VERTEX AND SOURCE DETERMINE THE BLOCK

VARIETY OF AN INDECOMPOSABLE MODULE

David J. Benson and Markus Linckelmann

Abstract. The block variety V_{G,b}(M) of a finitely generated indecomposable module M over the block algebra of a p-block b of a finite group G, introduced in [15], can be computed in terms of a vertex and a source of M. We use this to show that V_{G,b}(M) is connected, and that every closed homogeneous subvariety of the affine variety V_{G,b} defined by block cohomology H^{∗}(G, b) (cf. [14]) is the variety of a module over the block algebra. This is analogous to the corresponding statements on Carlson’s cohomology varieties in [8].

1. Introduction

The theory of the cohomology variety V_{G}(M) for a finitely generated module M over the modular group algebra of a finite group G was developed by Carlson [6], [7],

[8] and others back in the nineteen eighties. It has played a major role in recent years in the modular representation theory of finite groups. One of the problems with the theory has been that it has been much more fruitful as a way of proving theorems about the principal block than about more general blocks.

To address this problem, the second author [15] recently introduced the concept of the block variety V_{G,b}(M) of a finitely generated module M for a block b of a finite group G. The definition of the ambient variety V_{G,b} involves a variation of the Cartan–Eilenberg stable element method in ordinary cohomology for the Brauer pairs in the block. But the definition of V_{G,b}(M) involves going through the Hochschild cohomology of the block as an intermediate step. In [17], the dependence on Hochschild cohomology for the calculation of block varieties is removed. It is shown in that paper that the block variety of M is the image in V_{G,b} of the variety at the level of the defect group. But first, the module must be cut down using the idempotent for the source algebra of the block, in the sense of Puig. An example is given there, to show that this step is really necessary.

In this paper, we describe how to calculate the block variety of a module in terms of the vertex and source of the module. We show that it is given as the image in V_{G,b} of the variety of the source of M as a module for the vertex. Actually, there is a subtlety here: the vertex and source are only well defined up to G-conjugacy, and we must be careful how we choose a P -conjugacy class.

Taken together with the stratification theorem for V_{G,b} [17] and Carlson’s theory of rank varieties, our theorem implies that V_{G,b}(M) can be calculated without knowing anything about cohomology, just as was the case for V_{G}(M).

Our theorem enables us to prove some basic properties for V_{G,b}(M), including the analog of Carlson’s theorem [8] that the variety of an indecomposable module is

Typeset by AMS-TEX 1

connected. We also show that every closed homogeneous subvariety of V_{G,b} can be realised as V_{G,b}(N) for a suitably chosen module N. Finally, we show that the block variety of a module survives the Green correspondence with respect to a subgroup which controls fusion of Brauer pairs in the block.

Throughout this paper, k is an algebraically closed field of prime characteristic p. Let G be a finite group. A block of G is a primitive idempotent b in Z(kG). Following Alperin–Broué [1], a b-Brauer pair is a pair (Q, e) consisting of a p-subgroup Q of G and a block e of C_{G}(Q) such that Br_{Q}(b)e = e, where Br_{Q} : (kG)^{Q} → C_{G}(Q) is the Brauer homomorphism [4]. It is shown in [1], that the set of b-Brauer pairs is a partially ordered G-set such that G acts transitively on the set of maximal b-Brauer pairs. If (P, e) is a maximal b-Brauer pair, then P is called a defect group of b (this notion is due to Brauer, although he did not state it in these terms). If P is a defect group of b, then P is maximal with respect to Br_{P} (b) ≠ 0; in particular, there is a primitive idempotent i ∈ (kGb)^{P} such that Br_{P} (i) ≠ 0. Such an idempotent is called a source idempotent of b (with respect to P). By Broué–Puig [5, Theorem 1.8], for any subgroup Q of P, there is a unique block e_{Q} of C_{G}(Q) such that Br_{Q}(i)e_{Q} ≠ 0. Then (Q, e_{Q}) is a b-Brauer pair, and e_{Q} is also the unique block of C_{G}(Q) such that (Q, e_{Q}) ⊆ (P, e_{P}).

We denote by F_{G,b} the category whose objects are the subgroups of P, and whose morphisms are the sets of group homomorphisms φ : Q → R for which there exists an element x ∈ G satisfying ^{x}(Q, e_{Q}) ⊆ (R, e_{R}) and φ(u) = xux^{−1} for all u ∈ Q. The block cohomology, introduced in [14], is defined by

H^{∗}(G, b) = lim_{⟵}_{Q∈F_{G,b}} H^{∗}(Q, k) .

We identify H^{∗}(G, b) with the graded subalgebra of all “stable elements” ζ ∈ H^{∗}(P, k) satisfying res_{R}^{P}(ζ) = res_{φ}(ζ) for any morphism φ : R → P belonging to the category F_{G,b}.

The algebra H^{∗}(G, b) is finitely generated graded commutative, and we denote by V_{G,b} the associated affine variety, called the block variety of b (cf. [15]). Following [15], for any finitely generated kGb-module M, there is a canonical graded algebra homomorphism H^{∗}(G, b) → Ext_{kGb}^{∗}(M, M), defined using Hochschild cohomology of the block, whose kernel detemines the subvariety V_{G,b}(M) of V_{G,b}. If b is the principal block, then H^{∗}(G, b) ≅ H^{∗}(G, k), and we write then V_{G} and V_{G}(M) instead of V_{G,b} and V_{G,b}(M), respectively (that is, in the principal block case, the block varieties coincide with Carlson’s cohomology varieties, introduced in [6], [7]).

For any subgroup Q of P, the inclusion H^{∗}(G, b) ⊆ H^{∗}(P, k) followed by the restriction res_{Q}^{P} : H^{∗}(P, k) → H^{∗}(Q, k) is a graded algebra homomorphism, denoted by r_{Q} : H^{∗}(G, b) → H^{∗}(Q, k). We denote by r_{Q}^{∗} : V_{Q} → V_{G,b} the induced map on varieties. We refer to Benson [3] and Evens [10] for background material on cohomology varieties of modules and Thévenaz [20] for block theory.

Theorem 1.1. Let G be a finite group, let b be a block of G, let P be a defect group of b, and let i ∈ (kGb)^{P} be a source idempotent. Let M be an indecomposable kGbmodule. There is a vertex Q of M contained in P and a source U of M such that U is a direct summand of Res_{Q}^{P}(iM) and such that M is isomorphic to a direct summand of kGi ⊗_{kQ} U. For any such choice of a vertex and source of M we have

V_{G,b}(M) = r_{Q}^{∗}(V_{Q}(U)).

The proof of Theorem 1.1 will be given in Section 3.

Corollary 1.2. For a finitely generated indecomposable kGb-module M, the variety V_{G,b}(M) is connected.

Proof. The variety V_{Q}(U) is connected by Carlson’s theorem [8], and hence so is its image under the map r_{Q}^{∗}, which is V_{G,b}(M) by Theorem 1.1. □

Corollary 1.3. Every closed homogeneous subvariety V of V_{G,b} is of the form V_{G,b}(N) for some finitely generated kGb-module N.

Proof. The inverse image in V_{P} of V under the map r_{P}^{∗} : V_{P} → V_{G,b} is a closed homogeneous subvariety of V_{P}; thus, by Carlson [8], there is a finitely generated kP - module U such that r_{P}^{∗} (V_{P} (U)) = V. Using Green’s indecomposability theorem, we may write U = ⨁_{j∈J} Ind_{Q_{j}}^{P}(U_{j}), where J is a finite indexing set, and where for any j ∈ J, Q_{j} is a subgroup of P and U_{j} is an indecomposable kQ_{j}-module having Q_{j} as vertex. When viewed as kQ_{j}-kQ_{j}-bimodule, ikGi has a direct summand isomorphic to kQ_{j}, where j ∈ J (see e.g. [16, 6.1(iii)]). Thus for any j ∈ J the left kGb-module kGi ⊗_{kQ_{j}} U_{j} has an indecomposable direct summand M_{j} such that the restriction of

iM_{j} to kQ_{j} has U_{j} as direct summand. Note that the kGb-module kGi ⊗_{kQ_{j}} U_{j} is a

direct summand of kG ⊗_{kQ_{j}} U_{j} = Ind_{Q_{j}}^{G}(U_{j}). Thus M_{j} has Q_{j} as vertex and U_{j} as

source, and V_{G,b}(M_{j}) = r_{Q_{j}}^{∗}(V_{Q_{j}}(U_{j})) = r_{P}^{∗} (V_{P} (Ind_{Q_{j}}^{P}(U_{j}))) by Lemma 2.2 below. Thus N = ⨁_{j∈J} M_{j} satisfies V_{G,b}(N) = V. □

The next corollary states that the block variety of a module survives the Green correspondence with respect to a subgroup which controls fusion of Brauer pairs.

Corollary 1.4. Suppose the vertex Q of M is normal P. Set H = N_{G}(Q) and denote by c the unique block of H such that e_{Q}c = e_{Q}. If F_{G,b} = F_{H,c} then V_{G,b}(M) = V_{H,c}(f(M)), where f(M) is the Green correspondent of M.

Proof. Since Q is normal in P, the pair (P, e_{P}) is also a maximal c-Brauer pair. The condition F_{G,b} = F_{H,c} implies the equality H^{∗}(G, b) = H^{∗}(H, c) as graded subalgebras of H^{∗}(P, k). It is well known that the Green correspondent f(M) of M belongs to the block c of H. Since every vertex and source of f(M) is also a vertex and source of M, the statement follows from Theorem 1.1. □

Note that if Q = P is abelian, we have F_{G,b} = F_{H,c}, because in blocks with abelian defect groups, the normaliser of a defect group controls fusion (cf. Alperin– Broué [1]). Hence Corollary 1.4 applies in that situation. This particular case of 1.4 has independently been proved by H. Kawai [12], who also observed [17, 2.1] (cf. 2.3 below).

2. Auxiliary results

Lemma 2.1. Let G be a finite group, let H be a subgroup of G and let M be a finitely generated kG-module. We have (res_{H}^{G})^{∗}(V_{H}(res_{H}^{G}(M))) ⊆ V_{G}(M).

Proof. Let ζ ∈ H^{∗}(G, k) such that ζ ⊗ Id_{M} is zero in Ext_{kG}^{∗}(M, M). Then res_{H}^{G}(ζ) ⊗ Id_{M} is zero in Ext_{kH}^{∗}(Res_{H}^{G}(M), Res_{H}^{G}(M)). Thus the annihilator of Ext_{kG}^{∗}(M, M) in H^{∗}(G, k) is contained in the inverse image in H^{∗}(G, k) of the annihilator in H^{∗}(H, k) of Ext_{kH}^{∗}(Res_{H}^{G}(M), Res_{H}^{G}(M)). Passing to varieties yields the statement. □

Lemma 2.2. Let G be a finite group, let H be a subgroup of G, and let N be a finitely generated kH-module. We have

(res_{H}^{G})^{∗}(V_{H}(N)) = V_{G}(Ind_{H}^{G}(N)).

Proof. This is a standard property of varieties for modules, and can be found for example in Evens [10], Proposition 8.2.4 or Benson [2], Theorem 2.26.9. □

Theorem 2.3. 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 r_{P} : H^{∗}(G, b) → H^{∗}(P, k) induces a finite surjective morphism r_{P}^{∗} : V_{P} → V_{G,b}, and for any finitely generated kGb-module M we have V_{G,b}(M) = r_{P}^{∗} (V_{P} (iM)) , where iM is considered as kP -module.

Proof. See Linckelmann [17], Theorem 2.1 or Kawai [12]. □

It has been shown by Puig [19] that one can recover the category F_{G,b} from the kP - kP -bimodule structure of the source algebra ikGi of a block b (see also Linckelmann [16, 7.7] for a proof). For any two subgroups Q, R of P and any group homomorphism φ : R → Q we denote by _{φ}kQ the kR-kQ-bimodule kQ endowed with the regular action of kQ on the right and the action of kR on the left obtained from restricting the regular action of kQ to kR via φ.

Proposition 2.4. Let G be a finite group, let b be a block of G, let P be a defect group of b and let i ∈ (kGb)^{P} be a source idempotent. Let Q be a subgroup of P. Every indecomposable direct summand of the source algebra ikGi as kP -kQ-bimodule is isomorphic to kP ⊗_{kR}_{φ}kQ for some subgroup R of P and some morphism φ : R → Q

in F_{G,b}.

Finally, we need the following statement from Linckelmann [13], which is about a certain choice of vertices and sources of indecomposable modules (implying that vertices and sources of indecomposable modules can be detected at the source algebra level).

Proposition 2.5. Let G be a finite group, let b be a block of G, let P be a defect group of b and let i ∈ (kGb)^{P} be a source idempotent of b. Any indecomposable kGbmodule M has a vertex Q contained in P and a kQ-source U such that U is a direct summand of Res_{Q}^{P}(iM) and M is a direct summand of kGi ⊗_{kQ} U.

Proof. See 6.3 of Linckelmann [13]. □

3. Proof of Theorem 1.1

By Proposition 2.5 above there is a vertex Q of M and a kQ-source U of M such that U is a direct summand of Res_{Q}^{P}(iM) and such that M is a direct summand of kGi ⊗_{kQ} U. By Proposition 2.4, we can write

ikGi ≅ ⨁_{(R,φ)} kP ⊗_{kR} _{φ}kQ ,

where (R, φ) runs over a family of pairs consisting of a subgroup R of P and a morphism φ : R → Q belonging to the category F_{G,b}.

Then iM is, as kP -module, a direct summand of

ikGi ⊗_{kQ} U = ⨁_{(R,φ)} Ind_{R}^{P}(_{φ}U) ,

where (R, φ) is as before. Thus

V_{P} (iM) ⊆ V_{P} (ikGi ⊗_{kQ} U) = ⋃_{(R,φ)} V_{P} (Ind_{R}^{P}(_{φ}U)) .

By Lemma 2.2, we have V_{P} (Ind_{R}^{P}(_{φ}U)) = (res_{R}^{P})^{∗}(V_{R}(_{φ}U)), for any (R, φ) as above. Using Theorem 2.3 we get

V_{G,b}(M) = r_{P}^{∗} (V_{P} (iM)) ⊆ ⨁_{(R,φ)} r_{R}^{∗}(V_{R}(_{φ}U)) .

For any (R, φ) occurring in this union, we have r_{R}^{∗}(V_{R}(_{φ}U)) = r_{φ(R)}^{∗}(Res_{φ(R)}^{Q}(U)) ⊆ r_{Q}^{∗}(V_{Q}(U)), where the first equality uses the stability of the elements in H^{∗}(G, b) with respect to morphisms in F_{G,b}, and where the second inclusion follows from Lemma 2.1. Together, we get V_{G,b}(M) ⊆ r_{Q}^{∗}(V_{Q}(U)). The other inclusion follows trivially from the fact that U is a direct summand of iM as kQ-module. □

4. A question

A theorem of Hida [11] gives a converse to Carlson’s connectedness theorem. It states that given a connected closed homogeneous subvariety V of V_{G}, there exists an indecomposable finitely generated kG-module N such that V_{G}(N) = V. An obvious question is whether the corresponding statement holds for a connected closed homogeneous subvariety V of the block variety V_{G,b}. One might try to prove this statement by the method used to prove Corollary 1.3. The problem is that it is not at all clear whether there is a connected closed homogeneous subvariety of V_{P} whose image in V_{G,b} is equal to V. An easy modification of the proof of Corollary 1.3 shows that if V is the image of such a subvariety of V_{P} then there exists a finitely generated indecomposable kGb-module N such that V_{G,b}(N) = V.

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David J. Benson University of Georgia Department of Mathematics Athens, Georgia 30602

U.S.A.

Markus Linckelmann Ohio State University Math Tower 231, West 18th Avenue Columbus, Ohio 43210

U.S.A.