Hochschild and block cohomology varieties are isomorphic

Markus Linckelmann

March 18, 2009

Abstract We show that the varieties of the Hochschild cohomology of a block algebra and its block cohomology are isomorphic, implying positive answers to questions of Pakianathan and Witherspoon in [18] and [19]. We obtain as consequences that the cohomology H^{∗}(G; k) of a finite group G with coefficients in a field k of characteristic p is a quotient of the Hochschild cohomology of the principal block of kG by a nilpotent ideal, and that the canonical map from the Hochschild cohomology of a block algebra to the graded center of its bounded derived module category has a nilpotent kernel.

1 Introduction

In [18], Pakianathan and Witherspoon raised the question whether the Hochschild cohomology HH^{∗}(B) of a block algebra B of a finite group G over an algebraically closed field k of prime characteristic p and the corresponding block cohomology H^{∗}(B) are isomorphic modulo nilpotent ideals. They showed this in a number of cases, including for blocks with cyclic defect groups and blocks of Frobenius groups; for blocks with an abelian defect group P and an abelian inertial quotient acting freely on P − {1} this follows also from [9, 1.4]. Both H^{∗}(B) and HH^{∗}(B) are graded commutative Noetherian rings, and hence their maximal ideal spectra are affine varieties, denoted V_{B} and X_{B}, respectively. In [19], Pakianathan and Witherspoon described a stratification for the variety X_{B}, similar to Quillen’s group cohomology stratification, and deduced from this that the varieties X_{B} and V_{B} are at least homeomorphic for blocks of principal type. The purpose of this paper is to show that varieties X_{B} and V_{B} are isomorphic in general. As in [19], a key ingredient is a detection result for nilpotent elements in certain Ext-algebras, due to Siegel [19, Appendix, 5.1]. Other ingredients are the techniques developed in [19] applied to source algebras rather than block algebras, combined with the transfer technology from [10], in particular the canonical map [10, Theorem 5.6.(iii)] from block cohomology H^{∗}(B) to Hochschild cohomology HH^{∗}(B) of a block algebra B of a finite group.

Theorem 1.1. Let B be a block algebra of a finite group G over an algebraically closed field k of prime characteristic p with defect group P. The canonical graded algebra homomorphism τ from block cohomology H^{∗}(B) to the Hochschild cohomology HH^{∗}(B) induces an isomorphism modulo nilpotent ideals, or equivalently, induces an isomorphism of varieties X_{B} ≅ V_{B}.

The interest of Theorem 1.1 lies in the fact that V_{B} is an invariant of the fusion system F of the block B while X_{B} is an invariant of the Morita equivalence class of B, and the relationship between the two is far from understood, to say the least. In one way or another, most prominent conjectures in block theory try to predict how invariants of the module category of B and invariants of the local structure of B determine each other. The isomorphism X_{B} ≅ V_{B} implies in particular that Quillen’s stratification (extended to blocks in [13]) carries over to X_{B}, where it coincides for blocks of principal type with the stratification described in [19]. If B is the principal block of kG then τ is split injective, with retraction HH^{∗}(B) → H^{∗}(G, k) induced by the functor − ⊗_{B} k, and thus the above result has the following immediate consequence:

Corollary 1.2. If B is the principal block of kG then the functor − ⊗_{B} k induces a split surjective algebra homomorphism HH^{∗}(B) → H^{∗}(G, k) with nilpotent kernel.

Since the principal block of kG is defined over the prime field 𝔽_{p}, this statement remains true even if k is not algebraically closed. For non principal blocks it is not known whether τ is a split monomorphism, and hence we do not know whether block cohomology is a quotient of Hochschild cohomology. We list further easy consequences of Theorem 1.1 which are true for arbitrary blocks. The isomorphism in Theorem 1.1 restricts to the corresponding varieties of a finitely generated B-module M. The functor − ⊗_{B} M induces a unitary algebra homomorphism μ_{M} : HH^{∗}(B) → Ext_{B}^{∗}(M, M); we denote by X_{B}(V) the maximal ideal spectrum of HH^{∗}(B)/ker(μ_{M}), and by V_{B}(M) the maximal ideal spectrum of H^{∗}(B)/ker(μ_{M} ∘ τ).

Corollary 1.3. For any finitely generated B-module M, restriction along τ induces an isomorphism of varieties X_{B}(M) ≅ V_{B}(M).

In other words, for block algebras the theory of varieties associated with Hochschild cohomology, as developed by Snashall and Solberg in [25], coincides with that using block cohomology in [11]. If the module M has a source of dimension prime to p then V_{B}(M) = V_{B}, and hence Corollary 1.3 implies the following.

Corollary 1.4. For any finitely generated B-module M having a source of dimension prime to p, the canonical homomorphism HH^{∗}(B) → Ext_{B}^{∗}(M, M) induced by the functor − ⊗_{B} M has a nilpotent kernel.

Following notation in [17] we denote by Z^{∗}(D^{b}(mod(B))) the graded center of the bounded derived category D^{b}(mod(B)) of the category of finitely generated B-modules mod(B).

Corollary 1.5. The canonical algebra homomorphism HH^{∗}(B) → Z^{∗}(D^{b}(mod(B))) has a nilpotent kernel.

The positive part of HH^{∗}(B) is invariant under stable equivalences of Morita type, and thus block varieties are invariant under stable equivalences of Morita type:

Corollary 1.6. Let B, C be block algebras of finite groups G, H, respectively, and let M be a C-B-bimodule which is finitely generated projective as left and right module, and which induces a stable equivalence of Morita type between B and C. Then for any finitely generated B-module U we have an isomorphism of varieties V_{B}(U) ≅ V_{C}(M ⊗_{B} U). In particular, V_{B} ≅ V_{C}.

If the C-B-bimodule M inducing a stable equivalence of Morita type between the blocks B, C of G, H has a trivial source as k(H × G)-module then M induces more precisely an isomorphism of the block cohomology rings H^{∗}(B) ≅ H^{∗}(C), but it is not known in general whether a stable equivalence of Morita type induces such an isomorphism; this is not even known for arbitrary Morita equivalences. The main step in the proof of Theorem 1.1 consists in showing first a slightly weaker statement, namely that τ induces an inseparable isogeny:

Theorem 1.7. Let A be a source algebra of a block with defect group P and fusion system F. Identify the block cohomology H^{∗}(B) with the subalgebra of F-stable elements in H^{∗}(P; k) and denote by ρ : H^{∗}(B) → lim_{E} (H^{∗}(E; k)) the map induced by the product of the restriction maps from P to E ∈ E, where E is the full subcategory of F consisting of all elementary abelian subgroups of P. Denote by τ : H^{∗}(B) → HH^{∗}(A) the canonical injective graded algebra homomorphism. Then there is a graded algebra homomorphism σ making the following diagram commutative

\xymatrix{HH^{∗}(A) \ar[rr]^{σ} &  & lim_{E} (H^{∗}(E; k)) \\  & H^{∗}(B) \ar[ul]^{τ} \ar[ur]_{ρ}}

such that ker(σ) is nilpotent; in particular, σ induces an inseparable isogeny.

Section 2 contains some background material and technical statements on product structures in certain Ext-algebras of interior P -algebras, which is used in Section 3 in order to define the map σ in Theorem 1.7. In Section 4 this material is specialised to source algebras and then used to prove Theorem 1.7 and Corollary 1.2. The proof of Theorem 1.1 and the remaining corollaries are given in Section 5.

2 Products in Ext-algebras of interior P -algebras

Let G be a finite group and k a field. By Eckmannn-Shapiro’s lemma applied to the diagonal subgroup ΔG = {(x, x)| x ∈ G} of G × G there is a graded k-linear isomorphism HH^{∗}(kG) ≅ H^{∗}(ΔG; kG). Siegel and Witherspoon showed in [24, 3.1] that the product structure of HH^{∗}(kG) can be identified in H^{∗}(ΔG; kG) as being given by the cup product followed by multiplication in kG. In this section we show that this carries over to arbitrary interior algebras over some finite group - which we will denote by P, since the results of this section will be applied to source algebras of blocks with a defect group P. All tensor products without subscripts are tensor products over the base field k, which for now need not be algebraically closed or of positive characteristic. Let A be an interior P -algebra with structural homomorphism σ : P → A^{×}. If j is an idempotent in the fixpoint subalgebra A^{ΔP} with respect to the conjugation action then jAj is again an interior P -algebra, with structural homomorphism sending u ∈ P to σ(u)j = jσ(u) = jσ(u)j. The algebra A ⊗ kP is an interior kΔP -algebra, with structural homomorphism sending (u, u) to σ(u) ⊗ u, where u ∈ P. We view Aj as A ⊗ kP -module with a ⊗ u acting on b ∈ jAj by abσ(u^{−1}), where a ∈ A and u ∈ P. This restricts to a kΔP -module structure on jAj with (u, u) acting on b ∈ jAj by σ(u)bσ(u^{−1}). A suitable version of Eckmann-Shapiro for restriction from A ⊗ kP to kΔP via the bimodule Aj ⊗ kP yields a graded k-linear isomorphism Ext_{A⊗kP}^{∗} (Aj, Aj) ≅ H^{∗}(ΔP; jAj) described in more detail in the proof of 2.1 below. The left side is an Ext-algebra, so has a product. The right side has a product, too, given by the cup product and multiplication in jAj as follows: denote by V a projective resolution of k as trivial kΔP -module, let μ ∈ H^{m}(ΔP; jAj) and ν ∈ H^{n}(ΔP; jAj) be represented by chain maps (denoted abusively by the same letters) from V to jAj viewed as complexe of kΔP -modules concentrated in degree m and n, respectively. Then μ ⊗ ν is chain map from V ⊗ V to jAj ⊗ jAj, the latter viewed as complex of kΔP -modules concentrated in degree m + n. Precomposing this with the homotopy equivalence V ≃ V ⊗ V lifting the isomorphism k ≅ k ⊗ k and composing this with the map jAj ⊗ jAj → jAj given by multiplication in A yields an element in H^{m+n}(kΔP; jAj). We consider H^{∗}(kΔP; jAj) endowed with this product. Note that in order to define this product structure on H^{∗}(ΔP; jAj) we only need the structure of A as ΔP -algebra - this definition does not make use of the interior P -algebra structure.

Proposition 2.1. Let P be a finite group, A a finite-dimensional interior P -algebra over a field k and j an idempotent in A^{ΔP}. There is a canonical isomorphism of graded k-algebras

Ext_{A⊗kP}^{∗} (Aj, Aj) ≅ H^{∗}(ΔP; jAj)

Proof. Denote by σ : P → A^{×} the structural homomorphism of A. We first show that we may assume that j = 1. Let W be a projective resolution of jAj as jAj ⊗ kP -module. The restriction of W on the left is then a projective resolution of the projective jAj-module jAj, hence split. This implies that the complex Aj ⊗_{jAj} W is a projective resolution of Aj ⊗_{jAj} jAj ≅ Aj as A ⊗ kP -module; multiplying this resolution by j on the left yields again the resolution W. Thus adjunction via the A-jAj-bimodule Aj shows that multiplication by j yields a graded k-algebra isomorphism Ext_{A⊗kP}^{∗} (Aj, Aj) ≅ Ext_{jAj⊗kP}^{∗} (jAj, jAj). This shows that after replacing A by jAj we may assume j = 1. Note that there is a canonical isomorphism of A ⊗ kP -modules

A ≅ (A ⊗ kP) ⊗_{kΔP} k

sending a ∈ A to (a ⊗ 1_{P}) ⊗ 1_{k}, with inverse sending (a ⊗ u) ⊗ λ to λaσ(u^{−1}). Denote by V a projective resolution of the trivial kΔP -module k. Set U = (A ⊗ kP) ⊗_{kΔP} V. Then, via the above isomorphism, U is a projective resolution of A as A ⊗ kP -module. Adjunction via the A⊗kP -kΔP -bimodule A⊗kP yields a graded k-linear isomorphism Ext_{A⊗kP}^{∗} (A, A) ≅ H^{∗}(ΔP; A). In order to show that this is an algebra isomorphism we describe this map explicitly in terms of adjunction counits. Denote by _{ΔP} U the restriction of U to kΔP; this is a projective resolution of A as kΔP -module. The unitary map ι : k → A, viewed as homomorphism of kΔP -modules, is the adjunction counit of induction to A ⊗ kP followed by restriction back to kΔP evaluated at k. This map lifts to a chain map of complexes of kΔP -modules adj : V → _{ΔP} U, uniquely up to homotopy, and we may choose this chain map to send y ∈ V to (1_{A} ⊗ 1_{kP}) ⊗ y, and again this is the appropriate adjunction counit evaluated at V. Multiplication in A is in particular a homomorphism of kΔP -modules from A ⊗ A to A, and hence lifts uniquely, up to homotopy, to a chain map π : _{ΔP} U ⊗_{ΔP} U → _{ΔP} U. The obvious commutative diagram of kΔP -modules with vertical maps given by multiplication in A

[figure: A ⊗ k; Id⊗ι; A ⊗ A; A; A]

lifts to a homotopy commutative diagram of chain complexes of kΔP -modules

(1)

[figure: _{ΔP} U ⊗ V; Id⊗adj; ≃; _{ΔP} U ⊗_{ΔP} U; π; _{ΔP} U; _{ΔP} U]

For any chain map of complexes of A ⊗ kP -modules τ : U → U′, multiplication by A on the left of these two complexes yields a commutative diagram of chain complexes of A ⊗ kP -modules

\xymatrix{A ⊗ U \ar[r]^{Id⊗τ} \ar[d] & A ⊗ U′ \ar[d] \\ U \ar[r]_{τ} & U′}

where the vertical maps send a ⊗ u and a ⊗ u′ to au and au′, respectively, for a ∈ A, u ∈ U and u′ ∈ U′; the commutativity uses the fact that τ is a homomorphism of chain complexes of left A-modules. This diagram lifts, uniquely up to homotopy, to a homotopy commutative diagram of chain complexes of A ⊗ kP -modules of the form

(2)

\xymatrix{U ⊗ U \ar[r]^{Id⊗τ} \ar[d]_{π} & U ⊗ U′ \ar[d]^{π′} \\ U \ar[r]_{τ} & U′}

Here the A ⊗ kP -module structure in the first line is given by the left A-module structure on the left factor U in the tensor products and the right action of kP on the right factors. The k-linear graded isomorphism Ext_{A⊗kP}^{∗} (A, A) ≅ H^{∗}(ΔP; A) is induced by the adjunction map V → _{ΔP} U. More precisely, this isomorphism sends an element in Ext_{A⊗kP}^{m} (A, A) represented by a chain map

\xymatrix{ζ : U \ar[r] & U[m]}

to the element in H^{m}(ΔP; A) represented by the chain map

\xymatrix{μ = ζ ∘ adj : V \ar[r]^{adj} & _{ΔP} U \ar[r]^{ζ} & _{ΔP} U[m]}

where we have used the same letter ζ for the restriction of ζ to a kΔP -homomorphism. In order to show that this graded isomorphism is a homomorphism of algebras we need to show that if

\xymatrix{τ : U \ar[r] & U[n]}

is a chain map representing an element in Ext_{A⊗kP}^{n} (A, A) and if

\xymatrix{ν = τ ∘ adj : V \ar[r]^{adj} & _{ΔP} U \ar[r]^{τ} & _{ΔP} U[n]}

then the chain map

\xymatrix{U \ar[r]^{ζ} & U[m] \ar[r]^{τ[m]} & U[m + n]}

representing the product ζ · τ in Ext_{A⊗kP}^{m+n} (A, A) corresponds, via adjunction, to the chain map

\xymatrix{V ≅ V ⊗ V \ar[r]^{μ⊗ν} & _{ΔP} U[m] ⊗_{ΔP} U[n] \ar[r]^{π[m+n]} & _{ΔP} U[m + n]}

which represents the product of μ and ν in H^{m+n}(ΔP; A). That is, we have to show that the diagram of chain complexes of kΔP -modules

[figure: V; adj; _{ΔP} U; ζ; _{ΔP} U; τ[m]; _{ΔP} U[m + n]; V; ≃; V ⊗ V; μ⊗ν; _{ΔP} U[m] ⊗_{ΔP} U[n]; π[m+n]; _{ΔP} U[m + n]]

is chain homotopy commutative. Since μ = ζ ∘ adj this is equivalent to showing that the diagram

[figure: V; μ; ≃; _{ΔP} U[m]; τ[m]; _{ΔP} U[m + n]; V ⊗ V; μ⊗ν; _{ΔP} U[m] ⊗_{ΔP} U[n]; π[m+n]; _{ΔP} U[m + n]]

is chain homotopy commutative. By writing μ ⊗ ν = (Id ⊗ ν) ∘ (μ ⊗ Id) it follows that it suffices to show that the diagram

[figure: V; μ; ≃; _{ΔP} U[m]; τ[m]; ≃; _{ΔP} U[m + n]; V ⊗ V; μ⊗Id; _{ΔP} U[m] ⊗ V; Id⊗ν; _{ΔP} U[m] ⊗_{ΔP} U[n]; π[m+n]; _{ΔP} U[m + n]]

is chain homotopy commutative. The left square in this diagram is homotopy commutative because it lifts the obvious commutative square

\xymatrix{V \ar[r]^{μ} \ar[d]_{≅} & _{ΔP} U[m] \ar[d]^{≅} \\ V ⊗ k \ar[r]_{μ⊗Id} & _{ΔP} U[m] ⊗ k}

via the quasi-isomorphism V → k. Therefore it remains to show that the diagram

\xymatrix{_{ΔP} U[m] \ar[r]^{τ[m]} & _{ΔP} U[m + n] \\ _{ΔP} U[m] ⊗ V \ar[u]^{≃} \ar[r]_{Id⊗ν} & _{ΔP} U[m] ⊗_{ΔP} U[n] \ar[u]_{π[m+n]}}

is homotopy commutative. Using ν = τ ∘ adj (and shifting by −m) it suffices to show that the diagram

[figure: _{ΔP} U; _{ΔP} U; τ; _{ΔP} U[n]; _{ΔP} U ⊗ V; ≃; Id⊗adj; _{ΔP} U ⊗_{ΔP} U; π; Id⊗τ; _{ΔP} U ⊗_{ΔP} U[n]; π[n]]

is homotopy commutative. The left square in this diagram is homotopy commutative thanks to the diagram (1) above, and the right square is homotopy commutative thanks to (2). This completes the proof.

The following two results describe the obvious functoriality properties of the above isomorphism.

Proposition 2.2. Let P be a finite group and σ : A → B a homomorphism of finite-dimensional ΔP -algebras over a field k. The map η : H^{∗}(ΔP; A) → H^{∗}(ΔP; B) induced by σ is a homomorphism of graded k-algebras.

Proof. Let m, n be positive integers and V a projective resolution of k as kΔP -module. Let μ ∈ H^{m}(Δ; A), ν ∈ H^{n}(ΔP; A) represented by chain maps μ : V → A[m], ν : V → A[n]. The element η(μ) ∈ H^{m}(Δ; B) is represented by the chain map η(μ) = σ[m] ∘ μ : V → B[m]; similarly for η(ν). The result follows from the obvious homotopy commutative diagram

[figure: V; ≃; V ⊗ V; μ⊗ν; A[m] ⊗ A[n]; σ[m]⊗σ[n]; A[m + n]; σ[m+n]; V; ≃; V ⊗ V; η(μ)⊗ην; B[m] ⊗ B[n]; B[m + n]]

where the horizontal homotopy equivalences V ≃ V ⊗ V lift the isomorphism k ≅ k ⊗ k and where the horizontal maps in the right square are induced by multiplication in A and B, respectively. The commutativity of the right square in this diagram uses the fact that σ is a homomorphism of ΔP -algebras. The first row of this diagram composed with σ[m + n] represents η applied to the product of μ and ν, and the second row represents the product of η(μ) and η(ν), whence the result.

If σ : A → B above is actually a homomorphism of interior P -algebras then the algebra homomorphism induced by σ can be interpreted via the isomorphism of 2.1 as follows:

Proposition 2.3. Let P be a finite group and σ : A → B a homomorphism of finite-dimensional interior P -algebras over a field k. Consider B as a B-A-bimodule with the right A-module structure induced via σ. The functor B⊗_{A}− induces a graded k-algebra homomorphism β : Ext_{A⊗kP}^{∗} (A, A) → Ext_{B⊗kP}^{∗} (B, B), and the graded k-linear map η : H^{∗}(ΔP; A) → H^{∗}(ΔP; B) induced by σ is an algebra homomorphism making the following diagram of graded k-algebras commutative:

\xymatrix{Ext_{A⊗kP}^{∗} (A, A) \ar[r]^{≅} \ar[d]_{ϵ} & H^{∗}(ΔP; A) \ar[d]^{η} \\ Ext_{B⊗kP}^{∗} (B, B) \ar[r]_{≅} & H^{∗}(ΔP; B)}

where the horizontal isomorphisms are from 2.1.

Proof. We have an obvious isomorphism of B-A-bimodules B ⊗_{A} A ≅ B sending b ⊗ a to bσ(a). Let U be a projective resolution of A as A ⊗ kP -module and V a projective resolution of k as kΔP -module. As before denote by _{ΔP} U the restriction of U to a complex of ΔP -modules. The unitary map k → A, viewed as homomorphism of kΔP -modules, lifts to a chain map V → _{ΔP} U, uniquely up to homotopy. Since A is projective as left A-module, the restriction to A on the left of U yields a split complex of projective A-modules with homology isomorphic to A in degree zero. Thus B ⊗_{A} U is still a projective resolution of B as B ⊗ kP -module, and hence the functor B ⊗_{A} − induces indeed a homomorphism of graded k-algebras β as claimed. The unitary map k → B, again viewed as homomorphism of kΔP -modules, lifts to a chain map V → _{ΔP} (B ⊗_{A} U). The map σ, viewed as homomorphism of kΔP -modules, lifts to the chain map _{ΔP} U → _{ΔP} (B ⊗_{A} U) sending u ∈ U to 1_{B} ⊗ u. Let n be a nonnegative integer and ζ ∈ Ext_{A⊗kP}^{n} (A, A) represented by a chain map ζ : U → U[n]. Then β(ζ) is represented by the chain map Id⊗ζ : B ⊗_{A} U → B ⊗_{A} U[n]. Since σ is a unitary homomorphism, the above chain maps make the following diagram of chain complexes of kΔP -modules chain homotopy commutative:

[figure: V; _{ΔP} U; ζ; _{ΔP} U[n]; V; _{ΔP} (B ⊗_{A} U); Id⊗ζ; _{ΔP} (B ⊗_{A} U[n])]

The first row in this diagram represents the image of ζ in H^{n}(ΔP; A), and the second row represents the image of β(ζ) in H^{n}(ΔP; B), which implies the result.

Given a graded k-algebra H^{∗} we denote by Z(H^{∗}) the center of H^{∗} in the graded sense; that is, the component of Z(H^{∗}) in degree n consists of all ζ ∈ H^{n} with the property that for all m and all ξ ∈ H^{m} we have ζξ = (−1)^{nm}ξζ; this is also sometimes called graded center of H^{∗} - but since we use this terminology already in the context of triangulated categories, we simply call Z(H^{∗}) the center of H^{∗} as graded algebra. If char(k) = 2 this is the ususal center of H^{∗} as (ungraded) k-algebra; if char(k) > 2 then the even part of Z(H^{∗}) is the center of the even part of H^{∗}.

Lemma 2.4. Let P be a finite group and A a finite-dimensional interior P -algebra over a field k. For any subgroup E of P, the unitary map k → A induces a graded k-algebra homomorphism ι_{E}^{A} : H^{∗}(ΔE; k) → H^{∗}(ΔE; A), the induction functor (A ⊗ kE) ⊗_{kΔE} − induces a graded kalgebra homomorphism κ_{E}^{A} : H^{∗}(ΔE; k) → Ext_{A⊗kE}^{∗}(A, A), the functor A⊗_{kP} − applied to kP -kP - bimodules induces a graded k-algebra homomorphism HH^{∗}(kP) → Ext_{A⊗kP}^{∗} (A, A) and diagonal induction from P ≅ ΔP to P ×P induces a graded k-algebra homomorphism H^{∗}(P; k) → HH^{∗}(kP) making the following diagram commutative

[figure: H^{∗}(P; k); ≅; _{Ind{P ×P \atop ΔP}}; H^{∗}(ΔP; k); κ_{P}^{A}; H^{∗}(ΔP; k); ι_{P}^{A}; HH^{∗}(kP); A⊗_{kP} −; Ext_{A⊗kP}^{∗} (A; A); ≅; H^{∗}(ΔP; A)]

where the lower right horizontal isomorphism is from 2.1. Moreover, the images of κ_{E}^{A} and ι_{E}^{A} are contained in the graded centers of Ext_{A⊗kE}^{∗}(A, A) and H^{∗}(ΔE; A), respectively.

Proof. The unitary map k → A is a homomorphism of ΔP -algebras, hence induces an algebra homomorphisms ι_{E}^{A} : H^{∗}(ΔE; k) → H^{∗}(ΔE; A) by 2.3. Let V be a projective resolution of the trivial kΔE-module k. Since (A ⊗ kE) ⊗_{kΔE} k ≅ A as A ⊗ kE-module, the complex U = (A ⊗ kE)⊗_{kΔE}V is a projective resolution of A as A⊗kE-module, and hence the functor (A⊗kE)⊗_{kΔE}− induces an algebra homomorphism κ_{E}^{A} : H^{∗}(ΔE; k) → Ext_{A⊗kE}^{∗}(A, A). Transitivity of induction functors induced by tensoring with bimodules implies the commutativity of the left square of the diagram. For the right square, let n be an integer and ξ ∈ H^{n}(ΔE; k) represented by a chain map ξ : V → k[n]. Then κ_{E}^{A}(ξ) is represented by the chain map Id ⊗ ξ : U → A ⊗ k[n] = A[n]. The isomorphism from 2.1 maps κ_{E}^{A}(ξ) to the element in H^{n}(ΔE; A) represented by the chain map V → _{ΔE}U obtained from precomposing Id ⊗ ξ with the inclusion V → _{ΔE}U sending v ∈ V to 1 ⊗ v, which clearly is a chain map representing ι_{E}^{A}(τ). This shows the commutativity of the right square in the diagram. The last statement follows from [19, Lemma 5.7].

We will need the following special case of a result of Snashall and Solberg:

Proposition 2.5 (cf. [25, Theorem 1.1]). Let P be a finite group and A a finite-dimensional interior P -algebra over a field k such that A is projective as right kP -module, let E be a subgroup of P and j an idempotent in A^{ΔE}. The image of the canonical graded k-algebra homomorphism HH^{∗}(A) → Ext_{A⊗kE}^{∗}(Aj, Aj) induced by tensoring with the A-E-bimodule Aj is contained in the center Z(Ext_{A⊗kE}^{∗}(Aj, Aj)) of the graded algebra Ext_{A⊗kE}^{∗}(Aj, Aj).

3 On the Brauer construction in cohomology

Let k be an algebraically closed field of prime characteristic p. Let P be a finite group. Following the notation in [19, §2] and [24, §3] we denote by H̄^{∗}(P; k) the quotient of H^{∗}(P; k) by the ideal ∑_{Q<P} tr_{Q}^{P}(H^{∗}(Q; k)), where Q runs over the proper subgroups of P and where tr_{Q}^{P} : H^{∗}(Q; k) → H^{∗}(P; k) is the usual transfer map in group cohomology. If P is a not a p-group it is well-known that then H^{∗}(P; k) = 0, and if E is an elementary abelian p-group then H̄^{∗}(E; k) ≅ H^{∗}(E; k) by [6, 6.3.4]. For M a kP -module we denote by H̄^{∗}(P; M) the quotient of H^{∗}(P; M) by the subspace ∑_{Q<P} tr_{Q}^{P}(H^{∗}(Q; Res_{Q}^{P}(M))); in degree zero this is the usual Brauer construction applied to M. The canonical H^{∗}(P; k)-module structure on H^{∗}(P; M) induces a H̄^{∗}(P; k)-module structure on H̄^{∗}(P; M). The main purpose of this section is to prove that under suitable hypotheses, the canonical map ι_{E}^{A} : H^{∗}(ΔE; k) → Z(H^{∗}(ΔE; A)) from 2.4 is a split monomorphism.

Proposition 3.1. Let P be a finite p-group and A a finite-dimensional ΔP -algebra over k. Suppose that A has a ΔP -stable k-basis X such that X^{ΔP} ≠ ∅, or equivalently, such that A(ΔP) ≠ {0}. Let E be a subgroup of P such that Z(A(ΔE)) is local, or equivalently, such that the algebra A(ΔE) is indecomposable. There is a unique graded k-algebra homomorphism π_{E}^{A} : Z(H^{∗}(ΔE; A)) → H̄^{∗}(ΔE; k) such that π_{E}^{A} ∘ ι_{E}^{A} : H^{∗}(ΔE; k) → H̄^{∗}(ΔE; k) is equal to the canonical surjection, and such that π_{E}^{A} factors through the canonical map Z(H^{∗}(ΔE; A)) → Z(H̄^{∗}(ΔE; A)). In particular, if E is elementary abelian then π_{E}^{A} ∘ ι_{E}^{A} = Id_{H^{∗}(ΔE;k)}.

Before getting into the details of the proof - which we break up into two lemmas - we briefly describe where the hypotheses come into play. With the notation above, denote by X^{ΔE} the subset of all basis elements in X which are fixed by all elements in ΔE. As kΔE-module, A has a decomposition of the form A = k[X^{ΔE}] ⊕ k[X − X^{ΔE}]. The Brauer homomorphism Br_{ΔE}^{A} : A^{ΔE} → A(ΔE) maps X^{ΔE} to a k-basis of A(ΔE), and every indecomposable direct summand of k[X − X^{ΔE}] is induced from a proper subgroup of ΔE. Thus, as graded k-modules, we have an isomorphism H̄^{∗}(ΔE; A(ΔE)) ≅ H̄^{∗}(ΔE; A), where A(ΔE) is viewed as kΔE-module with the trivial action of ΔE. Both sides in this isomorphism have canonical algebra structures. The next result shows that there is a canonical choice for this isomorphism which is an isomorphism of graded k-algebras. We view A^{ΔE} and A(ΔE) as ΔE-algebras with the trivial action of ΔP.

Lemma 3.2. Let P be a finite p-group and A a finite-dimensional ΔP -algebra over k. Suppose that A has a ΔP -stable k-basis X such that X^{ΔP} ≠ ∅, or equivalently, such that A(ΔP) ≠ {0}. Let E be a subgroup of P. The inclusion A^{ΔE} → A induces a graded k-algebra homomorphism ϵ : H^{∗}(ΔE; A^{ΔE}) → H^{∗}(ΔE; A), the Brauer homomorphism Br_{ΔE}^{A} : A^{ΔE} → A(ΔE) induces a surjective graded k-algebra homomorphism β : H^{∗}(ΔE; A^{ΔE}) → H^{∗}(ΔE; A(ΔE)) and there is a unique graded k-algebra isomorphism β̄ : H̄^{∗}(ΔE; A(ΔE)) ≅ H̄^{∗}(ΔE; A) making the following diagram of graded k-algebra homomorphisms commutative:

[figure: H^{∗}(ΔE; A^{ΔE}); β; ϵ; H^{∗}(ΔE; A(ΔE)); H^{∗}(ΔE; A); H̄^{∗}(ΔE; A(ΔE)); β̄; ≅; H̄^{∗}(ΔE; A); H̄^{∗}(ΔE; k); ῑ_{E}^{A(ΔE)}; H̄^{∗}(ΔE; k); ῑ_{E}^{A}]

where the downward vertical arrows are the canonical surjections and the upward vertical arrows are induced by ι_{E}^{A(ΔE)} and ι_{E}^{A}, respectively.

Proof. Let X be a ΔP -stable k-basis of A. The decomposition A = k[X^{ΔE}] ⊕ k[X − X^{ΔE}] as kΔE-modules yields, upon taking ΔE-fixpoints in both summands, a decomposition of k-vector spaces

A^{ΔE} = k[X^{ΔE}] ⊕ (k[X − X^{ΔE}])^{ΔE}

Note that the second summand is equal to the ideal ker(Br_{ΔE}^{A}) and that the first summand is non zero, by the assumptions. Furthermore, since Br_{ΔE}^{A} induces an isomorphism of kΔEmodules (with trivial action) k[X^{ΔE}] ≅ A(ΔE), the map β is surjective. Any element ζ ∈ H^{∗}(ΔE; A^{ΔE}) is the sum of its two components ζ_{0} ∈ H^{∗}(ΔE; X^{ΔE}) and ζ_{1} ∈ H^{∗}(ΔE; ker(Br_{ΔE}^{A})). The component ζ_{1} is contained in the kernel of both β and of ι composed with the surjection H^{∗}(ΔE; A) → H̄^{∗}(ΔE; A). Similarly, the component ζ_{0} is in the kernel of β composed with H^{∗}(ΔE; A(ΔE)) → H̄^{∗}(ΔE; A(ΔE)) if and only if ζ_{0} is in the kernel of ϵ composed with H^{∗}(ΔE; A) → H̄^{∗}(ΔE; A). This shows that there is an injective graded k-algebra homomorphism from H̄^{∗}(ΔE; A(ΔE)) to H̄^{∗}(ΔE; A) making the upper pentagon in the diagram commutative; since both sides have equal dimensions in each degree this is an isomorphism of graded k-algebras. The lower square is commutative since the isomorphism β̄ is induced by unitary algebra homomorphisms between the three algebras A(ΔE), A^{ΔE} and A.

This result plays back the calculation of H̄^{∗}(ΔE; A) to the same calculation with A replaced by A(ΔE), viewed as ΔE-algebra with trivial action. In that case one can be more precise; the following Proposition is a minor generalisation of [24, 3.2]:

Lemma 3.3. Let E be a finite p-group and A a finite-dimensional k-algebra, considered as ΔEalgebra with ΔP acting trivially on A. For any subgroup E of P there are graded k-algebra isomorphisms

A^{op} ⊗ H^{∗}(ΔE; k) ≅ H^{∗}(ΔE; A)
A^{op} ⊗ H̄^{∗}(ΔE; k) ≅ H̄^{∗}(ΔE; A)

induced by the maps sending c ∈ A to right multiplication by c on A and by the graded algebra homomorphism ι_{E}^{A} : H^{∗}(ΔE; k) → H^{∗}(ΔE; A).

Proof. Since ΔE acts trivially on A, standard homological algebra results imply that there is a k-linear isomorphism H^{∗}(ΔE; k) ⊗ A ≅ H^{∗}(ΔE; A) sending ζ ⊗ a to ι_{E}^{A}(ζ)a, where ι_{E}^{A} is as in 2.4 and where a ∈ A is viewed as element in H^{0}(ΔE; A), whence the first isomorphism. Applying this isomorphism to subgroups of E and observing that these commute with transfer yields the second isomorphism A^{op} ⊗ H̄^{∗}(ΔE; k) ≅ H̄^{∗}(ΔE; A), which concludes the proof.

Proof of Proposition 3.1. We construct π_{E}^{A} as follows. The canonical surjection H^{∗}(ΔE; A) → H̄^{∗}(ΔE; A) induces a graded k-algebra homomorphism

Z(H^{∗}(ΔE; A)) → Z(H̄^{∗}(ΔE; A))

We then compose this map with the isomorphisms

Z(H̄^{∗}(ΔE; A)) ≅ Z(H̄^{∗}(ΔE; A(ΔE))) ≅ Z(A(ΔE)) ⊗ H̄^{∗}(ΔE; k)

induced by the isomorphisms in the two previous lemmas. Since Z(A(ΔE)) is local, there is a unique unitary algebra homomorphism Z(A(ΔE)) → k. This map induces hence a graded algebra homomorphism

Z(A(ΔE)) ⊗ H̄^{∗}(ΔE; A) → H̄^{∗}(ΔE; A)

The composition of the above maps is then denoted by π_{E}^{A}. It follows from the previous lemma that ι_{E}^{A} ∘ π_{E}^{A} induces the canonical surjection H^{∗}(ΔE; k) → H̄^{∗}(ΔE; k). The uniqueness of π_{E}^{A} is an easy consequence of the uniqueness of the unitary homomorphism from Z(A(ΔE)) to k.

4 Fusion and cohomology

Fusion in a block algebra is induced by conjugation of Brauer pairs with elements of the underlying group. By a result of Puig [20], in a source algebra A of a block with defect group P, fusion can be identified as given by conjugation on images of subgroups of P in A with invertible elements of A, modulo a crucial technical detail: one has to consider for a subgroup Q of P not the image of Q in A but the images of Q in jAj, where j is a primitive local idempotent in A^{ΔQ}. This technicality is the background for the following result. As in the previous section, k is an algebraically closed field of prime characteristic p. Given a finite p-group P and an interior P -algebra, we denote for any subgroup E of P and any idempotent j ∈ A^{ΔE} by

δ_{(E,j)} : HH^{∗}(A) ⟶ H^{∗}(ΔE; jAj)

the composition of the “restriction” map HH^{∗}(A) → Ext_{A⊗kE}(Aj, Aj) induced by the the functor −⊗_{A}Aj followed by the canonical isomorphism Ext_{A⊗kE}(Aj, Aj) ≅ H^{∗}(ΔE; jAj) from Proposition 2.1. If j = 1 we write δ_{E} instead of δ_{(E,1)}. By Lemma 2.5, the image of δ_{(E,j)} is contained in Z(H^{∗}(ΔE; jAj)), and we abusively denote the induced map from HH^{∗}(A) to Z(H^{∗}(ΔE; jAj)) again by δ_{(E,j)}.

Proposition 4.1. Let P be a finite p-group and A a finite-dimensional interior P -algebra over k such that A is projective as left and right kP -module. Let E, E′ be subgroups of P and j ∈ A^{ΔE}, j′ ∈ A^{ΔE′} be idempotents. Suppose that there is a group isomorphism φ : E ≅ E′ and an invertible element c in A such that for all e ∈ E we have cejc^{−1} = φ(e)j′. Then the algebra isomorphism α : jAj ≅ j′Aj′ sending a ∈ jAj to cac^{−1} ∈ j′Aj′ induces an isomorphism of graded k-algebras α^{∗} : H^{∗}(ΔE′; j′Aj′) ≅ H^{∗}(ΔE; jAj) making the following diagram of graded k-algebras commutative:

[figure: HH^{∗}(A); δ_{(E′,j′)}; H^{∗}(ΔE′; j′Aj′); α^{∗}; H^{∗}(ΔE′; k); ι_{E′}^{j′Aj′}; _{res_{φ}}; HH^{∗}(A); δ_{(E,j)}; H^{∗}(ΔE; jAj); H^{∗}(ΔE; k); ι_{E}^{jAj}]

Proof. The commutativity of the right square is a trivial consequence of the fact that α is unitary. The hypotheses imply that the isomorphism α sends the image of E in jAj to the image of E′ in j′Aj′. The commutativity of the left square in the above diagram expresses merely the fact that the elements in HH^{∗}(A) are “invariant under conjugation” by c because they are represented by chain maps of A-A-bimodules. Here are the rather tedious details. Let V, V′ be projective resolutions of k as kΔE-module, kΔE′-module, respectively. Denote by Δφ : ΔE ≅ ΔE′ the isomorphism induced by φ and by _{Δφ}(V′) the chain complex of ΔE-modules obtained from restricting V′ along Δφ. Then _{Δφ}(V′) is also a projective resolution of the trivial kΔE-module, hence there is a unique (up to homotopy) homotopy equivalence of chain complexes of kΔE-modules ν : V ≅ _{Δφ}(V′). Since jc^{−1} = c^{−1}j′, right multiplication by c^{−1} induces an A-homomorphism γ : Aj ≅ Aj′. Since jec^{−1} = c^{−1}j′φ(e) for all e ∈ E, the map γ is more precisely an isomorphism of A-kE-bimodules Aj ≅ (Aj′)_{φ}, where as before the right kE-module structure of (Aj′)_{φ} is obtained from restricting the right kE′-module structure of Aj′ via the group homomorphism φ : E → E′. Let X be a projective resolution of A as A-A-bimodule. Then Xj, Xj′ are projective resolutions of Aj, Aj′ as an A-kE-bimodule, A-kE′-bimodule, respectively, because A is projective as a right kEmodule and as a right kE′-module. Right multiplication by c^{−1} on Xj yields an isomorphism γ̂ : Xj ≅ (Xj′)_{φ} of chain complexes of A-kE-bimodules which lifts the isomorphism γ. Let n ≥ 0 be an integer and let ζ ∈ HH^{n}(A) be represented by a chain map of complexes of A-A-bimodules (denoted by the same letter) ζ : X → A[n]. Clearly ζ sends Xj to Aj[n] and Xj′ to Aj′[n]; thus the images of ζ in Ext_{A⊗kE}^{n}(Aj, Aj) and Ext_{A⊗kG′}^{n}(Aj′, Aj′) are obtained from restricting ζ to Xj and Xj′, respectively. The maps γ, γ̂ induce an isomorphism

γ^{∗} : Ext_{A⊗kE′}^{∗}(Aj′, Aj′) ≅ Ext_{A⊗kE}^{∗}(Aj, Aj)

given by γ^{∗}(ξ) = γ[n]^{−1} ∘ ξ ∘ γ̂, for any ξ ∈ Ext_{A⊗kE′}^{n}(Aj′, Aj′), represented by a chain map ξ : Xj′ → Aj′[n]. Explicitly, for x ∈ Xj we have γ^{∗}(ξ)(x) = ξ(xc^{−1})c. In particular, if ξ is the image of an element ζ ∈ HH^{n}(A) then γ^{∗}(ξ)(x) = ζ(xc^{−1})c = ζ(x), which shows that the following diagram of graded k-algebras is commutative:

[figure: HH^{∗}(A); δ_{(E′,j′)}; Ext_{A⊗kE′}^{∗}(Aj′, Aj′); γ^{∗}; HH^{∗}(A); δ_{(E,j)}; Ext_{A⊗kE}^{∗}(Aj, Aj)]

Set U = (Aj ⊗ kE) ⊗_{kΔE} V and U′ = (Aj ⊗ kE′) ⊗_{kΔE′} V′; these are projective resolutions of (Aj ⊗ kE) ⊗_{kΔE} k ≅ Aj and of (Aj′ ⊗ kE′) ⊗_{kΔE′} k ≅ Aj′, hence uniquely homotopy equivalent to Xj and Xj′ as complexes of A ⊗ kE-modules and A ⊗ kE′-modules, respectively. Through the above isomorphisms, the isomorphism γ : Aj ≅ Aj′ gets identified with the isomorphism

(γ ⊗ φ) ⊗ Id : (Aj ⊗ kE) ⊗_{kΔE} k ≅ (Aj′ ⊗ kE′) ⊗_{kΔE′} k

This isomorphism lifts in the obvious way to a homotopy equivalence

γ̃ = (γ ⊗ φ) ⊗ ν : U ≃ (U′)_{φ}

of complexes of A-kE-bimodules. Thus both γ̂, γ̃ are lifts of γ, and hence make the diagram

\xymatrix{Xj \ar[r]^{≃} \ar[d]_{γ̂} & U \ar[d]^{γ̃} \\ (Xj′)_{φ} \ar[r]_{≃} & (U′)_{φ}}

homotopy commutative, where the horizontal homotopy equivalences lift the identity maps on Aj, Aj′, respectively. Thus, using the projective resolutions U, U′ instead of Xj, Xj′, it follows that the isomorphism γ^{∗} is given by

γ(ξ) = γ[n]^{−1} ∘ ξ ∘ γ̃

for n ≥ 0 and ξ ∈ Ext_{A⊗kE′}^{n}(Aj′, Aj′), represented by a chain map ξ : U′ → Aj′[n]. We have to compare the images of ξ and γ^{∗}(ξ) in H^{n}(ΔE′; j′Aj′) and H^{n}(ΔE; jAj), respectively. Through the standard adjunction, the image of ξ in H^{n}(ΔE′; j′Aj′) is represented by the map ξ̄: V′ → j′Aj′[n] given by ξ̄(v′) = ξ((j′ ⊗ 1) ⊗ v′), where v′ ∈ V′. Similarly, the image of γ^{∗}(ξ) is the chain map γ̄^{∗}(ξ) : V → jAj[n] given by γ̄(ξ)(v) = γ^{∗}(ξ)((j ⊗ 1) ⊗ v) = (γ[n]^{−1} ∘ ξ ∘ γ̃)((j ⊗ 1) ⊗ v) = (γ[n]^{−1} ∘ ξ)((jc^{−1} ⊗ 1) ⊗ ν(v)) = ξ(c^{−1}j′ ⊗ 1) ⊗ ν(v))c = c^{−1} · (ξ((j′ ⊗ 1) ⊗ ν(v))) · c, where v ∈ V. In other words, we have a commutative diagram

\xymatrix{V \ar[r]^{ν} \ar[d]_{γ̄^{∗}(ξ)} & V′ \ar[d]^{ξ̄} \\ jAj[n] \ar[r]_{α[n]} & j′Aj′[n]}

This shows that the canonical isomorphisms from Proposition 2.1 make the following diagram commutative:

\xymatrix{Ext_{A⊗kE′}^{∗}(Aj′, Aj′) \ar[r]^{≅} \ar[d]_{γ^{∗}} & H^{∗}(ΔE′; j′Aj′) \ar[d]^{α^{∗}} \\ Ext_{A⊗kE}^{∗}(Aj, Aj) \ar[r]_{≅} & H^{∗}(ΔE; jAj)}

Together with the first commutative diagram this completes the proof.

In order to relate the maps δ_{(E,j)}, δ_{(E,j′)} for the same group E but different idempotents j, j′ in A^{ΔE}, we will need the following observation.

Proposition 4.2. Let P be a finite p-group and A a finite-dimensional interior P -algebra over k. Let E be a subgroup of P and j an idempotent in A^{ΔE}. The k-linear map A → jAj sending a ∈ A to jaj is a split surjective homomorphism of kΔE-modules and induces a split surjective graded k-linear map π : H^{∗}(ΔE; A) → H^{∗}(ΔE; jAj). The restriction of π to Z(H^{∗}(ΔE; A)) induces a graded k-algebra homomorphism, still denoted π : Z(H^{∗}(ΔE; A)) → Z(H^{∗}(ΔE; jAj)) making the following diagram of graded k-algebras commutative:

[figure: HH^{∗}(A); δ_{E}; Z(H^{∗}(ΔE; A)); π; H^{∗}(ΔE; k); ι_{E}^{A}; HH^{∗}(A); δ_{(E,j)}; Z(H^{∗}(ΔE; jAj)); H^{∗}(ΔE; k); ι_{E}^{jAj}]

Proof. The commutativity of the right square is an immediate consequence of the fact that π sends 1_{A} to j. For commutativity of the left square we will use the elementary ring theoretic fact that multiplication by an idempotent e in a ring R induces a ring homomorphism Z(R) → Z(R)e ⊆ Z(eRe), and hence Z(R) ⊆ Z(eRe) × Z((1 − e)R(1 − e)). The k-linear map A → jAj sending a ∈ A to jaj is a kΔE-homomorphism because j belongs to A^{ΔE}, and it has the inclusion jAj → A as section, hence induces indeed a split surjective graded k-linear map π : H^{∗}(ΔE; A) → H^{∗}(ΔE; jAj). We have H^{0}(ΔE; A) = (A^{ΔE})^{op}, and hence j can be identified with an idempotent in H^{0}(ΔE; A). One easily checks that the map π is induced by left and right multiplication with this idempotent, hence induces an algebra homomorphism Z(H^{∗}(ΔE; A)) → Z(ΔE; jAj)) by the remarks at the beginning of the proof extended to centers of graded algebras in the obvious way; it is a trivial verification that π makes the diagram in the statement commutative.

The last statement of this section is about the compatibility of the maps δ_{E} with restriction to subgroups:

Proposition 4.3. Let P be a finite p-group and A a finite-dimensional interior P -algebra over k. Let E, F be subgroups of P such that F ≤ E. Restriction from ΔE to ΔF induces a graded kalgebra homomorphism res_{ΔF}^{ΔE} : H^{∗}(ΔE; A) → H^{∗}(ΔF; A) making the following diagram of graded k-algebras commutative:

[figure: HH^{∗}(A); δ_{E}; H^{∗}(ΔE; A); _{res{ΔE \atop ΔF}}; H^{∗}(ΔE; k); ι_{E}^{A}; _{res{ΔE \atop ΔF}}; HH^{∗}(A); δ_{F}; H^{∗}(ΔF; A); H^{∗}(ΔF; k); ι_{F}^{A}]

Proof. Trivial verification, based on the fact that restrition is transitive and functorial.

5 Proof of Theorem 1.7

We start with the construction of the homomorphism σ in 1.7; the required block theoretic background material can be found in various sources such as [26], [8], or also [16]. For the rest of this section we use the following notation. Let G be a finite group and b a block of kG. The corresponding block algebra is denoted B = kGb. Let (P, e_{P}) be a maximal (G, b)-Brauer pair and i a primitive idempotent in (kGb)^{ΔP} such that Br_{ΔP} (i)e_{P} = Br_{ΔP} (i) ≠ 0; that is, i is a source idempotent of b and the interior P -algebra A = ikGi is a source algebra of b. The algebras A and B are Morita equivalent via the bimodules kGi and ikG. By [1, 3.4], for any subgroup Q of P there is a unique block e_{Q} of kC_{G}(Q) such that (Q, e_{Q}) ≤ (P, e_{P}), and by [4, 1.8], the block e_{Q} is the unique block of kC_{G}(Q) whose block algebra contains Br_{ΔQ}(i), or equivalently, Br_{ΔQ}(i)e_{Q} = Br_{ΔQ}(i). The fusion system on P determined by A has as morphisms all group homomorphisms Q → R induced by conjugation with an element x ∈ G satisfying ^{x}(Q, e_{Q}) ≤ (R, e_{R}), for any two subgroups Q, R of P. As a consequence of [4, Theorem 1.8], for any subgroup Q of P the algebra A(ΔQ) is indecomposable, and hence Z(A(ΔQ)) is local. If Q is a fully F-centralised subgroup of P one can be slightly more precise: in that case C_{P} (Q) is a defect group of the block e_{Q} of kC_{G}(Q), and the algebras kC_{G}(Q)e_{Q} and A(ΔQ) are Morita equivalent (see for instance [16, 4.6]), which implies that Z(A(ΔQ)) ≅ Z(kC_{G}(Q)e_{Q}). The source algebra A has a P -P -stable k-basis with at least one ΔP -fixpoint (because A(ΔP) ≠ {0}), and hence we can define for any elementary abelian subgroup E of P, a graded k-algebra homomorphism

σ_{E} : HH^{∗}(A) ⟶ H^{∗}(E; k)

as composition

\xymatrix{HH^{∗}(A) \ar[r]^{δ_{E}^{A}} & Z(H^{∗}(ΔE; A)) \ar[r]^{π_{E}^{A}} & H^{∗}(ΔE; k) \ar[r]^{≅} & H^{∗}(E; k)}

where δ_{E}^{A} is as at the beginning of section 4, π_{E}^{A} is as in 3.1, and where the isomorphism H^{∗}(ΔE; k) ≅ H^{∗}(E; k) is induced by the obvious identification ΔE ≅ E. We then define

σ : HH^{∗}(A) ⟶ ∏_{E∈E} H^{∗}(E; k)

as product of the homomorphisms σ_{E}, the product taken over the full subcategory E of F consisting of all elementary abelian subgroups of P. By the definition of block cohomology we have H^{∗}(B) = lim_{F}(H^{∗}(Q; k)), which we identify to its image of F-stable elements in H^{∗}(P; k). The algebra homomorphism ρ : H^{∗}(B) → lim_{E}(H^{∗}(E; k)) in the statement of 1.7 is induced by the product of the maps ρ_{E} : H^{∗}(B) → H^{∗}(E; k), obtained from restricting res_{E}^{P} to H^{∗}(B), with E running over E. By Quillen’s stratification, extended to blocks in [13, 3.5, 4.2] and to arbitrary fusion systems in [3, 5.1, 5.2], the map ρ induces an inseparable isogeny. We identify lim_{E}(H^{∗}(E; k)) to its canonical image in ∏_{E∈E} H^{∗}(E; k). For any subgroup E of P, denote by

ρ_{ΔE} : H^{∗}(B) ⟶ H^{∗}(ΔE; k)

the composition of the inclusion H^{∗}(B) ⊆ H^{∗}(P; k) followed by the canonical identification H^{∗}(P; k) ≅ H^{∗}(ΔP; k), followed by the restriction res_{ΔE}^{ΔP} : H^{∗}(ΔP; k) → H^{∗}(ΔE; k). Equivalently, ρ_{ΔE} is equal to the composition of ρ_{E} and the canonical identification H^{∗}(E; k) ≅ H^{∗}(ΔE; k). We need to show that σ extends ρ, that ker(σ) is nilpotent and that Im(σ) is contained in lim_{E}(H^{∗}(E; k)). We start with a Lemma describing the canonical map τ : H^{∗}(B) → HH^{∗}(A).

Lemma 5.1. Let E be a subgroup of P. The following diagram of graded k-algebras is commutative:

\xymatrix{H^{∗}(B) \ar[r]^{ρ_{ΔE}} \ar[d]_{τ} & H^{∗}(ΔE; k) \ar[d]^{ι_{E}^{A}} \\ HH^{∗}(A) \ar[r]_{δ_{E}^{A}} & Z(H^{∗}(ΔE; A))}

Proof. We prove this first for E = P. The map τ is defined in terms of maps induced by diagonal induction from ΔP to P ×P followed by the normalised transfer map Tr_{A_{P}} : HH^{∗}(kP) → HH^{∗}(A) determined by A viewed as A-kP -bimodule (denoted by A_{P}). We briefly recall the definition and relevant properties of Tr_{A_{P}} from [10, §3]. We have algebra homomorphisms HH^{∗}(A) → Ext_{A⊗kP}^{∗} (A, A) and HH^{∗}(kP) → Ext_{A⊗kP}^{∗} (A, A) induced by the restriction and induction A⊗_{kP} −, respectively. These homomorphisms are injective; indeed, this is an easy consequence of the wellknown fact that A is isomorphic to a direct summand of A⊗_{kP} A as A-A-bimodule; see for instance [16, 4.2]. Following [10, 3.1.(iii)], an element ξ ∈ HH^{n}(kP) is called A_{P} -stable if there is an element ζ ∈ HH^{n}(A) such that ζ ⊗ Id_{A_{P}} = Id_{A_{P}} ⊗ ξ in Ext_{A⊗kP}^{n} (A, A). Moreover, in that case we have ζ = Tr_{A_{P}} (ξ); one can take this as definition of Tr_{A_{P}} in this case. Thus the subalgebra HH_{A_{P}}^{∗} (kP) of A_{P} -stable elements in HH^{∗}(kP) yields a pullback diagram of the form

\xymatrix{HH_{A_{P}}^{∗} (kP) \ar[d]^{_{Tr_{A_{P}}}} \ar[r] & HH^{∗}(kP) \ar[d]_{A⊗_{kP} −} \\ HH^{∗}(A) \ar[r] & Ext_{A⊗kP}^{∗} (A, A)}

The map H^{∗}(P; k) → HH^{∗}(kP) given by “diagonal” induction from P ≅ ΔP to P × P, composed with the map HH^{∗}(kP) → Ext_{A⊗kP} (A, A) induced by A ⊗_{kP} − is equal to ι_{P}^{A}, modulo the identification H^{∗}(P; k) ≅ H^{∗}(ΔP, k). Moreover, by [10, 5.4], this map sends H^{∗}(B) to HH_{A_{P}}^{∗} (kP), and composing this with the normalised transfer map T_{A_{P}} is, by definition, the algebra homomorphism τ. As before, the diagonal induction functor from P ≅ ΔP to P × P induces a graded homomorphism H^{∗}(P; k) → HH^{∗}(kP). Using the above maps, the commutative diagram from 2.4 and the isomorphism from 2.1 we get a commutative diagram

[figure: H^{∗}(B); τ; H^{∗}(B); H^{∗}(P; k); ≅; H^{∗}(ΔP; k); ι_{P}^{A}; HH_{A_{P}}^{∗} (kP); _{Tr_{A_{P}}}; HH^{∗}(kP); HH^{∗}(A); Ext_{A⊗kP}^{∗} (A, A); ≅; HH^{∗}(ΔP; A); HH^{∗}(A); δ_{P}^{A}; H^{∗}(ΔP; A)]

Since the images of δ_{P}^{A}, ι_{P}^{A} are contained in Z(H^{∗}(ΔP; k)), the outer square of this diagram yields the commutative square of the statement in the case E = P. The general case follows from combining the outer commutative square of the previous diagram with the diagram from 4.3, together with the fact that the images of δ_{E}^{A}, ι_{E}^{A} are contained in Z(H^{∗}(ΔE; A)).

Proposition 5.2. With the notation above, we have ρ = σ ∘ τ.

Proof. Let E be a subgroup of P. Lemma 5.1 yields a commutative diagram

[figure: H^{∗}(B); ρ_{ΔE}; H^{∗}(ΔE; k); ι_{E}^{A}; H^{∗}(ΔE; k); H^{∗}(B); τ; HH^{∗}(A); δ_{E}^{A}; Z(H^{∗}(ΔE; A)); π_{E}^{A}; H^{∗}(ΔE; k)]

Modulo identifying ΔE and E, the first row of the diagram above combined yields ρ_{E}, and the second row yields σ_{E} ∘ τ, whence the result.

Proposition 5.3. With the notation above, the kernel of σ is nilpotent.

Proof. Let ζ ∈ HH^{∗}(A) be an element in the kernel of σ. Since HH^{∗}(A) ≅ Z(A) is local and since HH^{∗}(A) is graded commutative, we may assume that σ is homogeneous of positive degree. For any elementary abelian subgroup E of P the kernel of the map

Z(H̄^{∗}(ΔE; A)) ≅ Z(A(ΔE)) ⊗ H^{∗}(ΔE; k) → H^{∗}(ΔE; k)

is nilpotent because Z(A(ΔE)) is local. Thus, after replacing ζ by some power, we may assume that the image ζ̄_{E} of ζ in H̄^{∗}(ΔE; A) is zero. Since the restriction map HH^{∗}(A) → Ext_{A⊗kP}^{∗} (A, A) is injective, in order to show that ζ is nilpotent it suffices, by Siegel’s result in [19, 5.3], to show that the image ζ_{E} of ζ in H^{∗}(ΔE; A) is nilpotent, for any elementary abelian subgroup E of P. We show this by induction over the order of E, following the arguments in the proof of [19, 2.5]. For E = 1 there is nothing to prove. Assume that E is nontrivial. After replacing ζ by some power, we may assume that ζ_{F} = 0 for any proper subgroup F of E. For a ≥ 1 denote by S(E, a) the set of subgroups of index p^{a} in E. We show inductively that

(ζ_{E})^{p^{a−1}} ∈ ∑_{F ∈S(E,a)} tr_{F}^{E}(H^{∗}(ΔF; A))

for any positive integer a. For a = 1 this follows from the fact that ζ̄_{E} = 0. Suppose that a ≥ 1 and that we can write

(ζ_{E})^{p^{a−1}} = ∑_{F ∈S(E,a)} tr_{F}^{E}(ξ_{F})

for suitable ξ_{F} ∈ H^{∗}(ΔF; A). Then the usual adjunctions yield

(ζ_{E})^{2} = ∑_{F,F′∈S(E,a)} tr_{F}^{E}(ξ_{F})tr_{F′}^{E}(ξ_{F′}) = ∑_{F,F′∈S(E,a)} tr_{F}^{E}(ξ_{F} · Res_{F}^{E}(tr_{F′}^{E}(ξ_{F′})))

For any F ∈ S(E, a) we have

0 = ζ_{F} = Res_{F}^{E}(ζ_{E}) = ∑_{F′∈S(E,a)} Res_{F}^{E}(tr_{F′}^{E}(ξ_{F′}))

For F′ ≠ F the Mackey formula implies Res_{F}^{E}(tr_{F′}^{E}(ξ_{F′})) ∈ ∑_{F′′<F} tr_{F′′}^{F}(H^{∗}(ΔF′′; A)). Since ζ_{F} = 0, this implies that also Res_{F}^{E}(tr_{F}^{E}(ξ_{F})) ∈ ∑_{F′′<F} tr_{F′′}^{F}(H^{∗}(ΔF′′; A)). Thus (ζ_{E})^{2} ∈ ∑_{F ∈S(E,a+1)} tr_{F}^{E}(H^{∗}(ΔF; A)). This induction comes to a halt when p^{a} is the order of E, showing that (ζ_{E})^{p^{a}} is a trace from the trivial subgroup 1, hence zero.

In order to show that the image of σ is contained in lim_{E}(H^{∗}(E; k)) we use 4.2 to give an alternative description of the map σ_{E} : HH^{∗}(A) → H^{∗}(E; k), where E is an elementary abelian subgroup of P. If j is a primitive idempotent in A^{ΔE} satisfying Br_{ΔE}(j) ≠ 0, we define a map

σ_{(E,j)} : HH^{∗}(A) ⟶ H^{∗}(E; k)

essentially as σ_{E}, but with A replaced by jAj. Explicitly, σ_{(E,j)} is the composition of δ_{(E,j)}^{A}, followed by π_{E}^{jAj}, and finally composed with the obvious isomorphism induced by the identification ΔE ≅ E:

\xymatrix{HH^{∗}(A) \ar[r]^{δ_{(E,j)}^{A}} & H^{∗}(ΔE; jAj) \ar[r]^{π_{E}^{jAj}} & H^{∗}(ΔE; k) \ar[r]^{≅} & H^{∗}(E; k)}

Note that this makes sense: the map π_{E}^{jAj} is well-defined as (jAj)(ΔE) is local.

Lemma 5.4. Let E be an elementary abelian subgroup of P and let j be a primitive idempotent in A^{ΔE} satisfying Br_{ΔE}(j) ≠ 0. Then σ_{(E,j)} = σ_{E}; in particular, σ_{(E,j)} does not depend on the local point of E on A to which j belongs. More precisely, we have a commutative diagram of graded algebras

[figure: HH^{∗}(A); δ_{E}^{A}; Z(H^{∗}(ΔE; A)); π_{E}^{A}; H^{∗}(ΔE; k); HH^{∗}(A); δ_{E}^{jAj}; Z(H^{∗}(ΔE; jAj)); π_{E}^{jAj}; H^{∗}(ΔE; k)]

where the vertical map is induced by the map A → jAj given by left and right multiplication with j.

Proof. The commutativity of the left square follows from the left square in 4.2, and the commutativity of the right square follows from the right square in 4.2, together with the fact that π_{E}^{A} is a retraction for ι_{E}^{A} (and similarly for jAj instead of A).

Proposition 5.5. With the notation above, we have Im(σ) ⊆ lim_{E}(H^{∗}(E; k)).

Proof. We have to show that if φ : E → E′ is a morphism in F between two elementary abelian subgroups E, E′ of P then σ_{E} = res_{φ} ∘σ_{E′}. It suffices to do this in the cases where φ is an inclusion or an isomorphism. For inclusions this is a trivial consequence of 4.3, combined with the fact that π_{E}^{A} is a retraction for ι_{E}^{A}, by 3.1. We may assume that φ is an isomorphism. Puig’s characterisation of fusion in source algebras in [20, 2.12, 3.1], in the form as presented in [12, 7.5] implies that there are primitive idempotents j ∈ A^{ΔE}, j′ ∈ A^{ΔE′} satisfying Br_{ΔE}(j) ≠ 0, Br_{ΔE′}(j′) ≠ 0 and there is an element c ∈ A^{×} such that ^{c}(uj) = φ(u)j′ for all u ∈ E. This is the situation dealt with in 4.1, implying that σ_{(E,j)} = res_{φ} ∘ σ_{(E′,j′)}, where we use again 3.1. But then the previous lemma implies that σ_{E} = res_{φ} ∘ σ_{E′}, whence the result.

Proof of Theorem 1.7. The three propositions 5.2, 5.3, and 5.5, together with the fact that ρ induces an inseparable isogeny complete the proof of Theorem 1.7.

The proof of Corollary 1.2 uses the elementary fact that an inseparable isogeny induced by a split monomorphism is an isomorphism of varieties:

Lemma 5.6. Let R, S be rings and ρ : R → S, σ : S → R be ring homomorphisms such that σ ∘ ρ = Id_{R} and such that for any s ∈ S there is a positive integer n satisfying s^{n} ∈ Im(ρ). Then every element in ker(σ) is nilpotent.

Proof. Let s ∈ ker(σ). By the assumptions there is r ∈ R and n > 0 such that s^{n} = ρ(r). Applying σ yields 0 = σ(s)^{n} = σ(ρ(r)) = r, hence s^{n} = 0.

Proof of Corollary 1.2. If B is the principal block then τ is a split monomorphism with retraction induced by the functor − ⊗_{B} k. Thus Lemma 5.6 and Theorem 1.7 imply that this retraction has a kernel consisting of nilpotent elements. Since HH^{∗}(B) is Noetherian graded commutative it follows that the kernel of this retraction is a nilpotent ideal.

Remark 5.7. As pointed out earlier, this raises the question whether τ : H^{∗}(B) → HH^{∗}(A) is a split monomorphism as homomorphism of graded k-algebras for arbitrary blocks. If that were the case, the argument in the proof of 1.2 would immediately yield a proof of Theorem 1.1.

6 Proof of Theorem 1.1

We use the notation from the previous section; in particular, A is a source algebra of the block b of kG with defect group P. We denote by [A, A] the k-subspace of A spanned by the set of additive commutators [a, a′] = aa′ − a′a, where a, a′ ∈ A. The following elementary observation is an immediate consequence of well-known facts (which play a role in the context of bilinear forms on group algebras in [5]; see e.g. [26, §46]) and provides a weak linear analogue of the augmentation in principal blocks; we include a proof for the convenience of the reader.

Lemma 6.1. There is a k-linear map μ : A → k such that μ(1_{A}) = 1_{k} and such that J(A)+[A, A] ⊆ ker(μ). Any such map μ is then a homomorphism of kΔP -modules, and has, for any subgroup E of P, the following properties:

(i) There is a unique linear map μ̄_{E} : A(ΔE) → k satisfying μ|_{A^{ΔE}} = μ̄_{E} ∘ Br_{ΔE} : A^{ΔE} → k. (ii) We have μ̄_{E}(1_{A(ΔE)}) = 1_{k} and J(A(ΔE)) + [A(ΔE), A(ΔE)] ⊆ ker(μ̄_{E}). (iii) The restriction of μ̄_{E} to Z(A(ΔE)) is equal to the unique unitary algebra homomorphism Z(A(ΔE)) → k.

Proof. By [21, 14.6] or [26, (44.9)] the source algebra A has a simple module V of dimension prime to p. Define μ as composition of the canonical map A → End_{k}(V) followed by the map \frac{1}{dim_{k}(V)}tr_{V} : End_{k}(V) → k, where tr_{V} is the trace map. Then clearly μ(1_{A}) = 1_{k}. The kernel of the map A → End_{k}(V) contains J(A), hence so does the kernel of μ. Since tr_{V} is a symmetric linear map, ker(μ) contains [A, A]. Denoting by σ : P → A^{×} the structural homomorphism, for any u ∈ P and any a ∈ A we have ^{u}a − a = [σ(u)a, σ(u^{−1})] ∈ [A, A] ⊆ ker(μ), which implies that μ is a homomorphism of kΔP -modules. We observe next that the space J(A) + [A, A] contains every nilpotent element in A. Indeed, the image of [A, A] in the direct product of matrix algebras A/J(A) contains the product of all trace zero matrices, hence contains all nilpotent elements in A/J(A), from which this observation follows. Thus, in particular, for any subgroup E of P we have J(A^{ΔE}) ⊆ ker(μ) ∩ A^{ΔE}. Also, by [9, 2.5], we have ker(Br_{ΔE}) ⊆ [A, A] ⊆ ker(μ). Thus there is a unique linear map μ̄_{E} : A(ΔE) → k satisfying (i). Note that μ̄_{E}(1_{A(ΔE)}) = 1_{k}. The Brauer homomorphism Br_{ΔE} from A^{ΔE} to A(ΔE) is surjective, hence maps J(A^{ΔE}) onto J(A(ΔE)) and [A^{ΔE}, A^{ΔE}] onto [A(ΔE), A(ΔE)]. Thus ker(μ̄_{E}) contains J(A(ΔE)) + [A(ΔE), A(ΔE)], which shows (ii). This also shows that ker(μ̄_{E}) contains every nilpotent element in A(ΔE) by the above arguments. Thus ker(μ_{E}) contains J(Z(A(ΔE)). Since μ̄_{E} sends 1_{A(ΔE)} to 1_{k} and since Z(A(ΔE)) is local it follows that the restriction of μ_{E} to Z(A(ΔE)) is equal to the unique unitary algebra homomorphism from Z(A(ΔE)) to k, which completes the proof of (iii).

Fix a k-linear map μ : A → k satisfying μ(1_{A}) = 1_{k} and J(A) + [A, A] ⊆ ker(μ). Since the map μ is a homomorphism of kΔP -modules, it induces for any subgroup E of P a graded k-linear map

ν_{E} : H^{∗}(ΔE; A) → H^{∗}(ΔE; k)

In particular, the map ν_{P} is a k-linear (but not multiplicative) retraction for the algebra homomorphism ι_{P}^{A} : H^{∗}(ΔP; k) → H^{∗}(ΔP; A) induced by the unitary map k → A (cf. 2.4), and the purpose of what follows is to show that ν_{P} can be used to “lift” σ to a graded k-linear map HH^{∗}(A) → H^{∗}(B) which is a retraction of the map τ : H^{∗}(B) → HH^{∗}(A). We first lift σ to a map HH^{∗}(A) → H^{∗}(P; k) and then apply the projection H^{∗}(P; k) → H^{∗}(B) determined by a P -P -biset satisfying the fusion stability conditions in [3, 5.5]. As before, for any elementary abelian subgroup E of P, we denote by ι_{E}^{A} : H^{∗}(ΔE; k) → Z(H^{∗}(ΔE; A)) the graded map induced by the unitary map k → A, we denote by π_{E}^{A} : Z(H^{∗}(ΔE; A)) → H^{∗}(ΔE; k) the canonical retraction of ι_{E}^{A} from Proposition 3.1, and we denote by δ_{E}^{A} : HH^{∗}(A) → H^{∗}(ΔE; A) the composition of the map induced by restriction from A ⊗ A^{op} to A ⊗ kE, followed by the canonical isomorphism in Proposition 2.1, applied in the case j = 1_{A}.

Lemma 6.2. Let E be an elementary abelian subgroup of P. The restriction of ν_{E} to Z(H^{∗}(ΔE; A)) is equal to π_{E}^{A} : Z(H^{∗}(ΔE; A)) → H^{∗}(ΔE; k).

Proof. Since μ(1_{A}) = 1_{k} we have ν_{E} ∘ ι_{E}^{A} = Id_{H^{∗}(ΔE;k)}. In conjunction with 3.2, the commutative diagram (as diagram of of kΔE-modules)

\xymatrix{A \ar[ddr]^{μ} \\ A^{ΔE} \ar[u] \ar[d]_{_{Br{A \atop ΔE}}} \ar[dr] \\ A(ΔE) \ar[r]_{μ̄_{E}} & k}

yields, upon taking cohomology, a commutative diagram

\xymatrix{ & H^{∗}(ΔE; A) \ar[ddr]^{ν_{E}} \ar[ddl] \\  & H^{∗}(ΔE; A^{ΔE}) \ar[dr] \ar[d]^{β} \ar[u]_{ϵ} \\ H̄^{∗}(ΔE; A) \ar[r]^{≅}_{β̄} & H^{∗}(ΔE; A(ΔE)) \ar[r]_{ν̄_{E}} & H^{∗}(ΔE; k)}

where ϵ, β, β̄ are as in 3.2. In particular, ν_{E} factors through the canonical map Z(H^{∗}(ΔE; A)) → Z(H̄^{∗}(ΔE; A)) ≅ Z(A(ΔE))) ⊗ H^{∗}(ΔE; k). Since ν̄_{E} is induced by μ_{E}, it follows from 6.1 (iii) that the kernel of ν̄_{E} contains J(Z(A(ΔE))) ⊗ H^{∗}(ΔE; k). Thus the restriction of ν̄_{E} to Z(H^{∗}(ΔE; A(ΔE))) extends the unique unitary algebra homomorphism Z(A(ΔE)) → k, hence is equal to the unique unitary algebra homomorphism from Z(H^{∗}(ΔE; A(ΔE)) to H^{∗}(ΔE; k) which is a retraction for ι_{E}^{A(ΔE)}. This shows that the restriction of μ_{E} to Z(H^{∗}(ΔE; A)) is an algebra homomorphism, hence equal to π_{E}^{A} by the characterisation of π_{E}^{A} in 3.1.

Lemma 6.3. For any elementary abelian subgroup E of P, the following diagram is commutative:

[figure: HH^{∗}(A); δ_{P}^{A}; H^{∗}(ΔP; A); _{res{ΔP \atop ΔE}}; ν_{P}; H^{∗}(ΔP; k); _{res{ΔP \atop ΔE}}; HH^{∗}(A); δ_{E}^{A}; H^{∗}(ΔE; A); ν_{E}; H^{∗}(ΔE; k); ≅; HH^{∗}(A); σ_{E}; H^{∗}(E; k)]

Proof. The commutativity of the two upper squares of the diagram is a consequence of the transitivity and functoriality of restriction. The lower rectangle commutes by the previous lemma.

This lemma implies that ν_{P}, precomposed with δ_{P}^{A}, lifts σ. We do not know whether Im(ν_{P}) is contained in H^{∗}(B) = lim_{F}(H^{∗}(Q; k)) ⊆ H^{∗}(P; k), but we can force the issue by making use of a finite P -P -biset X satisfying the fusion stability properties as in [3, 5.5]. The associated transfer map

tr_{X} : H^{∗}(P; k) → H^{∗}(B)

is a graded k-linear projection of H^{∗}(P; k) onto H^{∗}(B); see for instance [14, 3.2], or [23].

Lemma 6.4. Denote by res_{E}^{P} : H^{∗}(P; k) → ∏_{E∈E} H^{∗}(E; k) the product of the restriction maps res_{E}^{P}, with E ∈ E. Identify lim_{E}(H^{∗}(E; k)) with its canonical image in ∏_{E∈E} H^{∗}(E; k). Let ζ ∈ H^{∗}(P; k) such that res_{E}^{P} (ζ) ∈ lim_{E}(H^{∗}(E; k)). Then res_{E}^{P} (ζ) = res_{E}^{P} (tr_{X}(ζ)).

Proof. Let E be an elementary abelian subgroup of P. The restriction of the biset X to an E-P -biset is a disjoint union of transitive E-P -bisets of the form

X = ∪_{(F,φ)} E ×_{(F,φ)} P

where (F, φ) runs over a family of pairs consisting of a subgroup F of E and a morphism φ ∈ Hom_{F}(F, P). Here E ×_{(F,φ)} P denotes the E-P -biset obtained by taking the quotient of E × P modulo the equivalence relation (uv, w) ∼ (u, φ(v)w), where u ∈ E, v ∈ F, w ∈ P. The hypothesis on ζ implies that for any such (F, φ) we have res_{φ}(ζ) = res_{E}^{P}(ζ). Thus

res_{E}^{P}(tr_{X}(ζ)) = ∑_{(F,φ)} tr_{F}^{P} (res_{φ}(ζ)) = ∑_{(F,φ)} tr_{F}^{P} (res_{F}^{P} (ζ)) = ∑_{(F,φ)}[E : F]res_{E}^{P}(ζ)

Since ∑_{(F,φ)}[E : F] = |X|/|P | ≡ 1(mod p), the result follows.

Proof of Theorem 1.1. Denote by λ : HH^{∗}(A) → H^{∗}(B) the composition of graded k-linear maps

λ : \xymatrix{HH^{∗}(A) \ar[r]^{δ_{P}^{A}} & H^{∗}(ΔP; A) \ar[r]^{ν_{P}} & H^{∗}(ΔP; k) \ar[r]^{≅} & H^{∗}(P; k) \ar[r]^{^{tr^{X}}} & H^{∗}(B)}

As before, we denote by τ the canonical algebra homomorphism from H^{∗}(B) to HH^{∗}(A). Lemma 5.1, applied to E = P, implies that there is a commutative diagram

\xymatrix{H^{∗}(B) \ar[d]_{ρ_{ΔP}} \ar[r]^{τ} & HH^{∗}(A) \ar[rr]^{λ} \ar[d]^{δ_{P}^{A}} &  & H^{∗}(B) \\ H^{∗}(ΔP; k) \ar[r]_{ι_{P}^{A}} & Z(H^{∗}(ΔP; A)) \ar[r]_{ν_{P}} & H^{∗}(ΔP; k) \ar[r]_{≅} & H^{∗}(P; k) \ar[u]_{_{tr_{X}}}}

Since ν_{P} ∘ ι_{P}^{A} = Id_{H^{∗}(ΔP;k)} and since tr_{X} restricts to the identity on H^{∗}(B), this implies that

λ ∘ τ = Id_{H^{∗}(B)}

and hence HH^{∗}(A) = Im(τ) ⊕ ker(λ). Since, by 6.3, ν_{P} ∘ δ_{P}^{A} lifts σ, it follows from 6.4 that λ lifts σ; in other words, we have a commutative diagram

[figure: HH^{∗}(A); λ; H^{∗}(B); _{res{F \atop E}}; HH^{∗}(A); σ; lim_{E}(H^{∗}(E; k))]

where res_{E}^{F} is induced by the product of the restrictiond from P to elementary abelian subgroups of P. Thus ker(λ) ⊆ ker(σ), hence HH^{∗}(A) = Im(τ) + ker(σ). By Theorem 1.7 (or Proposition 5.3), the ideal ker(σ) is nilpotent. This completes the proof of Theorem 1.1.

Proof of Corollary 1.3. Since A and B are Morita equivalent, we may replace B by A. Let M be a finitely generated A-module, and denote by μ_{M} : HH^{∗}(A) → Ext_{A}^{∗}(M, M) the graded algebra homomorphism induced by − ⊗_{A} M. The variety X_{B}(M) is, by definition, the maximal ideal spectrum of HH^{∗}(A)/ker(μ_{M}). By [11, 4.1], the variety V_{B} is the maximal ideal spectrum of H^{∗}(B)/ker(μ_{M} ∘ τ). Modulo nilpotent elements, the graded algebras HH^{∗}(A)/ker(μ_{M}) and H^{∗}(B)/ker(μ_{M} ∘ τ) are isomorphic, by Theorem 1.1, whence the result.

Proof of Corollary 1.4. This follows from Corollary 1.3 and [11, 2.3] or [7, Theorem 1.1].

Proof of Corollary 1.5. An element in the kernel of the canonical map HH^{∗}(A) → Z^{∗}(D^{b}(mod(A))) is in particular in the kernel of the map HH^{∗}(A) → Ext_{A}^{∗}(M, M) for any finitely generated Amodule M. Thus 1.5 follows from 1.4.

Proof of Corollary 1.6. It follows from [11, 5.1] that a stable equivalence of Morita type between block algebras B, C given by a C-B-bimodule M as in the statement of 1.6 induces an isomorphism of varieties X_{B}(U) ≅ X_{C}(M ⊗_{B} U), for any finitely generated B-module U. Combined with 1.3 this yields 1.6.

References

[1] J. Alperin, M. Broué, Local Methods in Block Theory, Ann. Math. 110 (1979), 143–157.

[2] D. J. Benson, Representations and Cohomology, Vol. I, Cambridge Studies in Advanced Mathematics 30, Cambridge University Press (1990)

[3] C. Broto, R. Levi, B. Oliver, The homotopy theory of fusion systems, J. Amer. Math. Soc. 16 (2003), 779–856.

[4] M. Broué and L. Puig, Characters and Local Structure in G-Algebras, J. Algebra 63 (1980), 306–317

[5] M. Broué, G. R. Robinson, Bilinear forms on G-algebras, J. Algebra 104 (1986), 377–396.

[6] L. Evens, Cohomology of Groups, Oxford University Press (1991).

[7] H. Kawai, Varieties for modules over a block of a finite group, Osaka J. Math. 40 (2003), 327–344.

[8] R. Kessar, Introduction to Block Theory, in: “Group Representation Theory” (eds. M. Geck, D. Testerman, J. Thévenaz), EPFL Press, Lausanne (2007), 47–77.

[9] R. Kessar, M.Linckelmann, On blocks with Frobenius inertial quotient, J. Algebra 249 (2002), 127–146.

[10] M. Linckelmann, Transfer in Hochschild cohomology of blocks of finite groups, Algebras Representation Theory 2 (1999), 107–135.

[11] M. Linckelmann, Varieties in block theory, J. Algebra 215 (1999), 460–480.

[12] M. Linckelmann, On splendid derived and stable equivalences between blocks of finite groups, J. Algebra 242 (2001), 819–843.

[13] M. Linckelmann, Quillen stratification for block varieties, J. Pure Applied Algebra 172 (2002), 257–270.

[14] M. Linckelmann, A note on the Schur multiplier of a fusion system, J. Algebra 296 (2006), 402–408.

[15] M. Linckelmann, Introduction to fusion systems, in: “Group Representation Theory” (eds. M. Geck, D. Testerman, J. Thévenaz), EPFL Press, Lausanne (2007), 79–113.

[16] M. Linckelmann, Trivial source bimodule rings for blocks and p-permutation equivalences, Trans. Amer. Math. Soc. 361 (2009), 1279–1316.

[17] M. Linckelmann, On graded centers and block cohomology, preprint, to appear in Proc. Edinburgh Math. Soc. (2009)

[18] J. Pakianathan, S. Witherspoon, Hochschild cohomology and Linckelmann cohomology for blocks of finite groups, J. Pure Applied Algebra 178 (2003), 87–100.

[19] J. Pakianathan, S. Witherspoon, Quillen stratification for Hochschild cohomology of blocks, Trans. Amer. Math. Soc. 358 (2005), 2897–2916.

[20] L. Puig, Local fusion in block source algebras, J. Algebra 104 (1986), 358–369.

[21] L. Puig, Pointed groups and construction of modules, J. Algebra 116 (1988), 7–129.

[22] D. Quillen The spectrum of an equivariant cohomology ring I/II, Ann. Math. 94 (1971), 549–572, 573–602.

[23] K. Ragnarsson, Classifying spectra of saturated fusion systems, Algebr. Geom. Topol. 6 (2006), 195–252.

[24] S. F. Siegel, S. Witherspoon, The Hochschild cohomology ring of a group algebra, Proc. London Math. Soc. 79 (1999), 131–157.

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

[26] J. Thévenaz, G-Algebras and Modular Representation Theory, Oxford Science Publications, Clarendon Press, Oxford (1995).