VARIETIES IN BLOCK THEORY

Markus Linckelmann November 1997

Typeset by AMS-TEX

1 Introduction

The content of this paper is motivated by the question, whether the cohomological variety V_{G}(U), as defined by J.F. Carlson in [9], of a bounded complex U of finitely generated kG−modules belonging to a block b of kG is an invariant of this block b, where G is a finite group and k an algebraically closed field of characteristic p.

We show that this is true if b is the principal block of kG, while this is not true in general - mainly because the definition of V_{G}(U) involves the cohomology ring H^{∗}(G, k) of G, which is an invariant of the principal block of kG. However, it is “not far” from being true in general: we define in section 4 a variety V_{G,b}(U) which comes along with a a finite surjective morphism

1.1.

V_{G,b}(U) ⟶ V_{G}(U)

and which is not only an invariant of U as complex of kGb−modules, but which is even invariant under splendid stable and derived equivalences (cf. [27], [18], [19]). The map in 1.1 is shown to be an isomorphism, if b is the principal block of kG.

The major ingredients for this are the following. In [20] we define for any block b of kG with defect pointed group P_{γ} a cohomology ring H(G, b, P_{γ}) of b and show that there is an embedding into the Hochschild cohomology ring of the block algebra kGb,

1.2.

H^{∗}(G, b, P_{γ}) ⟶ HH^{∗}(kGb),

which should be thought of as a generalization of the well-known embedding H^{∗}(G, k) ⟶ HH^{∗}(kG) induced by the “diagonal induction functor” Ind_{ΔG}^{G×G}. Next, the functor − ⊗_{kGb}U induces an algebra homomorphism HH^{∗}(kGb) ⟶ Ext_{kGb}^{∗}(U, U);

thus, by composing this with the homomorphism in 1.2 we obtain an algebra homomorphism

1.3.

H^{∗}(G, b, P_{γ}) ⟶ Ext_{kGb}^{∗}(U, U).

We denote by I_{G,b}^{∗}(U) the kernel of the homomorphism 1.3 and define V_{G,b}(U) to be the maximal ideal spectrum of H^{∗}(G, b, P_{γ})/I_{G,b}^{∗}(U). The maximal ideal spectrum V_{G}(b) of H^{∗}(G, b, P_{γ}) is called the variety of the block b. These definitions are, up to unique isomorphism, independent of the choice of P_{γ}, since all defect pointed groups of the block b are G−conjugate and N_{G}(P_{γ}) acts trivially on H^{∗}(G, b, P_{γ}). We observe then that the restriction from G to the defect group P of b induces an algebra homomorphism

1.4.

H^{∗}(G, k) ⟶ H^{∗}(G, b, P_{γ})

which is an isomorphism if b is the principal block of kG and which induces a finite surjective morphism as claimed in 1.1.

In section 5 we show that if H is another finite group and c a block of kH having also P as defect group, and if X is a splendid tilting complex of kGb−kHc−bimodules (or a splendid stable equivalence of Morita type), then the functor X ⊗_{kHc} − induces

an isomorphism of varieties 1.5.

V_{G,b}(X ⊗_{kHc} V) ≅ V_{H,c}(V)

for any bounded complex V of finitely generated kHc−modules. The sections 2 and 3 contain the required material on transfer maps in the Hochschild cohomology of symmetric algebras and the theory of blocks of finite groups, respectively. Finally, section 6 is an attempt to generalize the notion of the nucleus of G introduced in [5] to arbitrary blocks of finite groups.

Notation. All algebras and rings are associative with unit element, all modules are finitely generated unitary, and, if not stated otherwise, left modules. If A, B are algebras over a commutative ring R, by an A − B−bimodule M we mean a bimodule whose left and right R−module structures coincide; that is, we may consider M as A ⊗_{R} B^{0}−module, where B^{0} is the algebra obtained by endowing B with the opposite

product. The R−dual M^{∗} = Hom_{R}(M, R) becomes then a B −A−bimodule through (b.m^{∗}.a)(m) = m^{∗}(amb) for any a ∈ A, b ∈ B, m ∈ M and m^{∗} ∈ M^{∗}. If M is an A−A−bimodule, we denote by M^{A} the subspace of A−invariant elements in M; that is, M^{A} = {m ∈ M|am = ma for any a ∈ A}. For a finite group G we consider any RG − RG−bimodule N as R(G × G)−module with (x, y) ∈ G × G acting on n ∈ N as xny^{−1} (and vice versa).

Remember that an R−algebra A is symmetric if A is finitely generated projective as R−module and A ≅ A^{∗} as A−A−bimodules. The image of 1_{A} in A^{∗} under such an isomorphism is called a symmetrizing form on A. The group algebra RG is symmetric through the isomorphism RG ≅ (RG)^{∗} mapping x ∈ G to the unique linear form on RG sending x^{−1} to 1_{R} and any other element of G to zero. The symmetrizing form corresponding to this isomorphism maps 1_{G} to 1_{R} and any nontrivial element of G to zero; we call this the canonical symmetrizing form on RG. See [20,section 6] for a short account on some formal properties of symmetric algebras that we need here.

Our notation and sign conventions when dealing with complexes are as in [20,1.2].

2 Transfer maps for symmetric algebras

We describe here the material on transfer maps in Hochschild cohomology of symmetric algebras, developed in [20], that we need in this paper. See [6], [17] for analogous concepts for Hochschild- and cyclic homology, respectively, and [14] for a transfer in cohomology of Hopf algebras.

In this section, R is a commutative ring with unit element, A, B, C are symmetric R−algebras (cf. [20, 6.3]), X, X′ are bounded complexes of A − B−bimodules whose components are projective as left and right modules, f : X ⟶ X′ a chain homomorphism and Y is a bounded complex of B − C−bimodules whose components are projective as left and right modules. We denote by P_{X} ⟶_{μ_{X}} X a projective cover of

X; that is, P_{X} is a right bounded complex of projective A − B−bimodules and μ_{X} is a quasi-isomorphism (that is, μ_{X} is a chain homomorphism inducing an isomorphism on homology).

2.1 The functors X ⊗_{B} − and X^{∗} ⊗_{A} − between the categories C(A) and C(B) of

complexes of A−modules and B−modules, respectively, are adjoint to each other (cf.

[20, section 6]). More precisely, any choice of symetrizing forms s on A and t on B (cf. [20, 6.3]) gives rise to natural isomorphisms of bifunctors

Hom_{C(A)}(X ⊗_{B} −, −) ≅ Hom_{C(B)}(−, X^{∗} ⊗_{A} −) and

Hom_{C(B)}(X^{∗} ⊗_{A} −, −) ≅ Hom_{C(A)}(−, X ⊗_{B} −),

thus determine chain homomorphisms of complexes of bimodules

ϵ_{X} : B ⟶ X^{∗} ⊗_{A} X, η_{X} : X ⊗_{B} X^{∗} ⟶ A,

ϵ_{X^{∗}} : A ⟶ X ⊗_{B} X^{∗}, η_{X^{∗}} : X^{∗} ⊗_{A} X ⟶ B

representing the units and counits of this adjunction. Since for any projective resolution P_{A} of A as A − A−bimodule the total complex X^{∗} ⊗_{A} P_{A} ⊗_{A} X is a pro-

jective resolution of X^{∗} ⊗_{A} X, the above maps lift uniquely up to homotopy to chain

homomorphisms, still denoted by the same letters,

ϵ_{X} : P_{B} ⟶ X^{∗} ⊗_{A} P_{A} ⊗_{A} X, η_{X} : X ⊗_{B} P_{B} ⊗_{B} X^{∗} ⟶ P_{A},

ϵ_{X^{∗}} : P_{A} ⟶ X ⊗_{B} P_{B} ⊗_{B} X^{∗}, η_{X^{∗}} : X^{∗} ⊗_{A} P_{A} ⊗_{A} X ⟶ P_{B}.

2.2 The transfer map associated with X (with respect to the choice of the symmetrizing forms s, t) is the graded R−linear map

t_{X} : HH^{∗}(B) ⟶ HH^{∗}(A)

mapping, for any nonnegative integer n, the homotopy class of a chain map ζ : P_{B} ⟶ P_{B}[n] to the homotopy class of the composition of chain maps

P_{A} ⟶^{ϵ_{X^{∗}}} X ⊗_{B} P_{B} ⊗_{B} X^{∗} ⟶^{Id_{X}⊗ζ⊗Id_{X^{∗}}} X ⊗_{B} P_{B}[n] ⊗_{B} X^{∗} ⟶^{η_{X}[n]} P_{A}[n]

The map t_{X} does depend on the choice of the symmetrizing forms s and t since the adjunction maps in 2.1 do depend on this choice, but one can exactly tell in which way t_{X} depends on this choice (see [20, 2.10]).

2.3 We recall from [20, 2.11 and 2.12] the following basic properties of transfer maps:

(i) t_{X⊕X′} = t_{X} + t_{X′}.

(ii) t_{X⊗_{B}Y} = t_{X} ∘ t_{Y} .

(iii) t_{X} = ∑_{n∈ℤ}(−1)^{n}t_{X_{n}}.

(iv) t_{X[n]} = (−1)^{n}t_{X}.

(v) t_{C(f)} = t_{X′} − t_{X}. (vi) If X is acyclic then t_{X} = 0. (vii) If f is a quasi-isomorphism then t_{X} = t_{X′}.

2.4 In order to study the behaviour of t_{X} with respect to the multiplicative structure in the Hochschild cohomology, we introduce the notion of stable elements. An elements [ζ] ∈ HH^{∗}(A) is called X−stable ([20, 3.1]) if there is [τ] ∈ HH^{∗}(B) such that for any nonegative integer n, the following diagram is homotopy commutative:

P_{A} ⊗_{A} X ⟶^{≃} X ⊗_{B} P_{B}
ζ_{n}⊗Id_{X} ↓  ↓^{Id_{X}⊗τ_{n}}
P_{A}[n] ⊗_{A} X ⟶_{≃} X ⊗_{B} P_{B}[n]

where ζ_{n} and τ_{n} represent the degree n component of [ζ] and [τ], respectively, and where the horizontal maps are the homotopy equivalences lifting the natural isomorphism of complexes A ⊗_{A} X ≅ X ≅ X ⊗_{B} B.

2.5 We denote by π_{X} ∈ Z(A) the image of 1_{A} under the composition of A − A−bimodule homomorphisms

A ⟶^{ϵ_{X^{∗}}} X ⊗_{B} X^{∗} ⟶^{η_{X}} A

and call π_{X} the relatively X−projective element in Z(A); note that π_{X} depends again on the choice of the symmetrizing forms s and t (cf. [20, 3.2]).

If π_{X} is invertible in Z(A), we denote by

T_{X} : HH^{∗}(B) ⟶ HH^{∗}(A)

the graded R−linear map defined by T_{X} = (π_{X})^{−1}t_{X} and call T_{X} the normalized transfer map associated with X.

2.6 The set HH_{X}^{∗}(A) of X−stable elements is a graded subalgebra in HH^{∗}(A), and if π_{X} is invertible, the normalized transfer T_{X} induces a surjective graded R−algebra homomorphism

HH_{X^{∗}}^{∗}(B) ⟶ HH_{X}^{∗}(A).

Moreover, if both π_{X} and π_{X^{∗}} are invertible, then T_{X} and T_{X^{∗}} induce mutually inverse algebra isomorphisms between HH_{X}^{∗}(A) and HH_{X^{∗}}^{∗}(B) (cf. [20, 3.6]).

2.7 Stable elements in Hochschild cohomology satisfy a “cancellation property” ([20, 3.8]): if π_{Y} is invertible in Z(B) then HH_{X⊗_{B}Y}^{∗} (A) ⊂ HH_{X}^{∗}(A) and

T_{Y} (HH_{Y^{∗}⊗_{B}X^{∗}}^{∗}(C)) ⊂ HH_{X^{∗}}^{∗}(B). Also, for any direct summand X′ of the complex X

we have HH_{X}^{∗}(A) ⊂ HH_{X′}^{∗}(A).

2.8 We have the following connection between transfer maps in Hochschild cohomology and ordinary cohomology of finite groups: if G is a finite group and ΔG = {(x, x)}_{x∈G} ⊂ G × G, the “diagonal induction functor” Ind_{ΔG}^{G×G} maps the trivial RΔG−module R to the R(G × G)−module Ind_{ΔG}^{G×G}(R), which is, when viewed as RG − RG−bimodule, isomorphic to the regular bimodule RG. Thus Ind_{ΔG}^{G×G} maps a projective resolution of R as RΔG−module to a projective resolution of RG as RG − RG−bimodule and whence induces an injective graded R−algebra homomorphism (cf. [20,4.5])

δ_{G} : H^{∗}(G, R) ⟶ HH^{∗}(RG).

Moreover, by [20, 4.8], if H is any subgroup of G, we have

Im(δ_{G}) ⊂ HH_{(RG)_{H}}^{∗} (RG),

where (RG)_{H} is the regular RG−RG−bimodule RG restricted to RH on the right. We consider RG as symmetric R−algebra with respect to the canonical symmetrizing form RG ⟶ R mapping 1_{G} to 1_{R} and any nontrivial element of G to zero. Then the following diagrams are commutative (cf. [20, 4.6, 4.7]):

H^{∗}(G, R) ⟶^{res_{H}^{G}} H^{∗}(H, R) H^{∗}(H, R) ⟶^{t_{H}^{G}} H^{∗}(G, R)
δ_{G}↓  ↓^{δ_{H}}  ^{δ_{H}} ↓  ↓^{δ_{G}}
HH^{∗}(RG) ⟶_{t_{_{H} (RG)}} HH^{∗}(RH) HH^{∗}(RH) ⟶_{t_{(RG)_{H}}} HH^{∗}(RG)

where t_{H}^{G} denotes the usual transfer map on group cohomolgy.

2.9 For any bounded complex U of A−modules and any projective resolution P_{A} ⟶_{μ_{A}} A of A as A − A−bimodule, the total complex P_{A} ⊗_{A} U, together with the

chain map μ_{A} ⊗ Id_{U}, becomes a projective resolution of U. Thus the functor − ⊗_{A} U

induces an algebra homomorphism

α_{U} : HH^{∗}(A) ⟶ Ext_{A}^{∗}(U, U)

mapping the homotopy class of a chain map ζ : P_{A} ⟶ P_{A}[n] to that of ζ ⊗ Id_{U}, where n is a nonnegative integer.

If G is a finite group and U a bounded complex of RG−modules, for any complex V of RG−modules there is a natural isomorphism of complexes of RG−modules

Ind_{ΔG}^{G×G}(V) ⊗_{RG} U ≅ V ⊗_{R} U

mapping ((x, y) ⊗ v) ⊗ u to xv ⊗ xy^{−1}u, where x, y ∈ G, u ∈ U, v ∈ V, and where the complex of R(G × G)−modules Ind_{ΔG}^{G×G}(V) is considered as complex of RG − RG−bimodules according to our conventions introduced in section 1; that is, x ∈ G acts on the left and right of m ∈ Ind_{ΔG}^{G×G}(V) by x.m = (x, 1)m and m.x = (1, x^{−1})m, respectively. This isomorphism, applied to a projective resolution P_{R} of the trivial RG−module R instead of V, implies that the composition of R−algebra homomorphisms

H^{∗}(G, R) ⟶^{δ_{G}} HH^{∗}(RG) ⟶^{α_{U}} Ext_{RG}^{∗}(U, U)

is equal to the algebra homomorphism

γ_{U} : H^{∗}(G, R) ⟶ Ext_{RG}^{∗}(U, U)

given by the functor − ⊗_{R} U.

3 Quoted results on blocks and their cohomology

We sketch here briefly some basic concepts and results from block theory. Most of the material we present here holds in more general situations, but we restrict this section to what we need in this paper. In particular, since we are mostly interested in providing techniques for dealing with varieties, our ground ring will be a field (of prime characteristic), and we leave it to the reader to check, that all statements, including the results in the sections 4 and 5 below, could be done more generally over a complete discrete valuation ring.

The Brauer homomorphism described in 3.1 below goes back to work of R. Brauer and has since then been generalized to G−algebras [8] and modules [13]. The systematical treatment of the p−local structure of blocks of finite groups in terms of Brauer pairs and pointed groups starts with work of Alperin-Broué [1], Broué-Puig [8] and Puig [21]. See Thévenaz’ book [28] for a detailed exposition on block theory.

Let k be a field of prime characteristic p and G a finite group.

3.1 For any p−subgroup P of G the natural projection kG ⟶ kC_{G}(P) mapping x ∈ C_{G}(P) to x and x ∈ G − C_{G}(P) to zero restricts to a surjective algebra homomorphism

Br_{P}^{G} : (kG)^{P} ⟶ kC_{G}(P),

called the Brauer homomorphism of P in G. Here (kG)^{P} denotes the subalgebra of P −stable elements in kG with respect to the action of P by conjugation. If no confusion is possible we will write Br_{P} instead of Br_{P}^{G}. See [28, section 11] for more details and generalizations of this construction.

Recall from Puig [21] that a point of P on kG is a ((kG)^{P})^{×}−conjugacy class γ of primitive idempotents in (kG)^{P}; we say that γ is a local point of P on kG if Br_{P} (γ) ≠ 0. By standard theorems on lifting of idempotents, Br_{P} (γ) is then a conjugacy class of primitive idempotents in kC_{G}(P).

3.2 A block of kG is a primitive idempotent b in the center Z(kG) of the group algebra kG. The algebra kGb is then called the block algebra of the block b. A defect group of the block b is a minimal subgroup P of G such that the map kGb ⊗_{kP}

kGb ⟶ kGb induced by multiplication in kGb splits as homomorphism of kGb − kGb−bimodules. Equivalently, P is a maximal p−subgroup of G such that Br_{P} (b) ≠ 0. The defect groups of b form a G−conjugacy class of p−subgroups of G.

3.3 If P is a defect group of a block b of kG we have Br_{P} (b) ≠ 0, and therefore there is a primitive idempotent i ∈ (kGb)^{P} such that Br_{P} (i) ≠ 0. The ((kG)^{P})^{×}−conjugacy class γ of i in (kG)^{P} is then a local point of P on kG contained in kGb. The pair P_{γ} is called a defect pointed group of the block b. Again, G acts transitively by conjugation on the set of defect pointed groups of b (cf. [21, 1.2]), thus in particular, N_{G}(P) acts transitively on the set of local points of P on kG contained in kGb. The algebra ikGi, considered as interior P −algebra (cf. [21, 3.1]) via the group homomorphism P ⟶ (ikGi)^{×} mapping u ∈ P to ui is called a source algebra of the block b (cf. [21, 3.2]).

By the preceding remarks, up to automorphisms of P induced by N_{G}(P) all source algebras of b are isomorphic as interior P −algebras. The block algebra kGb and its source algebra ikGi are Morita equivalent through the kGb − ikGi−bimodule kGi and the ikGi − kGb−bimodule ikG (cf. [21, 3.5]). Moreover, the p−local structure of the block b (in terms of Brauer pairs or local pointed groups) is in fact an invariant of the source algebra ikGi (cf. [22]).

3.4 Let b be a block of G and P_{γ} a defect pointed group of b. Let i ∈ γ; that is, i is a primitive idempotent in (kGb)^{P} such that Br_{P} (i) ≠ 0. Then Br_{P} (i) is a primitive idempotent in kC_{G}(P) and thus there is a unique block e_{P} of kC_{G}(P) such that Br_{P} (i)e_{P} = Br_{P} (i). If Q is a subgroup of P, then in general Br_{Q}(i) need no longer be primitive in kC_{G}(Q), but we still have the following remarkable uniqueness property, due to Broué and Puig [8, 1.8]: for any subgroup Q of P there is a unique block e_{Q} of kC_{G}(Q) such that Br_{Q}(i)e_{Q} = Br_{Q}(i). Then Z(Q) is contained in any defect group of e_{Q}, and we say that the Brauer pair (Q, e_{Q}) is self-centralizing, if Z(Q) is a defect group of e_{Q} (cf. [28, section 41]).

For any two subgroups Q, R of P we denote by { \atop Hom}(Q, R) the set of equivalence classes of group homomorphisms from Q to R modulo inner automorphisms of R and for any group homomrphism φ : Q ⟶ R we denote by φ̃ its image in { \atop Hom}(Q, R). We denote by E_{G}((Q, e_{Q}), (R, e_{R})) the image in { \atop Hom}(Q, R) of all group homomorphisms φ : Q ⟶ R for which there is an element x ∈ G satisfying φ(u) = xux^{−1} and xe_{Q}x^{−1} = e_{xQx^{−1}}. In particular, E_{G}((Q, e_{Q}), (Q, e_{Q})) is the image of N_{G}(Q, e_{Q}) in the outer automorphism group of Q, whence isomorphic to N_{G}(Q, e_{Q})/QC_{G}(Q).

3.5 Let b be a block of kG and P_{γ} be a defect pointed group of b. Let i ∈ γ.

The cohomology ring of the block b (with respect to the defect pointed group P_{γ}) is the subring

H^{∗}(G, b, P_{γ})

of H^{∗}(P, k) of “stable elements with respect to the p−local structure of b ”; that is, H^{∗}(G, b, P_{γ}) consists of all [ζ] ∈ H^{∗}(P, k) satisfying res_{φ}([ζ]) = res_{Q}^{P}([ζ]) for any subgroup Q in P and any group homomorphism φ : Q ⟶ P such that φ̃ ∈ E_{G}((Q, e_{Q}), (P, e_{P})).

Any φ̃ ∈ E_{G}((Q, e_{Q}), (P, e_{P})) is, by Alperin’s fusion lemma (see e.g. [28, (48.3)]), a composition of ψ̃ ∈ E_{G}((R, e_{R}), (R, e_{R})) with R running over the set of subgroups of P such that (R, e_{R}) is self-centralizing.

Thus H^{∗}(G, b, P_{γ}) is equal to the subring consisting of all [ζ] ∈ H^{∗}(P, k) satisfying xres_{R}^{P}([ζ]) = res_{R}^{P}([ζ]) for all x ∈ N_{G}(R, e_{R}), where R runs over the set of subgroups of P such that (R, e_{R}) is self-centralizing.

This latter description, together with Puig’s results in [22] on ikGi−fusion shows that H^{∗}(G, b, P_{γ}) is an invariant of the source algebra ikGi of the block b. Also, the isomorphism class of the graded k−algebra H^{∗}(G, b, P_{γ}) does not depend on the choice of P_{γ} since all defect pointed groups of b are G−conjugate.

If b is the principal block of G then P is a Sylow-p−subgroup of G, for any subgroup Q of P the block e_{Q} is the principal block of kC_{G}(Q) and N_{G}(Q, e_{Q}) = N_{G}(Q); thus restriction from G to P induces an isomorphism H^{∗}(G, k) ≅ H^{∗}(G, b, P_{γ}) by the characterization of H^{∗}(G, k) in terms of stable elements in H^{∗}(P, k).

3.6 Blocks with “trivial p−local structure” are the so-called nilpotent blocks, introduced by Broué and Puig in [7]. With the notation of 3.5, the block b is nilpotent if P “controls fusion”; that is, if for any φ̃ ∈ E_{G}((Q, e_{Q}), (P, e_{P})) there is an element y ∈ P such that φ(u) = yuy^{−1} for all u ∈ Q. If k is large enough, this is equivalent to requiring that N_{G}(Q, e_{Q})/QC_{G}(Q) is a p−group for any subgroup Q of P. For k large enough, the structure of nilpotent blocks has been determined by Puig in [23]: if b is nilpotent, there is an indecomposable kP −endopermutation module N (cf. [11], [12]) with vertex P such that ikGi ≅ S ⊗_{k} kP as interior P −algebra, where S = End_{k}(N).

In particular, kGb is Morita equivalent to kP and H^{∗}(G, b, P_{γ}) ≅ H^{∗}(P, k).

Even if b is arbitrary, nilpotent blocks come in systematically: for any subgroup Q of P such that (Q, e_{Q}) is self-centralizing, the block e_{Q} of kC_{G}(Q) is nilpotent (this follows trivially from the fact that the defect group Z(Q) of e_{Q} lies in the center of kC_{G}(Q) and therefore there is no nontrivial fusion). In particular, e_{P} is always a nilpotent block of kC_{G}(P).

Furthermore, it is shown in [7] that G is p−nilpotent if and only if the principal block of kG is nilpotent.

3.7 The “diagonal embedding” δ_{G} : H^{∗}(G, k) ⟶ HH^{∗}(kG) has an analogue for any block b of kG, if we assume additionally that k is large enough for the block algebra kC_{G}(P)e_{P} to be split, which amounts to requiring that kC_{G}(P)e_{P} is Morita equivalent to kZ(P) as e_{P} is a nilpotent block.

Still with the notation of 3.5, we consider kGi as kGb − kP −bimodule and ikG as kP − kGb−bimodule. By [20, 5.6] the relative projective elements π_{kGi} and π_{ikG} are invertible in Z(kGb) and Z(kP), respectively (this is where we use that k is large enough). Moreover, by [20, 5.6(iii)], the composition of the inclusion H^{∗}(G, b, P_{γ}) ⊂ H^{∗}(P, k), the diagonal embedding δ_{P} : H^{∗}(P, k) ⟶ HH^{∗}(kP) and the normalized transfer map T_{kGi} : HH^{∗}(kP) ⟶ HH^{∗}(kGb) induce an injective algebra homomorphism

H^{∗}(G, b, P_{γ}) ⟶^{T_{kGi}∘δ_{P}} HH^{∗}(kGb)

whose image is contained in the subalgebra HH_{kGi}^{∗}(kGb) of kGi−stable elements in HH^{∗}(kGb). If b is the principal block of kG, this map is in fact equal to δ_{G} followed by the canonical projection from HH^{∗}(kG) onto HH^{∗}(kGb), as we will see in 4.2(ii) below.

4 Varieties for modules over a block algebra

We fix in this section an algebraically closed field k of prime characteristic p.

We remind the reader of the definition, due to J. F. Carlson [9], [10], of the cohomological variety V_{G}(U) of a bounded complex U of kG−modules, where G is a finite group: the functor − ⊗_{k} U induces an algebra homomorphism γ_{U} : H^{∗}(G, k) ⟶

Ext_{kG}^{∗}(U, U). Denote by I_{G}^{∗}(U) the kernel of γ_{U}. The variety V_{G}(U) is then defined to be the maximal ideal spectrum of the quotient H^{∗}(G, k)/I_{G}^{∗}(U). Note that γ_{U} = α_{U} ∘ δ_{P} by 2.9.

The case where U = k is the trivial kG−module had previously been considered by D. Quillen (see [24], [25]). See [3, Vol. II, section 5.1] for a more detailed historical overview on varieties in group representation theory and for an extensive bibliography on this subject.

Definition 4.1 Let G be a finite group, b a block of kG, P_{γ} a defect pointed group of b and let i ∈ γ. For any bounded complex U of kGb−modules denote by I_{G,b}^{∗}(U) the kernel in H^{∗}(G, b, P_{γ}) of the composition of k−algebra homomorphisms

H^{∗}(G, b, P_{γ}) ⟶^{T_{kGi}∘δ_{P}} HH^{∗}(kGb) ⟶^{α_{U}} Ext_{kG}^{∗}(U, U)

and let V_{G,b}(U) be the maximal ideal spectrum of H^{∗}(G, b, P_{γ})/I_{G,b}^{∗}(U).

The isomorphism class of the variety V_{G,b}(U) in 4.1 does not depend on the choice of P_{γ}. If b is the principal block, V_{G,b}(U) is just the cohomological variety V_{G}(U) (see 4.4 below).

The next theorem establishes a connection in general between V_{G,b}(U) and V_{G}(U).

Theorem 4.2. Let G be a finite group, b a block of kG, P_{γ} a defect pointed group of b and let i ∈ γ.

(i) The restriction res_{P}^{G} induces an algebra homomorphism

ρ_{b} : H^{∗}(G, k) ⟶ H^{∗}(G, b, P_{γ})

such that H^{∗}(G, b, P_{γ}) becomes Noetherian as a module over H^{∗}(G, k).

(ii) The diagram of graded k−algebra homomorphisms

H^{∗}(G, k) ⟶^{δ_{G}} HH^{∗}(kG)
ρ_{b}↓  ↓
H^{∗}(G, b, P_{γ}) ⟶^{T_{kGi}∘δ_{P}} HH^{∗}(kGb)

is commutative, where the right vertical map is the canonical projection induced by multiplication with b.

(iii) For any bounded complex U of kGb−modules the diagram of graded k−algebra homomorphisms

H^{∗}(G, k)  ⟶^{γ_{U}} Ext_{kG}^{∗}(U, U)
ρ_{b}↓  ↓≅
H^{∗}(G, b, P_{γ}) ⟶_{α_{U} ∘T_{kGi}∘δ_{P}} Ext_{kGb}^{∗}(U, U)

is commutative.

Before we prove 4.2, let us note some consequences. By a result of Gerstenhaber [15], the Hochschild cohomology of an associative ring is graded commutative. Using the fact that H^{∗}(P, k) is Noetherian over H^{∗}(G, k) via restriction from G to P (cf. [3, Vol. II, 4.2.5]) and that by a result of T. Holm in [16] the Hochschild cohomology ring HH^{∗}(kG) is Noetherian as module over H^{∗}(G, k) via the algebra homomorphism δ_{G}, the diagram in 4.2(ii) implies the following:

Corollary 4.3. (i) The algebra HH^{∗}(kGb) is Noetherian as module over H^{∗}(G, b, P_{γ}) through the homomorphism given by T_{kGi} ∘ δ_{P}.

(ii) The Krull dimensions of H^{∗}(G, b, P_{γ}), HH^{∗}(kGb) and H^{∗}(P, k) coincide and are whence all equal to the rank of P.

The fact that HH^{∗}(kGb) and H^{∗}(P, k) have same Krull dimension has previously been observed by S. Siegel, who communicated a short direct proof of this statement to the author.

Using standard results from commutative algebra, in terms of maximal ideal spectra, statement 4.2(iii) translates to:

Corollary 4.4. For any bounded complex U of kGb−modules we have I_{G}^{∗}(U) = ρ_{b}^{−1}(I_{G,b}^{∗}(U)); in particular, ρ_{b} induces a finite surjective map

V_{G,b}(U) ⟶ V_{G}(U).

Moreover, if b is the principal block of kG, the above map is an isomorphism.

Proof of 4.2. Clearly the restriction res_{P}^{G} maps H^{∗}(G, k) to H^{∗}(G, b, P_{γ}), thus induces an algebra homomorphism ρ_{b} : H^{∗}(G, k) ⟶ H^{∗}(G, b, P_{γ}). By [3, 4.2.5], H^{∗}(P, k) is Noetherian as module over H^{∗}(G, k) through restriction, thus H^{∗}(G, b, P_{γ}) is Noetherian as module over H^{∗}(G, k) through ρ_{b}. This proves (i).

Let S be a Sylow-p−subgroup of G containing P. Identify H^{∗}(G, k) to the subalgebra res_{S}^{G}(H^{∗}(G, k)) of H^{∗}(S, k) of G−stable elements. Since restriction res_{S}^{G} followed by the transfer map t_{S}^{G} is multiplication by the index [G : S] on H^{∗}(G, k), it follows from 2.8 (or [20,4.6]) that

[G : S]δ_{G} = t_{(kG)_{S}} ∘ δ_{S}|_{H^{∗}(G,k)}.

From [20,5.3] follows that

t_{(ikG)_{S}} ∘ δ_{S}|_{H^{∗}(G,k)} = \frac{dim_{k}(ikG)}{|S|}δ_{P} ∘ ρ_{b},

where (ikG)_{S} is ikG viewed as kP − kS−bimodule. Moreover, by 2.3(ii) we have

t_{(ikG)_{S}} = t_{ikG} ∘ t_{(kG)_{S}}.

By 2.7 and 2.8 we have

δ_{G}(H^{∗}(G, k)) ⊂ HH_{(kG)_{P}}^{∗} (kG) ⊂ HH_{kGi}^{∗}(kG)

and the projection HH^{∗}(kG) ⟶ HH^{∗}(kGb) maps clearly HH_{kGi}^{∗}(kG) to HH_{kGi}^{∗}(kGb) (with the notational abuse of considering kGi as kG − kP −bimodule in the first place and then as kGb − kP −bimodule in the second).

Thus, by 2.6, the composition T_{kGi} ∘ T_{ikG} restricts to the identity on the image of δ_{G}(H^{∗}(G, k)) in HH^{∗}(kGb). It follows that

δ_{P} ∘ ρ_{b} = (\frac{dim_{k}(ikG)}{|S|})^{−1}t_{ikG} ∘ t_{(kG)_{S}} ∘ δ_{S}|_{H^{∗}(G,k)} =

(\frac{dim_{k}(ikG)}{|S|})^{−1}[G : S]t_{ikG} ∘ δ_{G} = T_{ikG} ∘ δ_{G},

since π_{ikG} = \frac{dim_{k}(ikG)}{|G|} by [20,5.6(i)]. Applying T_{kGi} to this equality yields the commutativity of the diagram in statement (ii).

The last statement follows then easily from the observation that the map

HH^{∗}(kG) ⟶ Ext_{kG}^{∗}(U, U) ≅ Ext_{kGb}^{∗}(U, U) induced by the functor − ⊗_{kG} U fac-

tors through the natural projection HH^{∗}(kG) ⟶ HH^{∗}(kGb). This concludes the proof of 4.2.

5 Invariance properties of varieties of modules

We show in this section roughly speaking, that the varieties V_{G,b}(U) introduced in 4.1 are invariant under splendid stable and derived equivalences. This is based on the following general result:

Theorem 5.1. Let A, B be symmetric algebras over a commutative ring R with symmetrizing forms s ∈ A^{∗}, t ∈ B^{∗}, and let X be a bounded complex of A − B−bimodules whose components are projective as left and right modules.

If π_{X} is invertible in Z(A), for any bounded complex U of B−modules there is a commutative diagram of graded R−algebra homomorphisms

HH_{X^{∗}}^{∗}(B) ⟶^{α_{U}}  Ext_{B}^{∗}(U, U)
T_{X} ↓  ↓^{β_{X,U}}
HH_{X}^{∗}(A) ⟶_{α_{X{⊗ \atop B}U}} Ext_{A}^{∗}(X ⊗_{B} U, X ⊗_{B} U)

where the horizontal maps are induced by the functors − ⊗_{B} U and − ⊗_{A} (X ⊗_{B} U),

respectively, and where the right vertical map is induced by the functor X ⊗_{B} −.

Proof. Let n be a nonnegative integer, ζ : P_{A} ⟶ P_{A}[n] and τ : P_{B} ⟶ P_{B}[n] chain maps making the diagram

5.1.1

P_{A} ⊗_{A} X ⟶^{≃} X ⊗_{B} P_{B}
ζ⊗Id_{X} ↓  ↓^{Id_{X}⊗τ}
P_{A}[n] ⊗_{A} X ⟶_{≃} X ⊗_{B} P_{B}[n]

homotopy commutative, where the horizontal maps are homotopy equivalences lifting the natural isomorphism A ⊗_{A} X ≅ X ≅ X ⊗_{B} B (and its shift by degree n).

Then we have [ζ] = T_{X}([τ]) by [20,3.4(ii)]. Thus α_{X⊗_{B}U} ∘T_{X} maps [τ] to the element

of Ext_{A}^{n}(X ⊗_{B} U, X ⊗_{B} U) represented by the chain map

P_{A} ⊗_{A} X ⊗_{B} U ⟶^{ζ⊗Id_{X}⊗Id_{U}} P_{A}[n] ⊗_{A} X ⊗_{B} U

and β_{X,U} ∘ α_{U} maps [τ] to the element of Ext_{A}^{n}(X ⊗_{B} U, X ⊗_{B} U) represented by the

chain map

X ⊗_{B} P_{B} ⊗_{B} U ⟶^{Id_{X}⊗τ⊗Id_{U}} X ⊗_{B} P_{B}[n] ⊗_{B} U.

Thus both coincide as can be seen by tensoring the diagram 5.1.1 by − ⊗_{B} U.

Before we state our invariance theorem, we need some properties of relatively projective elements. We show that the relative projective elements coming from derived or stable equivalences of Morita type between block algebras are invertible not only with respect to the canonical symmetrizing forms but in fact with respect to any symmetrizing form; we state this in the following slightly more general form:

Proposition 5.2. Let A, B be indecomposable non simple symmetric algebras over a field k and let X be a bounded complex of A − B−bimodules whose components are projective as left and right modules. Suppose that

X ⊗_{B} X^{∗} ≃ A ⊕ U_{A}, where U_{A} is a bounded complex of projective A − A−bimodules,

and X^{∗} ⊗_{A}X ≃ B ⊕U_{B}, where U_{B} is a bounded complex of projective B −B−bimodules.

Then for any choice of symmetrizing forms on A and B, the corresponding relatively projective elements π_{X} and π_{X^{∗}} are invertible in Z(A) and Z(B), respectively.

Proof. If π_{X} is not invertible, the composition of adjunction maps A ⟶ X ⊗_{B}X^{∗} ⟶

A is a nilpotent A − A−endomorphism of A, since A is indecomposable as A − A−bimodule. However, modulo the thick subcategory of K^{b}(A) consisting of bounded complexes of projective A−modules (and similarly for B instead of A) the functor X⊗_{B}− induces an equivalence by the hypotheses on X, and so the image in the suitable

quotient category of the composition of adjunction maps A ⟶ X ⊗_{B} X^{∗} ⟶ A has to

be an isomorphism and thus cannot be nilpotent. This shows that π_{X} is invertible. The same argument shows that π_{X^{∗}} is invertible.

Note that the above proposition covers the situation of derived and stable equivalences of Morita type: if U_{A} and U_{B} are zero, the complex X is a two-sided tilting complex, frequently called Rickard complex in the literature, and also two-sided split endomorphism tilting complex in [27]. If the complex X is concentrated in degree zero (that is, X is a bimodule), we are in the situation of what is called a stable equivalence of Morita type, a concept due to M. Broué.

We collect now some elementary properties of symmetric subalgebras of a symmetric algebra that we are going to apply in 5.4 to the particular case of the subalgebra kP i (which is clearly isomorphic to kP, whence symmetric) of the source algebra ikGi of the block b, where the notation is as in 3.3 (or 5.4 below).

Lemma 5.3. Let A be a symmetric algebra over a commutative ring R with symmetrizing form s ∈ A^{∗} and let B be a unitary symmetric subalgebra of A such that the restriction s|_{B} ∈ B^{∗} is a symmetrizing form of B. Then the following hold.

(i) As B − B−bimodule, B is a direct summand of A, and B has a unique complement C in A contained in ker(s).

(ii) The projection π : A ⟶ B with kernel C maps any a ∈ A to the unique element b ∈ B satisfying a.s|_{B} = b.s|_{B}. Moreover, π maps Z(A) to Z(B).

(iii) If C^{B} ⊂ J(A^{B}) then π maps Z(A)^{×} to Z(B)^{×}.

Proof. Any R−linear form on A is equal to a.s for a uniquely determined a ∈ A, where a.s is defined by (a.s)(a′) = s(aa′) for all a′ ∈ A. Its restriction to B is thus equal to b.s|_{B} for a uniquely determined b ∈ B, as s|_{B} is a symmetrizing form for B. The map sending a ∈ A to b ∈ B defined this way is clearly a projection of A onto B as B − B−bimodule, and its kernel C is the unique complement of B in A contained in ker(s). This map sends A^{B} to B^{B} = Z(B), thus it sends in particular Z(A) to Z(B). This shows (i) and (ii). Let now z ∈ Z(A)^{×} and write z = π(z) + c for some c ∈ C. Similarly, write z^{−1} = π(z^{−1}) + c′ for some c′ ∈ C. Then in fact c, c′ ∈ C^{B}. Write cc′ = π(cc′) + d for some d ∈ C; again in fact d ∈ C^{B}. We have now 1_{A} = zz^{−1} = π(z)π(z^{−1}) + π(z)c′ + cπ(z^{−1}) + cc′. As π(1) = 1 we obtain from the previous equation that 1 = π(z)π(z^{−1}) + π(cc′). Therefore, if C^{B} ⊂ J(A^{B}) then π(cc′) = cc′ − d ∈ J(A^{B}), so in particular π(z) is invertible.

Observe that the projection π : A ⟶ B in 5.3(ii) is a B − B−homomorphism, thus the induced map Z(A) ⟶ Z(B) is R−linear, but not multiplicative in general. In particular, the induced map Z(A)^{×} ⟶ Z(B)^{×} in 5.3(iii) is not a group homomorphism in general.

Proposition 5.4. Let k be an algebraically closed field of prime characteristic p, let G be a finite group, b a block of kG and P_{γ} a defect pointed group of b. Let i ∈ γ. Denote by s ∈ (ikGi)^{∗} the canonical symmetrizing form on ikGi.

(i) We have s(i) ∈ k^{×} and s(ui) = 0 for any u ∈ P − {1}. In particular, the restriction of s to the subalgebra kP i of ikGi is a symmetrizing form of kP i.

(ii) The subalgebra kP i of ikGi has a unique complement C in ikGi as kP − kP −bimodule such that C ⊂ ker(s), and then C^{P} ⊂ J((ikGi)^{P}).

(iii) The projection π : ikGi ⟶ kP i with kernel C maps a ∈ ikGi to

s(i)^{−1} ∑_{u∈P}s(u^{−1}a)ui. Moreover, π maps Z(ikGi)^{×} to Z(kP)^{×}.

(iv) For any choice of symmetrizing forms on ikGi and on kP, the relatively ikG−projective element π_{ikG} is invertible in Z(kP).

Proof. (i) The fact that s(ui) is non zero if and only if u = 1_{P} is proved in [20,5.5]. This means that through the obvious isomorphism kP ≅ kP i, the restriction of s to kP is a non zero scalar multiple of the canonical symmetrizing form on kP, thus itself a symmetrizing form.

(ii) By (i) and 5.3(i), kP i has a unique complement in ikGi as claimed. Now applying the Brauer construction [28, section 11] yields, by a result of Puig [24, 14.5], that (ikGi)(P) ≅ kZ(P) (this isomorphism is also described in [28, (38.10)]), and clearly (kP i)(P) ≅ kZ(P). Thus C(P) = {0}, or, equivalently, C^{P} ⊂ ker(Br_{P}), where Br_{P} denotes the Brauer homomorphism on (ikGi)^{P} (cf. 3.1). Since i is primitive in (ikGi)^{P}, the latter is a local algebra. As Br_{P} (i) ≠ 0 the ideal ker(Br_{P}) of (ikGi)^{P} is therefore contained in J((ikGi)^{P}), which implies (ii).

(iii) Since C ⊂ ker(s), the given formula in (iii) maps C to zero, and whence coincides with π on C. Let x ∈ P. Then, by (i), for any u ∈ P we have s(u^{−1}xi) = 0 unless u = x. Thus the given formula is the identity on kP i, whence coincides again with π. The second statement in (iii) follows from (ii) and 5.3(iii).

(iv) With respect to the canonical symmetrizing forms on kP and ikGi, by [20,5.5], the adjunction map kP ⟶ ikG ⊗_{kGb} kGi ≅ ikGi sends u ∈ P to ui, and the adjunc-

tion map ikGi ≅ ikG ⊗_{kGb} kGi ⟶ kP sends a ∈ ikGi to ∑_{u∈P}s(u^{−1}a)u. Any other

symmetrizing form on ikGi is equal to z.s for a uniquely determined z ∈ Z(ikGi)^{×}. With respect to this new symmetrizing form on ikGi (and still the canonical form on kP), the first adjunction map kP ⟶ ikG ⊗_{kGb} kGi ≅ ikGi maps u ∈ P to zui,

while the second adjunction map remains unchanged (this follows from the formulae given in [20, 2.4]). Thus, the relatively ikG−projective element π_{ikG} with respect to this choice of symmetrizing forms is equal to ∑_{u∈P}s(u^{−1}zi)u. It follows from (iii)

that π_{ikG}i = s(i)π(z). Since π maps Z(ikGi)^{×} to Z(kP i)^{×}, indeed π_{ikG} is invertible. Modifying the symmetrizing form on kP has no influence on the property of π_{ikG} being invertible (this follows from [20, 3.2.2]), which completes the proof.

We state now the invariance theorem for the varieties V_{G,b}(U) with respect to splendid derived and stable equivalences, again in a slightly more general form analogously to 5.2.

Theorem 5.5. Let k be an algebraically closed field of prime characteristic p, let G, H be finite groups and b, c be blocks of kG, kH, respectively, having a common defect group P. Let γ, δ be local points of P on kGb, kHc, respectively, and choose i ∈ γ, j ∈ δ. For any subgroup Q of P denote by e_{Q} and f_{Q} the unique blocks of kC_{G}(Q) and kC_{H}(Q), respectively, satisfying Br_{Q}^{G}(i)e_{Q} = Br_{Q}^{G}(i) and Br_{Q}^{H}(j)f_{Q} = Br_{Q}^{H}(j).

Assume that E_{G}((Q, e_{Q}), (P, e_{P})) = E_{H}((Q, f_{Q}), (P, f_{P})) for any subgroup Q of P. Let X be a bounded complex of kGb − kHc−bimodules whose components are iso-

morphic to direct sums of direct summands of the bimodules kGi ⊗_{kQ} jkH, where Q

runs over the set of subgroups of P.

Assume that X ⊗_{kHc} X^{∗} ≃ kGb ⊕ U_{b}, where U_{b} is a bounded complex of projective

kGb−kGb−bimodules, and that X^{∗} ⊗_{kGb}X ≃ kHc⊕U_{c}, where U_{c} is a bounded complex

of projective kHc − kHc−bimodules.

(i) For any choice of symmetrizing forms on kGb and kHc, the relatively projective elements π_{X} and π_{X^{∗}} are invertible in Z(kGb) and Z(kHc), respectively. With respect to the canonical symmetrizing form on kP we have π_{iXj} = π_{jX^{∗}i} ∈ k^{×}1_{kP}, where iXj and its dual jX^{∗}i are considered as complexes of kP − kP −bimodules.

(ii) The map T_{kGi} ∘ δ_{P} sends H^{∗}(G, b, P_{γ}) to HH_{X}^{∗}(kGb) and T_{kHj} ∘ δ_{P} sends H^{∗}(H, c, P_{δ}) to HH_{X^{∗}}^{∗}(kHc), making the following diagram of graded k−algebras commutative:

H^{∗}(H, c, P_{δ}) ⟶^{T_{kHj}∘δ_{P}} HH_{X^{∗}}^{∗}(kHc)
Id↓  ↓^{T_{X}}
H^{∗}(G, b, P_{γ}) ⟶_{T_{kGi}∘δ_{P}} HH_{X}^{∗}(kGb)

(iii) For any bounded complex V of kHc−modules, the following diagram of graded k−algebras is commutative:

H^{∗}(H, c, P_{δ}) ⟶^{α_{V} ∘T_{kHj}∘δ_{P}}  Ext_{kHc}^{∗}(V, V)
Id↓  ↓^{β_{X,V}}
H^{∗}(G, b, P_{γ}) ⟶_{α_{X{⊗ \atop kHc}V} ∘T_{kGi}∘δ_{P}} Ext_{kGb}^{∗}(X ⊗_{kHc} V, X ⊗_{kHc} V)

In particular, we have I_{G,b}^{∗}(X ⊗_{kHc} V) = I_{H,c}^{∗}(V), and whence,

V_{G,b}(X ⊗_{kHc} V) = V_{H,c}(V).

Proof. Statement (i) follows from 5.2 and [20,5.7(i)]. For the proof of (ii), we first show that T_{kGi} ∘ δ_{P} maps H^{∗}(G, b, P_{γ}) to HH_{X}^{∗}(kGb). This is based on the cancellation properties for stable elements in 2.7. By [20,5.7(iii)] we have δ_{P} (H^{∗}(G, b, P_{γ})) ⊂ HH_{iXj}^{∗}(kP). As iXj ≅ iX ⊗_{kHc}kHj and π_{kHj} is invertible, the first of the cancellation

properties in 2.7 implies that HH_{iXj}^{∗}(kP) ⊂ HH_{iX}^{∗}(kP). As iX ≅ ikG ⊗_{kGb} X and

π_{kGi} is invertible, the second of the cancellation properties in 2.7 shows that T_{kGi} maps HH_{iX}^{∗}(kP) to HH_{X}^{∗}(kGb). A similar argument shows that T_{kHj} ∘ δ_{P} maps H^{∗}(H, c, P_{δ}) to HH_{X^{∗}}^{∗}(kHc).

Note that the diagram in (ii) does not depend on the choice of symmetrizing forms on kGb, kHc, as follows from [20,3.6(ii)].

Observe next that H^{∗}(G, b, P_{γ}) = H^{∗}(H, c, P_{δ}) as subalgebras of H^{∗}(P, k) by the assumptions. We consider kP endowed with the canonical symmetrizing form. By [20,3.2.3] we may choose a symmetrizing form on kHc such that π_{kHj} = 1_{kHc}, or equivalently, such that T_{kHj} = t_{kHj}. Since π_{X} is invertible for any choice of a symmetrizing form on kHc, we may again by [20,3.2.3] choose a symmetrizing form on kGb such that π_{X} = 1_{kGb}, or equivalently, such that T_{X} = t_{X}. Since we have not changed the symmetrizing form of kP, the relatively projective element π_{kGi} is still invertible in Z(kGb). Moreover, by 5.4(iv), the relatively projective element π_{ikG} is invertible in Z(kP). In order to show that the diagram in (ii) is commutative we have to show that the maps T_{X} ∘T_{kHj} ∘δ_{P} and T_{kGi}∘δ_{P} coincide on H^{∗}(H, c, P_{δ}) = H^{∗}(G, b, P_{γ}). This is equivalent to showing that the map T_{ikG}∘T_{X}∘T_{kHj} is the identity on δ_{P} (H^{∗}(H, c, P_{δ})). Now

T_{ikG} ∘ T_{X} ∘ T_{kHj} = (π_{ikG})^{−1}t_{ikG} ∘ t_{X} ∘ t_{kHj} = (π_{ikG})^{−1}t_{iXj}

and by [20,5.7(iv)], the map t_{iXj} acts as multiplication by π_{iXj} on δ_{P} (H^{∗}(H, c, P_{δ})). Thus it suffices to observe that π_{ikG} = π_{iXj}. Now with the notation from [20, 3.2], we have π_{iXj} = t_{iXj}^{0}(1_{kP}) = t_{ikG}^{0} ∘ t_{X}^{0} ∘ t_{kHj}^{0}(1_{kP}), and by our choice of symmetrizing forms, we have t_{kHj}^{0}(1_{kP}) = 1_{kHc} and t_{X}^{0}(1_{kHc}) = 1_{kGb}. Since t_{ikG}^{0}(1_{kGb}) = π_{ikG}, the proof of (ii) is complete.

The diagram in (iii) is just obtained by composing the diagram of (ii) together with the appropriate version of 5.1.

6 Some further remarks

Let k be an algebraically closed field of prime characteristic p and G a finite group. Benson, Carlson and Robinson introduce in [5] the nucleus of kG as the subvariety

Y_{G} of V_{G}(k) which is the union of the images of the maps (res_{H}^{G})^{∗} induced by the restriction maps res_{H}^{G} : H^{∗}(G, k) ⟶ H^{∗}(H, k), with H running over the set of subgroups of G for which C_{G}(H) is not p−nilpotent, with the convention Y_{G} = {0} if G is p−nilpotent.

We define now a nucleus Y_{G,b} of a block b of G and show that this coincides with Y_{G} if b is the principal block of G. Remind from 4.1 that V_{G}(b) is the variety of the block b; that is, the maximal ideal spectrum of the cohomology ring of b.

Definition 6.1 Let b be a block of kG and P_{γ} a defect pointed group of b. Let i ∈ γ. For any subgroup Q of P denote by e_{Q} the unique block of kC_{G}(Q) such that Br_{Q}(i)e_{Q} = Br_{Q}(i). The nucleus Y_{G,b} of the block b is the union of the images of all maps V_{Q}(k) ⟶ V_{G}(b) induced by the restriction maps res_{Q}^{P} from H^{∗}(G, b, P_{γ}) to H^{∗}(Q, k), where Q runs over the set of subgroups of P such that the block e_{Q} of kC_{G}(Q) is not nilpotent, with the convention Y_{G,b} = {0} if the block b is nilpotent.

Proposition 6.2. If b_{0} is the principal block of kG then restriction from G to a Sylow-p−subgroup P of G induces isomorphisms

V_{G}(b_{0}) ≅ V_{G}(k) and Y_{G,b_{0}} ≅ Y_{G} .

Proof. If b_{0} is the principal block of kG then the cohomology ring of b_{0} is precisely the image of the restriction to P of H^{∗}(G, k), thus V_{G}(b_{0}) ≅ V_{G}(k). Moreover, for any subgroup Q of P, the block e_{Q} as defined in 6.1 is the principal block of kC_{G}(Q). By [7], e_{Q} is nilpotent if and only if the group kC_{G}(Q) is p−nilpotent. Thus res_{P}^{G} induces an injective map Y_{G,b_{0}} ⟶ Y_{G}. This map is also surjective; indeed, if H is a subgroup of G such that C_{G}(H) is not p−nilpoptent, then for any Sylow-p−subgroup Q of H, the group C_{G}(Q) is not p−nilpotent. The statement follows.

Remark 6.3 Let b be a block of kG, P_{γ} a defect pointed group of b, let i ∈ γ and denote for any subgroup Q of P by e_{Q} the unique block of kC_{G}(Q) such that Br_{Q}(i)e_{Q} = Br_{Q}(i).

(i) We have Y_{G,b} = {0} if and only if e_{Q} is nilpotent for every non trivial subgroup Q of P.

(ii) The nucleus Y_{G,b} of b is an invariant of the p−local structure of b since both the cohomology ring of b and the property of e_{Q} to be nilpotent are so.

(iii) One might want to try to generalize the results in [4], [5] to arbitrary blocks. The principal obstacle at this stage is, that we do not yet have analogous concepts which would generalize the representation theoretical nucleus defined in [5, 10.1] or the concept of trivial homology modules [5, 2.1] to arbitrary blocks.

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Markus Linckelmann CNRS, Université Paris 7 UFR Mathématiques 2, place Jussieu 75251 Paris Cedex 05

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