QUILLEN STRATIFICATION FOR BLOCK VARIETIES

Markus Linckelmann

May 2001

Abstract. The classical results on stratifications for cohomology varieties of finite groups and their modules due to Quillen [19, 20] and Avrunin-Scott [3] carry over to the varieties associated with finitely-generated modules over p-blocks of finite groups, introduced in [16].

1 Introduction

Throughout this paper, k is an algebraically closed field of prime characteristic p. By a Theorem of Evens [12] and Venkov [22, 23], the cohomology ring H^{∗}(G, k) of a finite group G is a finitely generated graded commutative k−algebra. Thus its maximal ideal spectrum V_{G} is an affine variety. Quillen showed in [19, 20] that this variety has a stratification indexed by the conjugacy classes of non trivial elementary abelian p−subgroups of G. The cohomology variety V_{G}(M) of a finitely generated kG−module M, introduced by Carlson [9, 10], is the maximal ideal spectrum of the quotient of H^{∗}(G, k) by the kernel of the canonical graded algebra homomorphism H^{∗}(G, k) → Ext_{G}^{∗}(M, M) induced by tensoring with M over k. Avrunin and Scott showed in [3] that V_{G}(M) has a similar stratification, generalising Quillen’s results on V_{G}.

Since the definition of V_{G}(M) involves the cohomology ring H^{∗}(G, k) - which is an invariant of the principal block of kG - , if M belongs to a non principal block b of kG, the variety V_{G}(M) is not in general an invariant of M viewed as kGb−module. This motivates in [16] the definition of a variety V_{G,b}(M), obtained as the maximal ideal spectrum of the quotient of the block cohomology H^{∗}(G, b) by the kernel of the canonical map H^{∗}(G, b) → Ext_{kGb}^{∗}(M, M) defined in [15]. It is shown in [16, 4.4] that there is a finite surjective morphism V_{G,b}(M) → V_{G}(M); in particular, by a Theorem of Alperin and Evens [2], the dimension of V_{G,b}(M) is equal to the complexity of M. The above morphism V_{G,b}(M) → V_{G}(M) is an isomorphism if b is the principal block. Moreover, by [16, 5.5], the variety V_{G,b}(M) is invariant under splendid stable, derived or Morita equivalences.

The main result of Section 2 provides a way to compute the block varieties V_{G,b}(M) in terms of the truncated restriction iM of M, where i is a source idempotent in (kGb)^{P} for some defect group P of b and where iM is viewed as kP -module. Section 3 contains the technicalities related to the Evens norm map, which we are going to use in Section 4 in order to see that the proof of the stratification of Avrunin and

Typeset by AMS-TEX

Scott for V_{G}(M) in [3] can be adapted to get a stratification for V_{G,b}(M), which coincides with that of V_{G}(M) if b is the principal block of kG. Section 5 is finally devoted to describing an example which shows that it really matters to work with the truncated restriction Res_{P} (iM) and not just Res_{P}^{G}(M). This is because both the block cohomology H^{∗}(G, b) and the variety V_{G,b}(M) are defined with respect to the choice of a source idempotent i, uniquely up to unique isomorphism.

Acknowledgements. The author would like to thank D. Benson and J. Rickard for some very helpful conversations.

2 Block varieties and source idempotents

We describe briefly some aspects of the local structure of a block of a finite group in terms of Brauer pairs, introduced by Alperin and Broué in [1], and developed further in work of Broué and Puig [7] and Puig [18] (see Thévenaz [21] for a more detailed account).

Let G be a finite group and b a block of kG; that is, b is a primitive idempotent of Z(kG). Let P be a defect group of b. Then P is a maximal p−subgroup of G with the property that Br_{P} (b) ≠ 0, where Br_{P} : (kG)^{P} → kC_{G}(P) is the Brauer homomorphism [21, §11]. Thus there is a primitive idempotent i ∈ (kGb)^{P} satisfying Br_{P} (i) ≠ 0; any such idempotent is called a source idempotent of the block b. The algebra ikGi, together with the group homomorphism P → (ikGi)^{×} mapping u ∈ P to ui is a source algebra of the block b (cf. Puig [18]). By [7, 1.8], for any subgroup Q of P there is a unique block e_{Q} of kC_{G}(Q) satisfying Br_{Q}(i)e_{Q} = Br_{Q}(i).

The category F_{G,b} has as objects the set of subgroups of P; for any two subgroups Q, R, the set of morphisms from Q to R in F_{G,b} is the set of (necessarily injective) group homomorphisms φ : Q → R for which there is an element x ∈ G satisfying φ(u) = xux^{−1} for all u ∈ Q and satisfying ^{x}(Q, e_{Q}) ⊆ (R, e_{R}) (the latter condition is equivalent to xe_{Q}x^{−1} = e_{xQx^{−1}}). The category F_{G,b} is, up to canonical isomorphism of categories, independent of the choice of the source idempotent i.

The block cohomology of the block b of kG, introduced in [15], is the graded subalgebra H^{∗}(G, b) of H^{∗}(P, k) consisting of all ζ ∈ H^{∗}(P, k) which satisfy the stability conditions Res_{Q}^{P}(ζ) = Res_{φ}(ζ) for any subgroup Q of P and any group homomorphism φ : Q → P belonging to the category F_{G,b}. In other words, H^{∗}(G, b) = lim_{←} H^{∗}(Q, k), where the inverse limit is taken over the category F_{G,b}.

The restriction from G to P induces a homomorphism H^{∗}(G, k) → H^{∗}(G, b); if b is the principal block of kG, this is an isomorphism by the characterisation of H^{∗}(G, k) in terms of stable elements in [11]. Restriction from P to any subgroup Q of P induces a graded algebra homomorphism r_{Q} : H^{∗}(G, b) → H^{∗}(Q, k) whose image is contained in (H^{∗}(Q, k))^{N_{G}(Q,e_{Q})}.

The block cohomology algebra H^{∗}(G, b) is defined with respect to a choice of a defect group P and a source idempotent i. Since all pairs consisting of a defect group P of b and a ((kGb)^{P})^{×}−conjugacy class γ of source idempotents i in (kGb)^{P} are transitively permuted by the action of G by conjugation (cf. [18, 1.2]), H^{∗}(G, b) is defined in this way uniquely up to isomorphism, and it is in fact unique up to unique isomorphism, because the stabiliser N_{G}(P_{γ}) of such a pair acts trivially on H^{∗}(G, b).

This justifies the notation H^{∗}(G, b), which makes no mention of the choice of a source idempotent.

There is a canonical injective graded algebra homomorphism

ν : H^{∗}(G, b) ⟶ HH^{∗}(kGb)

from the block cohomology into the Hochschild cohomology of the block algebra kGb (cf. [15]) and for any finitely generated kGb−module M, tensoring by − ⊗_{kGb}M induces

a graded algebra homomorphism

α_{M} : HH^{∗}(kGb) ⟶ Ext_{kGb}^{∗}(M, M) .

The variety V_{G,b}(M) is defined in [16,4.1] as the maximal ideal spectrum of the quotient H^{∗}(G, b) by the kernel of α_{M} ∘ ν. In particular, V_{G,b}(M) is a subvariety of the maximal ideal spectrum V_{G,b} of H^{∗}(G, b), which is called the block variety of the block b. The cohomology variety V_{G}(M), introduced by Carlson [9, 10], is the maximal ideal spectrum of the quotient of H^{∗}(G, k) by the kernel of the homomorphism H^{∗}(G, k) → Ext_{kG}^{∗}(M, M) induced by the functor − ⊗_{k} M. The variety V_{G}(M) is a

subvariety of the maximal ideal spectrum V_{G} of H^{∗}(G, k). There is a finite surjective morphism V_{G,b}(M) → V_{G}(M), which is an isomorphism if b is the principal block (cf. [16,4.4]).

Again, this definition of V_{G,b}(M) depends on the choice of the defect group P and the source idempotent i, because both H^{∗}(G, b) and the algebra homomorphism ν depend on this choice. As above, V_{G,b}(M) is defined in this way uniquely up to unique isomorphism. The definition of ν involves the normalised transfer map T_{kGi} : HH^{∗}(kP) → HH^{∗}(kGb), and the welcome consequence of the following Theorem is, that one can compute V_{G,b}(M) without all this technology (which is, though, needed in the proof):

Theorem 2.1. Let G be a finite group, b a block of kG, P a defect group of b and i a source idempotent of b in (kGb)^{P}. The inclusion ι : H^{∗}(G, b) → H^{∗}(P, k) induces a finite surjective morphism ι^{∗} : V_{P} → V_{G,b}, and for any finitely generated kGb−module M we have

V_{G,b}(M) = ι^{∗}(V_{P} (iM)) ,

where iM is considered as kP −module.

Proof. By [16, 4.3], H^{∗}(P, k) is Noetherian as a module over H^{∗}(G, b), which implies that ι^{∗} is finite surjective. The homomorphism ν is, by [15, 5.6(iii)], defined as the unique graded algebra homomorphism which makes the following diagram commutative:

H^{∗}(G, b) ⟶^{ν} HH^{∗}(kGb)
ι↓  ↑^{T_{kGi}}
H^{∗}(P, k) ⟶_{δ_{P}} HH^{∗}(kP)

Here δ_{P} is the algebra homomorphism induced by the “diagonal induction” functor Ind_{ΔP}^{P ×P} (cf. [15, 4.5]) and T_{kGi} is the normalised transfer map defined in [15, 3.1], with respect to the kGb − kP −bimodule kGi; this makes sense as the relative projective element π_{kGi} is invertible (see [15, 3.1] and [15, 5.6]). By [15, 5.6(iii)] again, the image of ν lies actually in the subalgebra HH_{kGi}^{∗}(kGb) of kGi−stable elements in HH^{∗}(kGb) (cf. [15, 3.1(iii)]). Similarly, by [15, 5.6(ii)], the image of δ_{P} ∘ ι is contained in the subalgebra HH_{ikG}^{∗}(kP) of ikG−stable elements in HH^{∗}(kP); here ikG is viewed as kP − kGb−bimodule. Since on these subalgebras of stable elements, the normalised transfer T_{ikG} is inverse to the normalised transfer T_{kGi} by [15, 3.6 (iii)], we may reverse the right vertical arrow in the preceding diagram, in order to get a commutative diagram

H^{∗}(G, b) ⟶^{ν} HH^{∗}(kGb)
ι↓  ↓^{T_{ikG}}
H^{∗}(P, k) ⟶_{δ_{P}} HH^{∗}(kP)

Observe that iM ≅ ikG ⊗_{kGb} M. Thus applying [16, 5.1] to kP, kGb, ikG instead

of A, B, X, respectively, yields a commutative diagram of graded algebra homomorphisms

HH_{kGi}^{∗}(kGb) ⟶^{α_{M}} Ext_{kGb}^{∗}(M, M)
T_{ikG}↓  ↓^{β_{M}}
HH_{ikG}^{∗}(kP) ⟶_{α_{iM}} Ext_{kP}^{∗} (iM, iM)

The homomorphism β_{M} is induced by the functor ikG ⊗_{kGb} −. It follows from the

above remarks that by combining the two preceding commutative diagrams we get a commutative diagram of graded algebra homomorphisms

H^{∗}(G, b) ⟶^{α_{M} ∘ν} Ext_{kGb}^{∗}(M, M)
ι↓  ↓^{β_{M}}
H^{∗}(P, k) ⟶ Ext_{kP}^{∗} (iM, iM)

In this diagram the bottom horizontal arrow is equal to the composition α_{iM} ∘ δ_{P}, and this is, by [15, 2.9], equal to the homomorphism induced by tensoring with −⊗_{k}iM.

In other words, the top row in this diagram defines V_{G,b}(M), and the bottom row defines V_{P} (iM). In order to conclude the proof of the Theorem, it suffices to show that β_{M} is injective. This follows from a general property of source idempotents: the canonical map kGi ⊗_{kP} ikG → kGb induced by multiplication in kGb is split

as homomorphism of kGb − kGb−bimodules. Thus the identity functor is a direct summand of the composition of the “truncated” restriction functor ikG ⊗_{kGb}− and the

corresponding induction functor kGi ⊗_{kP} −. Therefore the composition of the algebra

homomorphisms

Ext_{kGb}^{∗}(M, M) → Ext_{kP}^{∗} (iM, iM) → Ext_{kGb}^{∗}(kGi ⊗_{kP} iM, kGi ⊗_{kP} iM)

induced by the functors ikG ⊗_{kGb} − and kGi ⊗_{kP} − is injective. But then in particular

the first of the two homomorphisms is injective, and that is just β_{M}. This proves 2.1. □

The above result provides a technique to carry over properties of the cohomology varieties at the level of p−subgroups to the variety V_{G,b}(M). We note one easy consequence (which could, of course, also be proved without using 2.1):

Corollary 2.2. Let M, M′ be finitely generated kGb−modules. We have V_{G,b}(M ⊕ M′) = V_{G,b}(M) ∪ V_{G,b}(M′)

Proof. By [5, 5.7.5] we have V_{P} (i(M ⊕M′)) = V_{P} (iM)∪V_{P} (iM′), and thus 2.2 follows from 2.1. □

Corollary 2.3. Let M be a finitely generated indecomposable kGb-module with P as vertex and a source of dimension prime to p. Then V_{G,b}(M) = V_{G,b}.

Proof. By [13, 6.1], some source of M is a direct summand of iM. Thus V_{P} (iM) = V_{P} by [5, 5.8.5]. Since ι^{∗} : V_{P} → V_{G,b} is surjective by 2.1, the statement follows. □

Remark 2.4. In the situation of Theorem 2.1, it is not true in general that V_{G,b}(M) coincides with ι^{∗}(Res_{P}^{G}(M)); that is, it really matters to “cut” the module M down by the source idempotent i. This phenomenon occurs if P has more than one conjugacy class of source idempotents, or equivalently, in Puig’s terminology, if P has more than one local point on kGb (cf. [18]). We describe an example in Section 5.

3 Norm maps for bisets

Let G be a finite group. If p is odd, we denote by H^{·}(G, k) the even part of H^{∗}(G, k); if p = 2 we set H^{·}(G, k) = H^{∗}(G, k). Thus H^{·}(G, k) is commutative since H^{∗}(G, k) is graded commutative (cf. [4, 3.2]). We refer to [5, 4.1] for the definition and general properties of the Evens norm map n_{H}^{G} : H^{·}(H, k) → H^{·}(G, k), where H is a subgroup of G. We use this to define for any two finite groups P, Q and any finite P -Q-biset X on which P, Q act regularly on the left and right, respectively, a norm map

n_{X} : H^{·}(Q, k) ⟶ H^{·}(P, k)

as follows. If X is transitive, then X is isomorphic to a biset of the form P ×_{R} _{φ}Q for

some subgroup R of P and an injective group homomorphism φ : R → Q. In that case, we set n_{X} = n_{R}^{P} ∘ res_{φ}, and in general, we set

n_{X} = ∏_{Y} n_{Y} ,

where Y runs over the set of transitive P -Q-subbisets in X (and where the product is taken in H^{∗}(P, k)).

Note that the exact sign of n_{R}^{P} and hence of n_{X} depends on the choice of a system of representatives of the right cosets of R in P (cf. [5, 4.1]), and so all statements on norm maps hold modulo keeping track of signs (but since the signs are irrelevant in the Propositions 3.4 and 3.5 below we do not insist on this aspect).

Lemma 3.1. Let P, Q, R be finite groups, let ψ : R → P be an injective group homomorphism and let X be a finite P -Q-biset on which P and Q act regularly on the left and on the right, respectively. We have

res_{ψ} ∘ n_{X} = n_{_{ψ}X} ,

where _{ψ}X is the R-Q-biset obtained from restricting X through ψ on the left.

Proof. We may assume that X is transitive as P -Q-biset, and then the result follows from the Mackey formula [5, 4.1.2(v)] for the Evens norm map. □

For the rest of this section, we fix the following notation. Let G be a finite group, b a block of kG, P a defect group of b and choose a source idempotent i ∈ (kGb)^{P}. For any subgroup Q of P, denote by e_{Q} the unique block of kC_{G}(Q) such that Br_{Q}(i)e_{Q} ≠ 0 (cf. [8, 1.8]).

As before, denote by F_{G,b} the category whose objects are the subgroups of P and whose morphisms, for any two subgroups Q, R of P, are the group homomorphisms φ : Q → R such that there is x ∈ G fulfilling φ(u) = xux^{−1} for all u ∈ Q and ^{x}(Q, e_{Q}) ⊆ (R, e_{R}). In particular, the automorphism group of Q in F_{G,b} corresponds to N_{G}(Q, e_{Q})/C_{G}(Q). Since inner automorphisms of Q act trivially on H^{·}(Q, k), the action of N_{G}(Q, e_{Q}) on H^{·}(Q, k) induces an action of the group W (Q) = N_{G}(Q, e_{Q})/QC_{G}(Q) on H^{·}(Q, k).

The following result is due to Broto, Levi and Oliver:

Proposition 3.2. ([6]) With the notation above, there is a finite P -P -biset X with the following properties.

(i) Every transitive subbiset of X is isomorphic to P ×_{Q}_{φ}P for some subgroup Q of

P and some group homomorphism φ : Q → P belonging to the category F_{G,b}.

(ii) |X|/|P | is prime to p.

(iii) For any subgroup Q of P and any group homomorphism φ : Q → P in F_{G,b}, the Q-P -bisets _{φ}X and _{Q}X are isomorphic.

The original motivation for constructing such a biset is an observation by Linckelmann and Webb, that its existence implies the existence of a stable summand B̂(G, b) of the classifying space BP_{+}^{∧} viewed as p-complete spectrum such that the cohomology of B̂(G, b) with coefficients in k is precisely the block cohomology H^{∗}(G, b).

Lemma 3.3. Let X be a finite P -P -biset fulfilling the conditions in 3.2. Then, for any subgroup Q of P, there is a Q-Q-subbiset of _{Q}X_{Q} isomorphic to Q.

Proof. It suffices to show that X has a P -P -subbiset isomorphic to P. By 3.2(i) and 3.2(ii), X has a P -P -subbiset isomorphic to _{φ}P for some automorphism φ of P in F_{G,b}. The stability condition 3.2(iii) implies the result. □

Proposition 3.4. Let X be a finite P -P -biset fulfilling the conditions in 3.2, let Q be a subgroup of P and let Y be the Q-Q-subbiset of _{Q}X_{Q} which is the union of all Q-Q-orbits of length |Q|.

(i) The image of the norm map n_{X_{Q}} : H^{·}(Q, k) → H^{·}(P, k) is contained in H^{·}(G, b).

(ii) The set Y is non empty.

(iii) For any ζ ∈ H^{·}(Q, k)^{W (Q)} such that res_{R}^{Q}(ζ) = 0 for any proper subgroup R of Q we have n_{_{Q}X_{Q}}(ζ) = n_{Y} (ζ).

(iv) For any ζ ∈ H^{·}(Q, k)^{W (Q)} we have n_{Y} (ζ) = ζ^{|Y |/|Q|}.

Proof. (i) Let R be a subgroup of P and let φ : R → P be a group homomorphism in F_{G,b}. Using 3.1 and 3.2(iii), we get res_{φ} ∘ n_{X_{Q}} = n_{_{φ}X_{Q}} = n_{_{R}X_{Q}} = res_{R}^{P} ∘ n_{X_{Q}}.

(ii) follows from 3.3.

(iii) Any Q-Q-orbit of _{Q}X_{Q} outside Y is isomorphic to Q ×_{R} _{ψ}Q for some proper

subgroup R of Q and some group homomorphism ψ : R → Q in F_{G,b}, from which the statement follows.

(iv) The number of Q-Q-orbits in Y is equal to |Y |/|Q|, and any such orbit is isomorphic to _{φ}Q for some automorphism φ of Q in F_{G,b}. Moreover, for any ζ ∈ H^{·}(Q, k)^{W (Q)} we have then n_{_{φ}Q}(ζ) = ζ, from which the result follows. □

We apply this to translate [4, 5.6.2] to block cohomology.

Proposition 3.5. Let X be a finite P -P -biset fulfilling the conditions in 3.2. Let E be an elementary abelian subgroup of P and let σ_{E} be a homogeneous element in H^{·}(E, k) satisfying res_{F}^{E}(σ_{E}) = 0 for any proper subgroup F of E. Let Y be the E-E-subbiset of _{E}X_{E} which is the union of all E-E-orbits of length |E|. Write |Y |/|E| = p^{a}m for some nonnegative intergers a, m, such that (p, m) = 1.

(i) For any η ∈ H^{·}(E, k)^{W (E)} there is η′ ∈ H^{·}(G, b) such that r_{E}(η′) = (σ_{E} · η)^{p^{a}}. (ii) There is an element ρ_{E} ∈ H^{·}(G, b) such that r_{E}(ρ_{E}) = (σ_{E})^{p^{a}} and such that

r_{F} (ρ_{E}) = 0 whenever F is an elementary abelian subgroup of P such that no Gconjugate of (E, e_{E}) is containend in (F, e_{F}).

Proof. (i) We may assume that η is homogeneous. Set ζ = n_{X_{E}}(1 + σ_{E} · η). By 3.4, we have ζ ∈ H^{·}(G, b). Moreover, r_{E}(ζ) = n_{_{E}X_{E}}(1 + σ_{E} · η) = n_{Y} (1 + σ_{E} · η) = (1 + σ_{E} · η)^{p^{a}m} = (1 + (σ_{E} · η)^{p^{a}})^{m} = 1 + m(σ_{E} · η)^{p^{a}} + τ, where τ is a sum of elements of degree stricly bigger than deg((σ_{A} · η)^{p^{a}}) = p^{a} · deg(σ_{E} · η). Define η′ to be the homogeneous part of ζ in degree p^{a} · deg(σ_{E} · η), divided by m.

(ii) Applying (i) to η = 1 yields a homogeneous element ρ_{E} ∈ H^{·}(G, b) such that r_{E}(ρ_{E}) = (σ_{E})^{p^{a}}. By the construction in (i), ρ_{E} is a scalar multiple of the homogenous part of n_{X_{E}}(1+σ_{E}) in degree p^{a} ·deg(σ_{E}). Let F be another elementary abelian subgroup of P. Then r_{F} (ρ_{E}) is a scalar multiple of the homogeneous part of n_{_{F} X_{E}}(1 + σ_{E}) in degree p^{a} · deg(σ_{E}). If (E, e_{E}) has no G-conjugate contained in (F, e_{F}), then the biset _{F} X_{E} is a union of transitive bisets of the form F ×_{H} _{ψ}E, where

H is a subgroup of F of order smaller than |E|, and where ψ : H → E is an injective group homomorphism. Thus n_{_{F} X_{E}}(σ_{E}) = 0, and so r_{F} (ρ_{E}) = 0. □

4 The Quillen stratification of V_{G,b}(M)

Throughout this section, let G be a finite group, let b be a block of kG and let P be a defect group of b. Choose a source idempotent i ∈ (kGb)^{P} and denote, for any subgroup Q of P, by e_{Q} the unique block of kC_{G}(Q) satisfying Br_{Q}(i)e_{Q} = Br_{Q}(i). For any subgroup Q of P, the graded algebra homomorphism r_{Q} (which is the inclusion H^{∗}(G, b) ↪ H^{∗}(P, k) followed by the restriction map res_{Q}^{P} : H^{∗}(P, k) → H^{∗}(Q, k)) induces a morphism of varieties

r_{Q}^{∗} : V_{Q} ⟶ V_{G,b} .

Recall that since H^{∗}(P, k) is Noetherian over H^{∗}(G, b) by [16, 4.3], the morphism r_{P}^{∗} = ι^{∗} : V_{P} → V_{G,b} is finite surjective. We will try to follow as closely as possible the lines of the presentation given in Benson [5, 5.6]; there are two major technical adjustments: the extensive use of Puig’s notion of local pointed groups [18] (for which we refer again to the account given in Thévenaz [21]) and the application of the Evens norm map with respect to a biset fulfilling the conditions in 3.2.

Definition 4.1. Let M be a finitely generated kGb-module. For any local pointed group Q_{δ} on kGb, we define the following subvarieties of V_{Q}:

V_{Q}^{+} = V_{Q} − ∪_{R<Q} (res_{R}^{Q})^{∗}(V_{R}), V_{Q}^{+}(iM) = V_{Q}(iM) ∩ V_{Q}^{+};

V_{Q_{δ}}(M) = V_{Q}(jM), where j ∈ δ, V_{Q_{δ}}^{+}(M) = V_{Q_{δ}}(M) ∩ V_{Q}^{+}.

Furthermore, we define the following subvarieties of V_{G,b}:

V_{G,Q} = r_{Q}^{∗}(V_{Q}), V_{G,Q}^{+} = r_{Q}^{∗}(V_{Q}^{+});

V_{G,Q}(M) = r_{Q}^{∗}(V_{Q}(iM)), V_{G,Q}^{+}(M) = r_{Q}^{∗}(V_{Q}^{+}(iM)).

V_{G,Q_{δ}}(M) = r_{Q}^{∗}(V_{Q_{δ}}(M)), V_{G,Q_{δ}}^{+}(M) = r_{Q}^{∗}(V_{Q_{δ}}^{+}(M)).

Finally, we set

W (Q) = N_{G}(Q, e_{Q})/QC_{G}(Q) and W (Q_{δ}) = N_{G}(Q_{δ})/QC_{G}(Q).

Theorem 4.2. Let M be a finitely generated kGb-module.

(i) The variety V_{G,b}(M) is the disjoint union of the locally closed subvarieties V_{G,E}^{+}(M), where E runs over a set of subgroups of P such that (E, e_{E}) runs over a set of representatives of the G-conjugacy classes of those b-Brauer pairs contained in (P, e_{P}) for which E is elementary abelian and C_{P} (E) is a defect group of the block e_{E}.

(ii) Let E be an elementary abelian subgroup of P such that C_{P} (E) is a defect group of e_{E}. The group W (E) acts on the variety V_{E}^{+}(iM), and r_{E}^{∗} induces an inseparable isogeny V_{E}^{+}(iM)/W (E) → V_{G,E}^{+}(M).

(iii) Let E be an elementary abelian subgroup of P such that C_{P} (E) is a defect group of e_{E}. Then V_{G,E}^{+}(M) is the union of the subvarieties V_{G,E_{δ}}^{+}(M), with δ running over the set of local points of E on kGb such that E_{δ} ⊆ P_{γ}.

If one specialises the above theorem to the principal block of kG, the statements (i) and (ii) in 4.2 are equivalent to the Quillen stratification due to Avrunin and Scott [3]. Since a subgroup E of P can have more than one local point on kGb, statement (iii) gives some additional information on the subvarieties V_{G,E}^{+}(M).

If one specialises 4.2 to the case where M is indecomposable with P as vertex and a source of dimension prime to p, then, by 2.4, we have V_{G,b}(M) = V_{G,b}, and thus 4.2 yields a stratification for the block variety V_{G,b}. The following Proposition describes in that case the subvarieties V_{G,b}^{+}(M) more precisely:

Proposition 4.3. Let M be a finitely generated indecomposable kGb-module with P as vertex and a source of dimension prime to p.

(i) For any subgroup Q of P we have V_{Q}(iM) = V_{Q} and V_{Q}^{+}(iM) = V_{Q}^{+}.

(ii) For any subgroup Q of P we have V_{G,Q}(M) = V_{G,Q} and V_{G,Q}^{+}(M) = V_{G,Q}^{+}.

Proof. By [13, 6.1], some indecomposable direct summand of iM as kP -module is a source of M. Thus, for any subgroup Q of P, the restriction of iM to kQ has a direct summand of dimension prime to p, by the assumptions. But then V_{Q}(iM) = V_{Q} (cf. [5, 5.8.5]), and the second equality in (i) follows from the first. The two equalities in (ii) follow from applying r_{Q}^{∗} to the equalities in (i). □

Combining 4.2 and 4.3 yields the obvious analogue for block cohomology of Quillen’s stratification in [19, 20]. We break up the proof of 4.2 into a series of Lemmas; we keep the notation introduced above.

Lemma 4.4. We have V_{G,b}(M) = ∪_{E} r_{E}^{∗}(V_{E}(iM)), where E runs over the set of elementary abelian subgroups of P.

Proof. By 2.1, we have V_{G,b}(M) = ι^{∗}(V_{P} (iM)). Thus the Lemma follows from [5, 5.7.4] applied to P and iM instead of G and M, respectively. □

Lemma 4.5. For any subgroup Q of P and any idempotent i′ ∈ (kGb)^{Q} we have V_{Q}(i′M) = ∪_{R_{ϵ}} (res_{R}^{Q})^{∗}(V_{R_{ϵ}}(M)), where R_{ϵ} runs over the set of local pointed groups

on i′kGi′ such that R ⊆ Q.

Proof. Choose a primitive decomposition J of i′ in (kGb)^{Q}. That is, i′ = ∑_{j∈J} j, and

the elements of J are pairwise orthogonal primitive idempotents in (kGb)^{Q}. Thus i′M = ⊕_{j∈J} jM as direct sum of kQ-modules, and hence V_{Q}(i′M) = ∪_{j∈J} V_{Q}(jM).

Let j ∈ J. Then the conjugacy class of j in ((i′kGi′)^{Q})^{×} is a (not necessarily local) point δ of Q on i′kGi′. Let R_{ϵ} be a defect pointed group of Q_{δ}. By [17, Cor. 1] (see also [21, (23.1)]), this means that there is l ∈ ϵ such that j = Tr_{R}^{Q}(l) and such that the different Q-conjugates ulu^{−1} of l are pairwise orthogonal as u runs over a set of representatives of the right R-cosets in Q. Thus jM ≅ Ind_{R}^{Q}(lM), and therefore V_{Q}(jM) = (res_{R}^{Q})^{∗}(V_{R}(lM)), from which the Lemma follows. □

Lemma 4.6. For any local pointed group Q_{δ} on kGb we have

V_{Q_{δ}}^{+}(M) = V_{Q_{δ}}(M) − ∪_{R_{ϵ}} (res_{R}^{Q})^{∗}(V_{R_{ϵ}}(M)) ,

where R_{ϵ} runs over the set of local pointed groups on kGb properly contained in Q_{δ}.

Proof. By [5, 5.7.7] we have (res_{R}^{Q})^{∗}(V_{R}(jM)) = (res_{R}^{Q})^{∗}(V_{R}) ∩ V_{Q}(jM), where j ∈ δ and where R is any subgroup of Q. Thus V_{Q_{δ}}^{+}(M) = V_{Q}^{+} ∩ V_{Q}(jM) = V_{Q}(jM) − ∪_{R<Q} ((res_{R}^{Q})^{∗}(V_{R})∩V_{Q}(jM)) = V_{Q}(jM)− ∪_{R<Q} (res_{R}^{Q})^{∗}(V_{R}(jM)), and now the Lemma

follows from 4.5 applied to the varieties V_{R}(jM) appearing in the last expression. □

Lemma 4.7. Let Q_{δ}, R_{ϵ} be local pointed groups on kGb contained in P_{γ}. If Q_{δ} and R_{ϵ} are G-conjugate, then V_{G,Q_{δ}}(M) = V_{G,R_{ϵ}}(M) and V_{G,Q_{δ}}^{+}(M) = V_{G,R_{ϵ}}^{+}(M).

Proof. Let x ∈ G such that R_{ϵ} = ^{x}(Q_{δ}). Then the group isomorphism φ : Q → R mapping u ∈ Q to xux^{−1} has the property that res_{Q}^{P}(ζ) = res_{φ}(ζ) for all ζ ∈ H^{∗}(G, b), and thus the morphisms r_{Q}^{∗} ∘ (res_{φ})^{∗} and r_{R}^{∗} from V_{R} to V_{G,b} are equal. Thus V_{G,Q_{δ}}(M) = V_{G,R_{ϵ}}(M). The second equality is clear. □

Lemma 4.8. We have V_{G,b}(M) = ∪_{E_{δ}} V_{G,E_{δ}}^{+}(M), where E_{δ} runs over a set of repre-

sentatives of the set of G-conjugacy classes of local pointed groups on kGb such that E_{δ} ⊆ P_{γ} and such that E is elementary abelian.

Proof. By 4.4 and 4.6, the variety V_{G,b}(M) is the union of the subvarieties V_{G,E_{δ}}^{+}(M), where E_{δ} runs over the set of all local pointed groups on kGb contained in P_{γ} such that E is elementary abelian. The Lemma follows now from 4.7. □

Lemma 4.9. For any subgroup Q of P we have V_{Q}^{+}(iM) = ∪_{δ} V_{Q_{δ}}^{+}(M) and

V_{G,Q}^{+}(M) = ∪_{δ} V_{G,Q_{δ}}^{+}(M) , where δ runs over the set of local points of Q on kGb such that Q_{δ} ⊆ P_{γ}.

Proof. By 4.5 we have V_{Q}(iM) = ∪_{R_{ϵ}} (res_{R}^{Q})^{∗}(V_{R_{ϵ}}(M)), where R_{ϵ} runs over the set of

local pointed groups on kGb such that R_{ϵ} ⊆ P_{γ} and R ⊆ Q. Intersecting with V_{Q}^{+} yields the first equality, and applying r_{Q}^{∗} yields the second equality. □

Proposition 4.10. We have V_{G,b}(M) = ∪_{E} V_{G,E}^{+}(M), and in particular, V_{G,b} =

∪_{E} V_{G,E}^{+}, where E runs over a set of subgroups of P such that (E, e_{E}) runs over a set

of representatives of the G-conjugacy classes of b-Brauer pairs contained in (P, e_{P}) for which E is elementary abelian and C_{P} (E) is a defect group of e_{E}.

Proof. Any b-Brauer pair is G-conjugate to a b-Brauer pair of the form (Q, e_{Q}) for some subgroup Q of P such that C_{P} (Q) is a defect group of e_{Q} (see [1, 4.5]). Thus the first equality follows from combining 4.8 and 4.9, and the second equality follows from the first and 4.3. □

Lemma 4.11. Let Q be a subgroup of P such that C_{P} (Q) is a defect group of the block e_{Q}. The action of N_{G}(Q, e_{Q}) on H^{∗}(Q, k) induces an action of W (Q) on V_{Q} and V_{Q}^{+}, which preserves the subvariety V_{Q}^{+}(iM).

Proof. Since QC_{G}(Q) acts trivially on H^{∗}(Q, k), the action of N_{G}(Q_{δ}) induces an action of W (Q_{δ}) on V_{Q}. This action preserves V_{Q}^{+}. The action of the group N_{G}(Q, e_{Q}) permutes the set of all local points of Q on kGb satisfying Br_{Q}(δ)e_{Q} ≠ 0. Since C_{P} (Q) is a defect group of e_{Q}, by [14, 3.3(iii)], for any local point δ of Q on kGb such that Br_{Q}(δ)e_{Q} ≠ 0 we have Q_{δ} ⊆ P_{γ}. Thus N_{G}(Q, e_{Q}) acts on the set of local points δ of Q on kGb fulfilling Q_{δ} ⊆ P_{γ}, and now 4.11 follows from 4.9. □

We have set up the machinery in such a way that the rest of the proof of 4.2 follows now exactly that of [5, 5.6.3].

Proof of Theorem 4.2. Let E be an elementary abelian subgroup of P such that C_{P} (E) is a defect group of e_{E}. By the argument in [5, 5.6], there is a homogeneous element σ_{E} ∈ H^{·}(E, k)^{W (E)} such that V_{E}^{+} consists of all maximal ideals in H^{∗}(E, k) not containing σ_{E}, and such that res_{F}^{E}(σ_{E}) = 0 for any proper subgroup F of E. Thus V_{E}^{+} can be identified to the maximal ideal spectrum of the algebra H^{·}(E, k)[σ_{E}^{−1}], obtained from localising H^{·}(E, k) at σ_{E}. By [5, 5.4.8], the quotient V_{E}^{+}/W (E) can be identified to the maximal ideal spectrum of (H^{·}(E, k)[σ_{E}^{−1}])^{W (E)}. Let ρ_{E} be the element in H^{·}(G, b) fulfilling 3.5(iv). Then V_{G,E}^{+} consists of all maximal ideals in H^{·}(G, b) containing ker(r_{E}) and not containing ρ_{E}. Since r_{E} maps ρ_{E} to a power of σ_{E}, r_{E} induces an algebra homomorphism

H^{·}(G, b)[ρ_{E}^{−1}] ⟶ (H^{·}(E, k)[σ_{E}^{−1}])^{W (E)}

such that, by 3.5, the image of this homomorphism contains a p^{a}-th power of every element in (H^{·}(E, k)[σ_{E}^{−1}])^{W (E)}. Upon taking varieties, this is equivalent to saying that r_{E}^{∗} induces an inseparable isogeny V_{E}^{+}/W (E) → V_{G,E}^{+}. Since W (E) acts on the subvariety V_{E}^{+}(iM) by 4.11, passing down to subvarieties proves (ii).

If F is another elementary abelian subgroup of P such that C_{P} (F) is a defect group of e_{F} and such that (F, e_{F}) contains no G-conjugate of (E, e_{E}), then ρ_{F} ∈ ker(r_{E}) by 3.5. By the above description of V_{G,E}^{+}, it follows that V_{G,E}^{+} and V_{G,F}^{+} are disjoint. Since V_{G,E}^{+}(M) is a subvariety of V_{G,E}^{+}, this concludes the proof of (i). Finally, statement (iii) is a particular case of 4.9. □

5 An example

Let p be a prime such that p ≥ 5 and let k be a field of characteristic p containing a primitive 3^{rd} root of unity. For any positive integer n denote by C_{n} a cyclic group of order n. Let Q be a finite non trivial abelian p−group and set P = Q × Q. We consider the group

G = (C_{3} × P) ⋊ C_{2} ,

where the non trivial element t of C_{2} acts on C_{3} × P by inverting the elements of C_{3} and by exchanging the two factors Q of P; that is, (u, v)^{t} = (v, u) for any (u, v) ∈ Q × Q = P.

Since the Sylow-p-subgroup P of G is normal, P is the defect group of any block of kG. The blocks of kG correspond bijectively to the G−orbits of blocks of kC_{G}(P). Since C_{G}(P) = C_{3}×P, the algebra kC_{G}(P) has three p−blocks e_{0}, e, e′, corresponding to the linear characters ζ_{0}, ζ, ζ′ of C_{3} with values in k, where we choose notation such that e_{0} is the principal block and hence ζ_{0} is the trivial character of C_{3}. Then e_{0} is G−stable, while the two blocks e, e′ get permuted by the action of the involution t.

It follows that kG has exactly two blocks, namely the principal block b_{0} = e_{0} and a unique non principal block b = e + e′.

The structure of the principal block b_{0} of kG is as follows: the canonical map G → P ⋊ C_{2} with kernel C_{3} = O_{p′}(G) induces an algebra isomorphism kGb_{0} ≅ k(P ⋊ C_{2}).

The non principal block b of kG is a nilpotent block (cf. [8]): since t permutes e and e′, we have N_{G}(P, e) = C_{G}(P). The pairs (P, e) and (P, e′) are exactly the two maximal b−Brauer pairs. The idempotents e, e′ are in fact source idempotents; that is, they remain primitive in (kGb)^{P}. To see this, observe first that e and e′ = e^{t} are orthogonal, and therefore ete = 0. Thus ekGe = kC_{G}(P)e. Since kC_{3}e ≅ k we have kC_{G}(P)e ≅ kP. This shows not only, that e is primitive in (kGb)^{P}, but also that the source algebra ekGe of b is isomorphic to kP. The same argument works for e′.

Since b is nilpotent, in particular the block cohomology of b is isomorphic to H^{∗}(P, k).

We define an indecomposable kGb−module M as follows. Consider kQ ⊗_{k} k as

kP −module through the canonical isomorphism kP ≅ kQ ⊗_{k} kQ. Extend kQ ⊗_{k} k to a

kC_{G}(P)−module by letting act any element c ∈ C_{3} as multiplication with the scalar ζ(c). In this way, kQ ⊗_{k} k becomes an indecomposable kC_{G}(Q)−module belonging to

the block e. Set

M = Ind_{C_{G}(P)}^{G}(kQ ⊗_{k} k) .

By Mackey’s formula, Res_{C_{G}(P)}^{G}(M) ≅ (kQ ⊗_{k} k) ⊕ ^{t}(kQ ⊗_{k} k) ≅ (kQ ⊗_{k} k) ⊕ (k ⊗_{k}

kQ). Since t exchanges these two summands, which are both indecomposable as kC_{G}(P)−modules, it follows that M is indeed indecomposable. Moreover, as kP −modules, we have

eM ≅ kQ ⊗_{k} k and e′M ≅ k ⊗_{k} kQ .

Through the Künneth isomorphism H^{∗}(P, k) ≅ H^{∗}(Q, k) ⊗_{k} H^{∗}(Q, k), the ideal

H^{+}(Q, k) ⊗_{k} H^{∗}(Q, k) is the annihilator of Ext_{kP}^{∗} (eM, eM), while the annihilator of

Ext_{kP}^{∗} (e′M, e′M) is H^{∗}(Q, k) ⊗_{k} H^{+}(Q, k), where H^{+}(Q, k) denotes the ideal gener-

ated by the elements of positive degree in H^{∗}(Q, k). Thus V_{P} (eM) and V_{P} (e′M) are different subvarieties of H^{∗}(P, k) = H^{∗}(G, b), or equivalently, V_{P} (M) ≠ V_{P} (eM).

References

1. J. L. Alperin, M. Broué, Local methods in block theory, Ann. Math. 110 (1979), 143–157.

2. J. L. Alperin, L. Evens, Representations, resolutions and Quillen’s dimension theorem, J. Pure Appl. Algebra 22 (1981), 1–9.

3. G. S. Avrunin, L. L. Scott, Quillen stratification for modules, Invent. Math. 66 (1982), 277–286.

4. D. J. Benson, Representations and Cohomology I, Cambridge University Press, Cambridge, UK, 1991.

5. D. J. Benson, Representations and Cohomology II, Cambridge University Press, Cambridge, UK, 1991.

6. C. Broto, R. Levi, R. Oliver, The homotopy theory of fusion systems, preprint (2001).

7. M. Broué, L. Puig, Characters and local structure in G-algebras, J. Algebra 63 (1980), 306–317.

8. M. Broué, L. Puig, A Frobenius theorem for blocks, Invent. Math. 56 (1980), 117–128.

9. J. F. Carlson, The complexity and varieties of modules, in: Lecture Notes in Mathematics 882 (1981), 415–422.

10. J. F. Carlson, Varieties and the cohomology ring of a module, J. Algebra 85 (1983), 104–143.

11. H. Cartan, S. Eilenberg, Homological Algebra, Princeton University Press, 1956.

12. L. Evens, The cohomology ring of a finite group, Trans. Amer. Math. Soc. 101 (1961), 224–239.

13. M. Linckelmann, The source algebras of blocks with a Klein four defect group, J. Algebra 167 (1994), 821–854.

14. M. Linckelmann, On derived equivalences and local structure of blocks of finite groups, Turkish J. Math. 22 (1998), 93–107.

15. M. Linckelmann, Transfer in Hochschild cohomology of blocks of finite groups, Algebras Rep. Theory 2 (1999), 107–135.

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

17. L. Puig, Sur un Théorème de Green, Math. Z. 166 (1979), 117–129.

18. L. Puig, Pointed groups and construction of characters, Math. Z. 176 (1981), 265–292.

19. D. Quillen, The spectrum of an equivariant cohomology ring I, Ann. Math. 94 (1971), 549–572.

20. D. Quillen, The spectrum of an equivariant cohomology ring II, Ann. Math. 94 (1971), 573–602.

21. J. Thévenaz, G-algebras and Modular Representation Theory, Oxford Science Publications, Clarendon, Oxford, 1995.

22. B. B. Venkov, Cohomology algebras for some classifying spaces, (Russian) Dokl. Akad. Nauk. SSSR 127 (1959), 943–944.

23. B. B. Venkov, Characteristic classes for finite groups, (Russian) Dokl. Akad. Nauk. SSSR 137 (1961), 1274–1277.

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

FRANCE