F)

be the composition of res_{F}^{G} and the following sequence of maps, where we note that the image of res_{F}^{G} is contained in the C_{G}(E)-invariant subalgebra:

H^{}(F, kG)^{C_{G}(E)} → H^{}(F, kC_{G}(E))^{C_{G}(E)} ⟶^{∼} (kC_{G}(E) ⊗ H^{}(F, k))^{C_{G}(E)} = A^{}(E, F).

The first map above is induced by the Brauer map br_{1}^{E}, and the second is the isomorphism of Proposition 2.3(i).

Lemma 3.5. For elementary abelian p-subgroups F < E of G, ϕ_{E,F} is an algebra homomorphism, and the following diagram commutes:

A^{}(E)_{Res_{E}_{F}} [figure] H^{}(G, kG) _{ϕ_{E}}  [figure] [figure]_{ϕ_{E,F}} [figure]_{ϕ_{F}}  [figure] [figure]A^{}(E, F) A^{}(F)_{Br_{E}_{F}} [figure]

Proof. The map ϕ_{E} is essentially res_{E}^{G} followed by the map induced by the Brauer map br_{1}^{E}. From this it is clear that the top square commutes. The bottom square also commutes as ϕ_{E,F} is essentially ϕ_{F} followed by the map induced by the Brauer map br_{F}^{E}. Finally all the maps in the diagram other than ϕ_{E,F} are known to be algebra maps, and so commutativity of the diagram forces ϕ_{E,F} to be an algebra map as well. □

We may now restrict the domains of Res_{F}^{E} and Br_{F}^{E} to obtain maps Res_{F}^{E} : Im(ϕ_{E}) → Im(ϕ_{E,F}) and Br_{F}^{E} : Im(ϕ_{F}) → Im(ϕ_{E,F}). We will next define auxiliary varieties X_{G,E,F} and Y_{G,E,F} that will be related to our previous varieties via these maps. Definition 3.6. Let Y_{G,E,F} = max(A^{}(E, F)) and X_{G,E,F} = max(Im(ϕ_{E,F})). By the same arguments as in the proof of Lemma 3.2, we have Y_{G,E,F} ≅ ∪_{B∈𝔅_{E}}k^{rk(F)}, a disjoint union of affine spaces of dimension equal to the rank of F. The map ϕ_{E,F} induces a map of varieties ϕ_{E,F}^{∗} : Y_{G,E,F} → X_{G,E,F}, Res_{F}^{E} induces (Res_{F}^{E})^{∗} : X_{G,E,F} → X_{G,E}, and Br_{F}^{E} induces (Br_{F}^{E})^{∗} : X_{G,E,F} → X_{G,F}.

The following theorem gives the fundamental picture: Theorem 3.7. If F < E are elementary abelian p-subgroups of G, then the following diagram of varieties commutes:

\xymatrix{X_{G,E} & Y_{G,E} \ar[l]_{ϕ_{E}^{∗}} \\ X_{G,E,F} \ar[d]_{(Br_{F}^{E})^{∗}} \ar[u]^{(Res_{F}^{E})^{∗}} & Y_{G,E,F} \ar[u]_{(Res_{F}^{E})^{∗}} \ar[l]_{ϕ_{E,F}^{∗}} \ar[d]^{(Br_{F}^{E})^{∗}} \\ X_{G,F} & Y_{G,F} \ar[l]_{ϕ_{F}^{∗}}}

The maps ϕ_{E}^{∗}, ϕ_{F}^{∗} , ϕ_{E,F}^{∗} are finite surjective maps. The maps (Res_{F}^{E})^{∗} are injective.

Proof. The commutativity of the diagram is a consequence of Lemma 3.5 and the comments in the paragraph following it. The injectivity of (Res_{F}^{E})^{∗} follows from the surjectivity of Res_{F}^{E} which in turn follows from the surjectivity of res_{F}^{E} : H^{}(E, k) → H^{}(F, k).

We have already seen that ϕ_{E}^{∗} and ϕ_{F}^{∗} are finite maps. Now A^{}(E) is finitely generated over Im(ϕ_{E}) and Res_{F}^{E} is surjective. Therefore A^{}(E, F) is finitely generated over Im(ϕ_{E,F}). It follows that ϕ_{E,F} is a finite map, and hence surjective as well. □

By Theorem 3.7, we may view X_{G,E,F} as the part of X_{G,E} coming from X_{G,F}. Notice that Y_{G,E,F} has the same number of components as Y_{G,E} in general but their dimension is the same as that of Y_{G,F}. The map (Res_{F}^{E})^{∗} : Y_{G,E,F} → Y_{G,E} may be defined componentwise for the blocks of E, the component associated to a block B ∈ 𝔅_{E} being the map (res_{F}^{E})^{∗} : max(H^{}(F, k)) → max(H^{}(E, k)). Thus we have a reasonably good understanding of the injective map (Res_{F}^{E})^{∗}.

However we will soon see that (Br_{F}^{E})^{∗} is neither injective nor surjective in general. It can happen that F < E but X_{G,F} does not inject into X_{G,E}, unlike the situation in the Quillen stratification of H^{}(G, k). We will give a simple example next to illustrate this.

Example 3.8. Let G = Σ_{3} and p = 2. Up to conjugacy, the only elementary abelian 2-subgroups are E = {1, (12)} and 1.

We have A^{}(E) = kE ⊗H^{}(E, k) and so Y_{G,E} = k. Similarly A^{}(1) = Z(kG)⊗k and so Y_{G,1} consists of two points (as kG has two blocks). By the same reasoning, Y_{G,E,1} consists of exactly one point. It follows that (Br_{1}^{E})^{∗} : Y_{G,E,1} → Y_{G,1} is not surjective.

In fact it will follow from Theorem 4.3, as N_{G}(E) = E, that ϕ_{1}^{∗}, ϕ_{E}^{∗}, ϕ_{E,1}^{∗} are injective and hence homeomorphisms onto their images. Thus we have X_{G,E} = k, X_{G,1} consists of two points and X_{G,E} ∩ X_{G,1} consists of one point. Consequently, the Hochschild variety of G is X_{G} = X_{G,E} ∪ X_{G,1} = k ∪ {∗}, where ∗ is an isolated point. Notice that the trivial elementary abelian group contributed essentially to X_{G}!

In terms of the block decomposition of kG, we have X_{G} = ∪X_{B} where X_{B} = max(HH^{∗}(B)), and B ranges over the two blocks of kG. The dimension of X_{B} is the p-rank of a defect group P of B, that is the rank of a maximal elementary abelian p-subgroup of P (see Theorem 4.2 below or [15, Cor. 4.3(ii)]). Thus our picture X_{G} = k ∐{∗} reflects the block structure of kG: It has two blocks, one semisimple block (whose variety is {∗}) and the principal block having variety k.

Now we work to understand better the map (Br_{F}^{E})^{∗} : Y_{G,E,F} → Y_{G,F}. The behaviour of this map is completely determined by (br_{F}^{E})^{∗} : max(Z(kC_{G}(E))) → max(Z(kC_{G}(F))), so we will study (br_{F}^{E})^{∗} first.

By [9, Prop. 56.16], Z(kC_{G}(E))/ rad(Z(kC_{G}(E))) ≅ ⊕_{b∈𝔅_{E}}k, where the primitive central idempotent e_{b} of kC_{G}(E) projects to zero in all factors except the one corresponding to B, where it projects to 1. Thus max(Z(kC_{G}(E)) consists of a finite number of points, one for each block b of E, and in this way we may identify max(Z(kC_{G}(E)) with the set 𝔅_{E} of blocks of kC_{G}(E). Specifically, we associate to the block b of kC_{G}(E) the maximal ideal M_{b} of Z(kC_{G}(E)) that does not contain e_{b}.

The algebra map br_{F}^{E} : Z(kC_{G}(F)) → Z(kC_{G}(E)) induces a map of varieties (br_{F}^{E})^{∗} : max(Z(kC_{G}(E))) → max(Z(kC_{G}(F))) that may be viewed as a map from 𝔅_{E} to 𝔅_{F} by the above comments. In fact (br_{F}^{E})^{∗} is the block correspondence of Alperin and Broué: By [1, Lemma 3.6(1)], if b is a block of kC_{G}(E), there is a unique block B of kC_{G}(F) such that br_{F}^{E}(e_{B})e_{b} = e_{b}. The corresponding statement for the associated varieties is that (br_{F}^{E})^{∗}(M_{b}) does not contain e_{B}, and hence must be M_{B}. Conversely, since br_{F}^{E} is an algebra map from Z(kC_{G}(F)) to Z(kC_{G}(E)), the image br_{F}^{E}(e_{B}) is a central idempotent of kC_{G}(E) and hence one of the following occurs:

(i) br_{F}^{E}(e_{B}) = 0, in which case M_{B} is not in the image of (br_{F}^{E})^{∗}, or (ii) br_{F}^{E}(e_{B}) = ∑_{b∈S} e_{b} for some S ⊆ 𝔅_{E}, so that (br_{F}^{E})^{∗}(M_{b}) = M_{B} for all b ∈ S.

A consequence of (i) is that (br_{F}^{E})^{∗} may not be surjective, and a consequence of (ii) is that (br_{F}^{E})^{∗} may not be injective. Therefore (Br_{F}^{E})^{∗} is in general neither injective nor surjective. The following lemma summarizes implications for the relationship between the varieties Y_{G,E,F} and Y_{G,F}. Lemma 3.9. Let F < E be elementary abelian p-subgroups of G. The Brauer map Br_{F}^{E} induces a map of varieties

(Br_{F}^{E})^{∗} : Y_{G,E,F} = ⋃_{B∈𝔅_{E}} k^{rk(F)} → Y_{G,F} = ⋃_{B∈𝔅_{F}} k^{rk(F)}

which identifies components according to the Alperin-Broué correspondence and is injective when restricted to any particular component.

At this point, we have some understanding of the maps

Y_{G,E} ↩^{(Res_{F}^{E})^{∗}} Y_{G,E,F} ⟶^{(Br_{F}^{E})^{∗}} Y_{G,F}

of Theorem 3.7. How well this carries over to the maps

X_{G,E} ↩^{(Res_{F}^{E})^{∗}} X_{G,E,F} ⟶^{(Br_{F}^{E})^{∗}} X_{G,F}

depends on how far the finite surjective maps ϕ_{E}^{∗}, ϕ_{F}^{∗} and ϕ_{E,F}^{∗} are from being injective. We address this question partially in the next section, and use the auxiliary varieties X_{G,E,F} to obtain a Quillen stratification of the Hochschild variety X_{G}. More precisely, we describe the stratification in terms of the blocks of kG.

4. Quillen stratification and consequences

In this section we will restrict our attention to the summands of H^{}(G, kG) corresponding to the blocks of kG. By standard arguments (see for example [22, §3]),

HH^{∗}(B) = Ext_{B⊗B^{op}}^{∗}(B, B) ≅ Ext_{kG⊗(kG)^{op}}^{∗}(kG, B) ≅ Ext_{kG}^{∗}(k, B),

that is HH^{∗}(B) ≅ H^{∗}(G, B) where B is a kG-module via conjugation. Let E be an elementary abelian p-subgroup of G. Recall that in degree 0, the map ϕ_{E} of (2.7) is essentially the Brauer map br_{1}^{E}.

Let B be a block of kG. We may apply ϕ_{E} to H^{}(G, B) via the embedding H^{}(G, B) ↪ H^{}(G, kG), and we will denote this map by ϕ_{B,E}. Similarly, let ϕ_{B,E,F} denote the map ϕ_{E,F} restricted to H^{}(G, B). Now H^{}(G, B) = H^{}(G, kGe_{B}) = H^{}(G, kG)e_{B} by identifying e_{B} with an element of H^{0}(G, kG) ≅ Z(kG). As ϕ_{E} is an algebra homomorphism, ϕ_{B,E}(H^{}(G, B)) = ϕ_{E}(H^{}(G, kG))ϕ_{E}(e_{B}) ⊆ (Z(kC_{G}(E)) ⊗ H^{}(E, k))(br_{1}^{E}(e_{B}) ⊗ 1). Thus we have

Im(ϕ_{B,E}) ⊆ br_{1}^{E}(e_{B})Z(kC_{G}(E)) ⊗ H^{}(E, k),

and further the image is an N_{G}(E)-invariant subalgebra.

Let

A_{B}(E) = br_{1}^{E}(e_{B})Z(kC_{G}(E)) ⊗ H^{}(E, k),
and A_{B}(E, F) = br_{1}^{E}(e_{B})Z(kC_{G}(E)) ⊗ H^{}(F, k).

We next define the corresponding varieties. Definition 4.1. The Hochschild variety of the block B of kG is X_{B} = max(H^{}(G, B)). Let Y_{B,E} = max(A_{B}(E)) and Y_{B,E,F} = max(A_{B}(E, F)). By restricting the domains of the maps (3.4), we have Res_{F}^{E} : A_{B}(E) → A_{B}(E, F) and Br_{F}^{E} : A_{B}(F) → A_{B}(E, F). These induce maps on varieties

Y_{B,F} ⟵^{(Br_{F}^{E})^{∗}} Y_{B,E,F} ↪^{(Res_{F}^{E})^{∗}} Y_{B,E}.

Let

Y_{B,E}^{+} = Y_{B,E} − ⋃_{F <E}(Res_{F}^{E})^{∗}(Y_{B,E,F}).

Similarly, let X_{B,E} = max(Im(ϕ_{B,E})), X_{B,E,F} = max(Im(ϕ_{B,E,F})), and

X_{B,E}^{+} = X_{B,E} − ⋃_{F <E}(Res_{F}^{E})^{∗}(X_{B,E,F}).

Note that by Theorem 3.7, X_{B,E}^{+} = ϕ_{B,E}^{∗}(Y_{B,E}^{+}).

The above observations and those at the end of the last section allow us to describe the varieties X_{B,E} in terms of the defect groups of B: These are the maximal psubgroups P of G such that br_{1}^{P} (e_{B}) ≠ 0 [1, (2.6)]. They are all conjugate in G, and if P is any p-subgroup, br_{1}^{P} (e_{B}) ≠ 0 if and only if P is contained in a defect group of B [1, (2.6)]. Thus X_{B,E} is nonempty if, and only if, E is contained in a defect group of B. The following theorem is the first part of the Quillen stratification, giving a more precise description of the Hochschild variety X_{B} of B. Theorem 4.2. Let B be a block of G and P a defect group of B. Then

X_{B} = ⋃_{E} X_{B,E}^{+},

a disjoint union of subvarieties, taken over a set of elementary abelian p-subgroups E of P, one from each G-conjugacy class of elementary abelian p-subgroups of G for which at least one member is contained in P. Further there are finite surjective maps Y_{B,E}^{+} → X_{B,E}^{+}, so that dim(X_{B,E}^{+}) = dim(Y_{B,E}^{+}) = rk(E), and dim(X_{B}) = rk(P).

We remark that dim(X_{B}) was already known to be equal to rk(P), as follows from [15, Cor. 4.3(ii)].

Proof. We use standard methods, such as in [7, §5.6]. For each E, let

σ_{E} = ∏_{0≠ζ∈H^{1}(E,𝔽_{p})}β(ζ),

where β is the Bockstein homomorphism, so that σ_{E} ∈ H^{}(E, k). By definition of the map Res_{F}^{E}, we now have Y_{B,E}^{+} = max (A_{B}(E) [br_{1}^{E}(e_{B}) ⊗ σ_{E}^{−1}]). Note that Y_{B,E}^{+} is empty if br_{1}^{E}(e_{B}) = 0, which is the case if no G-conjugate of E lies in P.

By [7, Lemma 5.6.2], for each elementary abelian p-subgroup E of G, there is an element ρ_{E} ∈ H^{}(G, k) such that res_{E}^{G}(ρ_{E}) = (σ_{E})^{p^{a}} for some a. Further, if F is an elementary abelian p-subgroup that is not contained in a conjugate of E then res_{E}^{G}(ρ_{F}) = 0. Let ψ_{B} : H^{}(G, k) → H^{}(G, B) be the map induced by the inclusion k ↪ B that sends 1 to e_{B}. We have ϕ_{B,E}(ψ_{B}(ρ_{F})) = br_{1}^{E}(e_{B}) ⊗ res_{E}^{G}(ρ_{F}) for all elementary abelian p-subgroups F ≤ E of G, which is nonzero if and only if E is contained in a conjugate of P and F is contained in a conjugate of E. It follows from these observations that

X_{B,E}^{+} = max(ϕ_{B,E}(H^{}(G, B)[ψ_{B}(ρ_{E})^{−1}])) = max(Im(ϕ_{B,E})[(br_{1}^{E}(e_{B}) ⊗ res_{E}^{G}(ρ_{E}))^{−1}]).

Now Im(ϕ_{B,E}) is an H^{}(G, B)-module via ϕ_{B,E}, and ψ_{B}(ρ_{F}) ∈ H^{}(G, B) acts on Im(ϕ_{B,E}) as br_{1}^{E}(e_{B}) ⊗ res_{E}^{G}(ρ_{F}) which is zero if no G-conjugate of F is contained in E. Thus the X_{B,E}^{+} are disjoint since given a pair of elementary abelian p-subgroups E and F which are not G-conjugate, at least one of them does not contain the other up to G-conjugacy.

Also note that the only maximal ideals M removed from X_{B,E} to form X_{B,E}^{+} are those which contain br_{1}^{E}(e_{B}) ⊗ res_{E}^{G}(ρ_{E}) or equivalently br_{1}^{E}(e_{B}) ⊗ (σ_{E})^{p^{a}}. Pulling M back to H^{∗}(G, B), we have an ideal containing Ker(ϕ_{B,E}) and ψ_{B}(ρ_{E}). This ideal is in the image (Res_{F}^{E})^{∗}(X_{B,E,F}) for some F < E and hence the ideal M contains Ker(ϕ_{B,E,F}) and Ker(ϕ_{B,F}). Thus M will be in X_{B,F} for some F < E. It follows that the union of X_{B,E}^{+}, over all elementary abelian p-subgroups E, is still X_{B}. □

We now give a more complete description of the varieties X_{B,E}^{+} in a special case. Let W_{G}(E) = N_{G}(E)/C_{G}(E), the Weyl group of E. Theorem 4.3. Let B be a block of G and E an elementary abelian p-subgroup of G such that br_{1}^{E}(e_{B}) = e_{b} for some block b of E. Then ϕ_{B,E}^{∗} induces an inseparable isogeny Y_{B,E}^{+}/W_{G}(E) → X_{B,E}^{+}.

The relationship between Y_{B,E}^{+} and X_{B,E}^{+} in general appears to be more subtle as the maps ϕ_{B,E} may be more complicated.

Proof. Under the hypothesis, A_{B}(E) = Z(b)⊗H^{}(E, k). Let γ_{B,E} : H^{}(G, k) → A_{B}(E) be the composition

H^{}(G, k) ⟶^{ψ_{B}} H^{}(G, B) ⟶^{ϕ_{B,E}} A_{B}(E),

where as above ψ_{B} is induced by the inclusion k ↪ B that sends 1 to e_{B}. Note that γ_{B,E} sends ζ ∈ H^{}(G, k) to br_{1}^{E}(e_{B}) ⊗ res_{E}^{G}(ζ). As elements of H^{}(G, k) and of H^{}(G, B) are invariant under conjugation by elements of G, we have the following containments:

(4.4)  Im(γ_{B,E}) ⊆ Im(ϕ_{B,E}) ⊆ A_{B}(E)^{W_{G}(E)}.

Every element in A_{B}(E)^{W_{G}(E)} may be expressed as e_{b}⊗ζ, for some ζ ∈ H^{}(E, k)^{W_{G}(E)}, plus an element in the radical. (This follows from writing x ∈ A_{B}(E)^{W_{G}(E)} as x = ∑_{i}(c_{i}e_{b} + r_{i}) ⊗ ζ_{i} where c_{i} ∈ k and r_{i} ∈ rad(Z(b)).)

By [7, Lemma 5.6.2], there is a nonnegative integer a such that ζ^{p^{a}} = res_{E}^{G}(ρ_{E}^{−1}ζ′) for some ζ′ ∈ H^{}(G, k) and ρ_{E} ∈ H^{}(G, k) as before with res_{E}^{G}(ρ_{E}) = (σ_{E})^{p^{a}}. Therefore (e_{b} ⊗ ζ)^{p^{a}} = e_{b} ⊗ ζ^{p^{a}} = e_{b} ⊗ res_{E}^{G}(ρ_{E}^{−1}ζ′) ∈ Im(γ_{B,E}) where we have extended γ_{B,E} to a map from H^{}(G, k)[ρ_{E}^{−1}] to (A_{B}(E)[e_{b}⊗σ_{E}^{−1}])^{W_{G}(E)}. Considering the sequence of subalgebras (4.4), this shows that (e_{b} ⊗ζ)^{p^{a}} is also contained in ϕ_{B,E}(H^{}(G, B)[ψ_{B}(ρ_{E})^{−1}]). Thus

((A_{B}(E)[e_{b} ⊗ σ_{E}^{−1}])^{W_{G}(E)})^{p^{a}} ⊆ Im(ϕ_{B,E})[e_{b} ⊗ σ_{E}^{−1}] ⊆ (A_{B}(E)[e_{b} ⊗ σ_{E}^{−1}])^{W_{G}(E)}.

Hence ϕ_{B,E} induces an inseparable isogeny

max((A_{B}(E)[e_{b} ⊗ σ_{E}^{−1}])^{W_{G}(E)}) = Y_{B,E}^{+}/W_{G}(E) → X_{B,E}^{+}.

□

The principal block B_{0} of G satisfies the hypotheses of Theorem 4.3 for each elementary abelian p-subgroup E of G, that is the defect groups of B_{0} are the Sylow p-subgroups, and br_{1}^{E}(e_{B_{0}}) = e_{b_{0}} for every elementary abelian p-subgroup E of G. We are therefore led to the following corollary, which proves a weak form of the conjecture [23, Conj. 1]. Corollary 4.5. Let B_{0} be the principal block of kG. Then max(HH^{}(B_{0})) is homeomorphic to max(H^{}(G, k)).

Proof. First note that all elementary abelian p-subgroups of G are G-conjugate to a subgroup of the Sylow p-subgroup P, the defect group of B_{0}.

By the proof of Theorem 4.3, the maps ϕ_{B_{0},E} together give a homomorphism ϕ_{B_{0}} : H^{}(G, B_{0}) → lim A_{B_{0}}(E) where this inverse limit is over the category of elementary abelian p-subgroups of G with morphisms given by inclusions and G-conjugations. Note that since br_{1}^{E}(e_{B_{0}}) = e_{b_{0}}, there is no ambiguity in defining the gluing maps in this inverse limit.

The proofs of Theorems 4.2 and 4.3 show that modulo radicals, ϕ_{B_{0}} induces an inseparable isogeny. However since br_{1}^{E}(e_{B_{0}}) = e_{b_{0}} for every elementary abelian psubgroup E, we see that modulo radicals, lim A_{B_{0}}(E) is isomorphic to lim H^{}(E, k). Quillen has shown [7, Thm. 5.6.3] that there is an inseparable isogeny between this latter inverse limit and H^{}(G, k). It follows that the varieties for H^{}(G, k) and H^{}(G, B_{0}) are homeomorphic.

□

More generally, Theorem 4.3 applies to a block of principal type, that is a block B such that for every p-subgroup P of G, br_{1}^{P} (e_{B}) is either 0 or a unique block idempotent e_{b}. For such a block B, Theorem 4.3 implies that the Hochschild variety X_{B} is homeomorphic to the subvariety of max(H^{}(G, k)) obtained by removing those pieces in the Quillen stratification corresponding to elementary abelian p-subgroups not contained in a defect group of B.

In the remainder of this section, we will speculate on related potential consequences of Theorems 4.2 and 4.3. Let (P, B_{P}) be a Sylow B-subpair of G, unique up to conjugacy [1], so that P is a defect group of B and B_{P} is a block of kC_{G}(P). If R is a subgroup of P, there is a unique block B_{R} of kC_{G}(R) such that (R, B_{R}) ≤ (P, B_{P}), where the partial order on subpairs is defined in [1]. Let N_{G}(B_{R}) be the subgroup of N_{G}(R) fixing B_{R} setwise, under conjugation. Definition 4.6 (Linckelmann [14, 15]). Let B be a block of kG with defect group P. The block cohomology ring of B is the subring LH^{∗}(B) of H^{∗}(P, k) consisting of all ζ ∈ H^{∗}(P, k) satisfying

g res_{R}^{P}(ζ) = res_{R}^{P}(ζ)

for all subgroups R of P, and all g ∈ N_{G}(B_{R}).

If B = B_{0} is the principal block, then LH^{∗}(B_{0}) ≅ H^{∗}(G, k). Thus Corollary 4.5 shows that max(HH^{∗}(B_{0})) is homeomorphic to max(LH^{∗}(B_{0})). We do not know whether the generalization of this statement to all blocks is true, so we raise it as a question. Question 4.7. Let B a block of kG. Are max(HH^{∗}(B)) and max(LH^{∗}(B)) homeomorphic?

In fact, the cohomology rings HH^{∗}(B) and LH^{∗}(B) are isomorphic, modulo their radicals, in many known cases as shown in [17], and the question was raised in that paper whether this is true in general. We do not know of a counterexample to either statement. It is possible that a detailed comparison of our stratification of the Hochschild variety max(HH^{∗}(B)) of a block B with Linckelmann’s stratification of the block variety max(LH^{∗}(B)) [16] would yield further information, potentially providing an answer to this question.

In Linckelmann’s paper [16], he also deals more generally with the support varieties of modules over a block B, defined via the block cohomology LH^{∗}(B). An analogous theory defined via the Hochschild cohomology HH^{∗}(B) was developed in unpublished work of Siegel [21], some of which has now been done in greater generality by Snashall and Solberg [24] for modules over Artinian algebras. The techniques we have developed in this paper should lead to a Quillen stratification of these support varieties of modules over B, defined via the Hochschild cohomology HH^{∗}(B), but we do not pursue this here.

ACKNOWLEDGMENT. The second author thanks M. Linckelmann for some very helpful discussions.

5. Appendix: Hochschild cohomology and elementary abelian subgroups

Stephen F. Siegel Department of Computer Science University of Massachusetts, Amherst, MA 01003-9264

                            siegel@cs.umass.edu

Main results. Throughout this section, G is a finite group and k is a (not necessarily algebraically closed) field of prime characteristic p. All k-algebras are assumed to have units.

By a G-equivariant k-algebra we mean a k-algebra A on which G acts as algebra automorphisms. A map of G-equivariant k-algebras is an algebra homomorphism which is also a kG-homomorphism. By a G-equivariant A-module we mean an Amodule M such that the structure map

μ: A ⊗ M → M

is a kG-homomorphism, i.e., g(ax) = (ga)(gx) for all g ∈ G, a ∈ A, and x ∈ M. A map of G-equivariant A-modules is an A-homomorphism which is also a kGhomomorphism.

In this setting, the cup product gives H^{∗}(G, A) the structure of a graded k-algebra, and H^{∗}(G, M) becomes an H^{∗}(G, A)-module.

The main result is the following Key Lemma: Lemma 5.1. Let A be a G-equivariant k-algebra, and ζ ∈ H^{∗}(G, A). Let M be a G-equivariant A-module. Suppose that, for each elementary abelian p-subgroup E of G, some power of res_{E}^{G}(ζ) ∈ H^{∗}(E, A) annihilates H^{∗}(E, M). Then some power of ζ annihilates H^{∗}(G, M).

Before proving Lemma 5.1, we look at several consequences. It will be seen that the Lemma allows one to easily obtain several of the well-known results in ordinary group cohomology, as well as analogous results for Hochschild cohomology, in a unified way.

The main application is in the following setting. Suppose N is a G-equivariant k-algebra, and that there is a map of G-equivariant algebras α: A → N. Then we may consider N as a G-equivariant A-module via μ(a ⊗ x) = α(a)x. In this case we can say something in both directions. Theorem 5.2. Let A and ζ be as in Lemma 5.1, and α: A → N as above. Then some power of ζ annihilates H^{∗}(G, N) if, and only if, for each elementary abelian p-subgroup E of G, some power of res_{E}^{G}(ζ) annihilates H^{∗}(E, N).

Proof. Since we are dealing with algebras with units, ζ^{n} annihilates H^{∗}(E, N) if, and only if, it annihilates 1 ∈ H^{0}(E, N). Assuming ζ^{n} annihilates H^{∗}(G, N), we have

res_{E}^{G}(ζ^{n}) · 1 = res_{E}^{G}(ζ^{n}) · res_{E}^{G}(1) = res_{E}^{G}(ζ^{n} · 1) = 0.

So ζ^{n} annihilates H^{∗}(E, N). The other direction follows from the Lemma. □

The special case of Theorem 5.2 in which N = A and α = 1 yields the following: Corollary 5.3. Let A be a G-equivariant k-algebra, and ζ ∈ H^{∗}(G, A). Then ζ is nilpotent if, and only if, res_{E}^{G}(ζ) is nilpotent for every elementary abelian subgroup E of G.

The case A = k is well-known and is a key step in the ordinary Quillen stratification for H^{∗}(G, k). Another case is where M is a kG-module and N = End_{k}(M). In that case there is a natural isomorphism

H^{∗}(G, N) ≅ Ext_{kG}^{∗}(M, M).

So Corollary 5.3 yields the well-known result that an element of Ext_{kG}^{∗}(M, M) is nilpotent if and only if its restriction to E is nilpotent for every elementary abelian E.

We may also take A = k and N = End_{k}(M) and obtain from Theorem 5.2 the weaker well-known result that for ζ ∈ H^{∗}(G, k), some power of ζ annihilates Ext_{kG}^{∗}(M, M) if and only if for all E, some power of res_{E}^{G}(ζ) annihilates Ext_{kE}^{∗}(M, M).

To see how this applies to Hochschild cohomology, recall that HH^{∗}(kG), the Hochschild cohomology of kG, may be identified with H^{∗}(G, kG), where kG is considered a module under conjugation. In this case, Corollary 5.3 yields Corollary 5.4. Let ζ ∈ HH^{∗}(kG). Then ζ is nilpotent if, and only if, its restriction to H^{∗}(E, kG) is nilpotent for every elementary abelian p-subgroup E of G.

We also get a module version for Hochschild cohomology, as follows. Let A = kG, let M be a kG-module and N = End_{k}(M). We let G act on A by conjugation, and on N by (gf)(x) = gf(g^{−1}x). Then the map A → N which takes g to the map (x ↦ gx), is G-equivariant. Hence Theorem 5.2 says the following: Corollary 5.5. Let M be a kG-module and ζ ∈ HH^{∗}(kG). Then some power of ζ annihilates Ext_{kG}^{∗}(M, M) if, and only if, for each elementary abelian p-subgroup E of G, some power of res_{E}^{G}(ζ) annihilates Ext_{kE}^{∗}(M, M).

Corollary 5.5 has an interpretation in terms of varieties for modules. For consider X_{G}, the (prime or maximal ideal) spectrum of HH^{∗}(kG). Let X_{G}(M) = V (I), the subvariety of X_{G} consisting of the (prime or maximal) ideals containing I, where I is the radical of the annihilator in HH^{∗}(kG) of Ext_{kG}^{∗}(M, M); we call X_{G}(M) the Hochschild support variety for M (cf. [24]). More generally, for any subgroup H of G, let X_{G,H}(M) = V (J), where J is the radical of the kernel of the composite

H^{∗}(G, kG) → Ext_{kG}^{∗}(M, M) → Ext_{kH}^{∗}(M, M),

in which the second map is restriction. We may think of X_{G,H}(M) as the part of X_{G}(M) coming from H. Now Corollary 5.5 is easily seen to be equivalent to the following: Corollary 5.6. Let M be a kG-module. Then

X_{G}(M) = ⋃_{E} X_{G,E}(M),

where the union is taken over all elementary abelian p-subgroups E of G.

The analogous result for ordinary cohomology is the well-known Alperin-Evens Theorem ([3]).

Results needed for proof. The proof of Lemma 5.1 follows the one for the usual case (k in place of A) given in [11, Cor. 8.3.4], which uses Serre’s Theorem on the product of Bocksteins, and the module version of the Quillen-Venkov Lemma. We just have to make a few adjustments along the way.

The first ingredient is this commutativity lemma. Lemma 5.7. Let G be a finite group, k a field, A a G-equivariant k-algebra, and η: k → A the map sending λ to λ1. Then the image of the algebra homomorphism

η^{∗}: H^{∗}(G, k) → H^{∗}(G, A)

is contained in the graded-center of H^{∗}(G, A).

Proof. There is a commutative diagram

\xymatrix{k ⊗ A \ar[r]^{η⊗1} \ar[d]^{η⊗1} & A ⊗ A \ar[r]^{τ} & A ⊗ A \ar[d]^{μ} \\ A ⊗ A \ar[rr]^{μ} &  & A}

where τ(x⊗y) = y⊗x and μ(x⊗y) = xy. The lemma then follows from the definition of the cup product. □

Next, a definition. Suppose P is a p-group and K is a maximal subgroup of P. Then K is normal and P/K is cyclic of order p, so H^{1}(P/K, k) ≅ k. Let α be a non-zero element of H^{1}(P/K, k) and define

β_{K} = β(inf_{P/K}^{P}(α)) ∈ H^{2}(P, k),

where β: H^{s}(P, k) → H^{s+1}(P, k) (s ≥ 0) is the Bockstein map and inf_{P/K}^{P} is the inflation map [11, §§1.1, 3.3]. Of course β_{K} is only well-defined up to non-zero scalar multiple, but the particular choice doesn’t matter for what follows.

Theorem 5.8 (Serre, [20]). Let P be a p-group. Then P is not elementary abelian if, and only if, there exist maximal subgroups K_{1}, …, K_{r} of P such that

β_{K_{1}} … β_{K_{r}} = 0

in H^{∗}(P, k).

For a simple proof, see [11, Thm. 6.4.1]. The statement there differs only in that it is stated in terms of integral cohomology, but as explained in the proof of [11, Cor. 6.4.2], this implies the mod-p version.

The other ingredient we need is a lemma of Alperin and Evens ([2, Lemma 4.1]), which generalizes to arbitrary modules the Quillen-Venkov Lemma ([19]):

Lemma 5.9 (Quillen-Venkov, Alperin-Evens). Let P be a p-group, K a maximal subgroup of P, and M a kP -module. Then the filtration of H^{∗}(P, M) associated to the Lyndon-Hochschild-Serre spectral sequence

H^{∗}(P/K, H^{∗}(K, M)) ⇒ H^{∗}(P, M)

satisfies

F^{2}H^{∗}(P, M) = β_{K}H^{∗}(P, M).

Proof of Lemma 5.1. We assume that for each elementary abelian p-subgroup E of G, some power of res_{E}^{G}(ζ) annihilates H^{∗}(E, M), and we will show that some power of ζ annihilates H^{∗}(G, M).

Let S be a Sylow p-subgroup of G. It suffices to show that some power of res_{S}^{G}(ζ) annihilates H^{∗}(S, M), since restriction from H^{∗}(G, M) to H^{∗}(S, M) is injective [6, Cor. 3.6.18]. We will do this by showing that some power of res_{P}^{G}(ζ) annihilates H^{∗}(P, M) for every subgroup P of S, by induction on |P |.

If |P | = 1 the result is clear. So suppose |P | > 1 and we have established the result for every proper subgroup of P. If P is elementary abelian, we are done by hypothesis. If not, then let K_{1}, …, K_{r} be maximal subgroups of P as in Theorem 5.8, and let β_{i} = β_{K_{i}}.

Let K = K_{1}. For any kP -module U, let E(U) denote the Lyndon-Hochschild-Serre spectral sequence

H^{∗}(P/K, H^{∗}(K, U)) ⇒ H^{∗}(P, U).

Now E(M) is a module over E(A) and over E(k), and the commutative diagram

\xymatrix{k ⊗ M \ar[r] \ar[d]^{η⊗1} & M \ar[d]^{1} \\ A ⊗ M \ar[r]^{μ} & M}

implies the following diagram commutes:

\xymatrix{E(k) ⊗ E(M) \ar[r] \ar[d]^{η^{∗}⊗1} & E(M) \ar[d]^{1} \\ E(A) ⊗ E(M) \ar[r] & E(M)}

Let χ = res_{P}^{G}(ζ). By induction, for some m = m_{1}, ξ = res_{K}^{P}(χ^{m}) ∈ E_{2}^{0,∗}(A) annihilates H^{∗}(K, M), and therefore E_{2}(M). Since ξ is in the image of res_{K}^{P}, ξ lives to E_{∞}(A), and so it annihilates E_{∞}(M) as well. This means that

χ^{m}F^{i}H^{∗}(P, M) ⊆ F^{i+1}H^{∗}(P, M)

for all i ≥ 0. In particular,

χ^{2m_{1}}H^{∗}(P, M) ⊆ F^{2}H^{∗}(P, M) = β_{1}H^{∗}(P, M),

by Lemma 5.9.

Applying the same reasoning to K = K_{2}, there is some m_{2} such that

χ^{2m_{2}}H^{∗}(P, M) ⊆ β_{2}H^{∗}(P, M).

So

χ^{2m_{1}+2m_{2}}H^{∗}(P, M) ⊆ χ^{2m_{2}}(β_{1}H^{∗}(P, M))
= (χ^{2m_{2}}η^{∗}(β_{1}))H^{∗}(P, M)
= (η^{∗}(β_{1})χ^{2m_{2}})H^{∗}(P, M)

since, by Lemma 5.7, η^{∗}(β_{1}) is in the center of H^{∗}(P, A). Hence

χ^{2m_{1}+2m_{2}}H^{∗}(P, M) ⊆ β_{1}(χ^{2m_{2}}H^{∗}(P, M)) ⊆ (β_{1}β_{2})H^{∗}(P, M).

Continuing in this way, we see

χ^{2m_{1}+···+2m_{r}}H^{∗}(P, M) ⊆ β_{1} … β_{r}H^{∗}(P, M) = 0.

That completes the inductive step, and the proof of Lemma 5.1.

ACKNOWLEDGMENT. The author thanks D. J. Benson for some clarifying conversations.

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Department of Mathematics, University of Rochester, Rochester, NY 14627

  E-mail address: jonpak@math.rochester.edu

Department of Mathematics and Computer Science, Amherst College, Amherst, MA 01002

  E-mail address: sjw@cs.amherst.edu