ON DERIVED EQUIVALENCES AND LOCAL STRUCTURE OF BLOCKS OF FINITE GROUPS

Markus Linckelmann September 1996

Typeset by AMS-TEX

1 On Rickard equivalences for blocks of finite groups

The types of equivalences between p−blocks of finite groups usually considered - namely Morita equivalences, derived and stable equivalences, as well as perfect isometries on the character level - all admit refinements to equivalences which in addition take into account the local structure of the considered blocks (this is explained in more details in Broué [4]). In particular, the notion of a derived equivalence refines to what we call splendid derived equivalence (called “splendid equivalence” in [14] or “Rickard equivalence with groups” in [4]), developed by Rickard in [14]. The desired compatibility property of a splendid Rickard equivalence with the local structure of the considered blocks is, however, essentially proved in [14] for principal blocks only.

Theorem 1.1 below suggests a slight modification of the definition of a splendid equivalence in [14] in order to get a compatibility for arbitrary blocks. We give our statements, explain them in a series of remarks and refer to the sections 2 and 3 for a more detailed description of the notions of Brauer pairs and Brauer homomorphisms that we use (and the proofs of all statements in this section are given in section 4 ).

Throughout this paper we fix a prime number p, a complete discrete valuation ring O having a residue field k = O/J(O) of characteristic p. By a block of a finite group G we mean a primitive idempotent b of the center Z(OG) of the group algebra OG of G over O and call then OGb the corresponding block algebra of b. Recall that a defect group of such a block b of G is a minimal subgroup P of G such that the obvious map OGb ⊗_{OP} OGb ⟶ OGb induced by multiplication in OGb splits as homomorphism

of OGb − OGb−bimodules. The defect groups of b form a unique conjugacy class of p−subgroups in G.

Theorem 1.1. Let G, H be finite groups, b a block of G, c a block of H, having a common defect group P. Let i ∈ (OGb)^{P} and j ∈ (OHc)^{P} be primitive idempotents such that Br_{P}^{G}(i) ≠ 0 and Br_{P}^{H}(j) ≠ 0. 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, such that Br_{Q}^{G}(i)e_{Q} ≠ 0 and Br_{Q}^{H}(j)f_{Q} ≠ 0.

Assume that for any two subgroups Q, R of P we have E_{G}((Q, e_{Q}), (R, e_{R})) = E_{H}((Q, f_{Q}), (R, f_{R})).

Let X be a Rickard tilting complex of OGb − OHc−bimodules. Assume that all terms of X are sums of direct summands of the OGb−OHc−bimodules OGi ⊗_{OQ}jOH,

where Q runs over the set of subgroups of P.

Then for any subgroup Q of P the complex e_{Q}X(Q)f_{Q} is a Rickard tilting complex of kC_{G}(Q)e_{Q} − kC_{H}(Q)f_{Q}−bimodules.

In the case of principal blocks, 1.1 is equivalent to Rickard’s theorem [14, 4.1] . Remember that if b is the principal block of the finite group G then Br_{Q}(b) = e_{Q} for any subgroup Q of P and e_{Q} is the principal block of kC_{G}(Q) (where the notation is as in 1.1; see e.g. [16, (40.17)(a)]). We restate this for completeness:

Corollary 1.2. (Rickard [14, 4.1]) With the notation and hypotheses of 1.1, suppose that Br_{Q}(b) = e_{Q} and Br_{Q}(c) = f_{Q} for any subgroup Q of P.

Let X be a Rickard tilting complex of OGb − OHc−bimodules all of whose terms are sums of direct summands of the bimodules OGb ⊗_{OQ} OHc, where Q runs over the

set of subgroups of P.

Then for any subgroup Q of P the complex X(Q) is a Rickard tilting complex of kC_{G}(Q)e_{Q} − kC_{H}(Q)f_{Q}− bimodules.

The next theorem connects 1.1 to Broué’s “tentative de définition” of Morita equivalences compatible with the local structure of the blocks [3, 6.3] via a technique due to Rickard (in [14] applied to p−nilpotent groups and in [9] extended to p−solvable groups). See also Puig [13, 7.9] for the stability condition occurring in the theorem below.

Theorem 1.3. Let G, H be finite groups, b a block of G and c a block of H having a common defect group P. Let i ∈ (OGb)^{P} and j ∈ (OHc)^{P} be primitive idempotents such that Br_{P}^{G}(i) ≠ 0 and Br_{P}^{H}(j) ≠ 0. 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, such that Br_{Q}^{G}(i)e_{Q} ≠ 0 and Br_{P}^{H}(j)f_{Q} ≠ 0.

Let V be an endo-permutation OP −module. Assume that there is a bounded complex X_{V} of permutation OP − modules such that X_{V} has homology concentrated in degree zero isomorphic to V and that the complex X_{V}^{∗} ⊗_{O} X_{V} is split as com-

plex of OP −modules with respect to the diagonal action of P. Assume finally that φX_{V} ≅ Res_{Q}^{P}(X_{V}) for any subgroup Q of P and any injective group homomorphism φ : Q ⟶ P such that φ̃ ∈ E_{G}((Q, e_{Q}), (P, e_{P})) ∪ E_{H}((Q, f_{Q}), (P, f_{P})).

If M is a direct summand of the OGb−OHc−bimodule OGi ⊗_{OP} Ind_{ΔP}^{P ×P} (V) ⊗_{OP} jOH

which induces a Morita equivalence between OGb and OHc then there is a direct summand X of the complex of OGb − OHc−bimodules OGi ⊗_{OP} Ind_{ΔP}^{P ×P} (X_{V}) ⊗_{OP} jOH

such that X is a Rickard tilting complex having homology concentrated in degree zero isomorphic to M, and in particular, all terms of X are isomorphic to direct sums of direct summands of the modules OGi ⊗_{OQ}jOH, where Q runs over the set of subgroups

of P.

Moreover, for any two subgroups Q, R of P we have E_{G}((R, e_{R}), (Q, e_{Q})) = E_{H}((R, f_{R}), (Q, f_{Q})).

Remarks.

1.4 With the notation of 1.1, the algebras A = iOGi and B = jOHj are source algebras of OGb and OHc, a concept due to Puig[10]. They are always considered as interior P −algebras; that is, together with the group homomorphisms P ⟶ A^{×} and P ⟶ B^{×} mapping u ∈ P to ui and uj, respectively. Recall from [10] that OGb and A are Morita equivalent via the bimodules OGi and iOG. However, kC_{G}(Q)e_{Q} and A(Q) need not be Morita equivalent for all subgroups Q of P. A sufficient condition for kC_{G}(Q)e_{Q} and A(Q) to be Morita equivalent is that C_{P} (Q) is a defect group of e_{Q} ; this condition can always be fulfilled by replacing Q by a suitable G−conjugate in P (see 3.3 below for details).

1.5 If Q is a subgroup of P, the idempotent i need no longer be primitive in (OGb)^{Q}, and hence Br_{Q}^{G}(i) need not be a primitive idempotent in kC_{G}(Q), so the fact that there is a unique block e_{Q} of kC_{G}(Q) satisfying Br_{Q}^{G}(i)e_{Q} ≠ 0 is non trivial; this is due to Broué and Puig [5, 1.8].

1.6 Recall from [3] that for any two symmetric O−algebras A, B a Rickard tilting complex T of A−B−bimodules (called split endomorphism tilting complex in [14]) is a bounded complex of finitely generated A−B−bimodules whose terms are projective as left and right modules, such that the total complexes T^{∗} ⊗_{A}T and T ⊗_{B}T^{∗} are homotopy

equivalent to B and A as complexes of B − B−bimodules and A − A−bimodules, respectively. Note that in 1.1 for any subgroup Q of P the OGb − OHc−bimodule OGi ⊗_{OQ} jOH is projective as left OGb−module and as right OHc−module, so this

condition for the terms of X is redundant.

1.7 With the notation of 1.1 , the indecomposable direct summands of OGi ⊗_{OQ}jOH

with Q running over the set of subgroups of P , viewed as O(G × H)−modules, are trivial source modules with vertex contained in ΔP = {(u, u)}_{u∈P}; indeed, OGi ⊗_{OQ}

jOH is a direct summand of OG ⊗_{OQ} OH ≅ Ind_{ΔQ}^{G×H}(O). Similarly, the bimodule M

in 1.3, when viewed as O(G × H)−module, has vertex ΔP and source V (assuming additionally that V is indecomposable).

1.8 If Q, R are subgroups of P, the set E_{G}((Q, e_{Q}), (R, e_{R})) is the set of all group homomorphisms Q ⟶ R modulo inner automorphisms of R which are induced by conjugation u ⟶ ^{x}u = xux^{−1} with an element x of G satisfying ^{x}(Q, e_{Q}) ⊂ (R, e_{R}) or equivalently, satisfying xQx^{−1} ⊂ R and xe_{Q}x^{−1} = e_{xQx^{−1}}. See 3.1 below for more details.

1.9 Recall that an endo-permutation OP −module (cf. Dade [6]) is an O−free OP −module V such that V^{∗} ⊗_{O} V is a permutation OP −module (i.e. has a

P −stable O−basis). The notation Ind_{ΔP}^{P ×P} (V) means, that we consider first V as OΔP −module through the obvious isomorphism ΔP = {(u, u)}_{u∈P} ≅ P and second the O(P × P)−module Ind_{ΔP}^{P ×P} (V) as OP − OP −bimodule with u ∈ P acting on the left as (u, 1) and on the right as (1, u^{−1}). The complex X_{V} in 1.3 is called an endosplit p−permutation resolution of V in [14]. For abelian P, Rickard proved in [14] that any endo-permutation kP −module “lifts” to an endo-permutation OP −module having an endo-split p−permutation resolution, using Dade’s classification in [7] of endo-permutation OP −modules for abelian P.

1.10 A complex X satisfying the hypotheses of 1.1 is going to be called a splendid tilting complex, analogously to the terminology in [14] (this is compatible with the terminology in [9]). One also might want to call a Morita equivalence induced by a bimodule M isomorphic to a direct summand of OGi ⊗_{OP} jOH a splendid Morita

equivalence. It has first been observed by L. Scott [15] and independently by L. Puig, that splendid Morita equivalences are precisely those Morita equivalences which arise from an isomorphism of the corresponding source algebras iOGi ≅ jOHj mapping ui to uj for all u ∈ P.

Notation. If not stated otherwise, an O−algebra is supposed to be associative, unitary, O−free of finite rank over O, and a module is a finitely generated left module.

For any O−algebra A we denote by J(A) its Jacobson radical, by A^{×} its group of invertible elements and by A^{0} its opposite algebra. If u ∈ A^{×} and a is an element or subset of A, we write ^{u}a = uau^{−1} and a^{u} = u^{−1}au.

For any two O−algebras A, B, an A − B−bimodule is a bimodule whose left and right O−module structure coincide and which hence can be considered as A⊗_{O}B^{0}−module. If P, Q are finite groups, φ : Q ⟶ P a group homomorphism

and A an interior P −algebra (i. e. an O−algebra endowed with a group homomorphism σ : P ⟶ A^{×}, see [10]) we may consider any A−module U as OQ−module via restriction through σ and φ, usually denoted by _{φ}U (if the structural homomorphism σ is clear from the context). If H is a subgroup of a finite group G we denote by [G/H] a system of representatives in G of the set of right transversals of H in G (similarly for double cosets), assuming implicitely, that in all statements where this notation occurs, it is verified that the statement does not depend on the choice of this set of representatives.

2 The Brauer construction

We recall briefly the definition and some properties of the Brauer construction, due to Brauer [2] in the case of the group algebra, generalized by Broué and Puig [5] to arbitrary G−algebras and also to modules in Feit [8,II.3].

2.1 Let P be a finite p−group and X an OP − OP −bimodule. For any subgroup Q of P we set as usual

X^{Q} = {x ∈ X|ux = xu for all u ∈ Q}

and for any further subgroup R of Q we set

X_{R}^{Q} = { ∑_{u∈[Q/R]}uxu^{−1}|x ∈ X^{R}}.

Clearly, X^{Q} defined above is an Z(OQ)−module and X_{R}^{Q} is a submodule of X^{Q}. We set (cf. [5, 1.2])

X(Q) = X^{Q}/(J(O)X^{Q} + ∑_{R<Q}X_{R}^{Q})

and denote by Br_{Q}^{X} : X^{Q} ⟶ X(Q) the canonical surjection.

Clearly a homomorphism of OQ − OQ−bimodules X ⟶ Y induces a homomorphism X(Q) ⟶ Y (Q) which makes this construction functorial in an obvious sense.

2.2 We will have to compute X(Q) only in situations where X has an O−basis B which is stable with respect to the “diagonal” action of Q on X; that is, which satisfies uBu^{−1} = B for all u ∈ Q. In that case, the orbit sums ∑_{u∈[Q/C_{Q}(x)]}uxu^{−1} ,

where x ∈ B, form an O−basis of X^{Q}. Non trivial orbit sums lie clearly in ker(Br_{Q}^{X}), and hence Br_{Q}^{X}(B^{Q}) is a k−basis of X(Q). In particular we have X(Q) = 0 if B contains no element of X^{Q}.

The preceding section applies obviously, if X = OG for some finite group G containing P; we get then X(Q) ≅ kC_{G}(Q) . In that case we write Br_{Q}^{G} instead of Br_{Q}^{OG} and even sometimes just Br_{Q}, if no confusion arises. We usually identify (OG)(Q) to kC_{G}(Q) and Br_{Q} to the surjective algebra homomorphism (OG)^{Q} ⟶ kC_{G}(Q) induced by the projection mapping x ∈ C_{G}(Q) to its image in kC_{G}(Q) and x ∈ G − C_{G}(Q) to 0.

2.3 More generally, if A is an interior P −algebra; that is, an O−algebra endowed with a group homomorphism σ : P ⟶ A^{×}, we may consider A as OP −OP −bimodule via σ. Since, for any subgroup Q of P, A^{Q} is a subalgebra of A and A_{R}^{Q} an ideal in A^{Q} for any subgroup R of Q, the quotient A(Q) becomes naturally a k−algebra.

Moreover, if X is an A−OQ−bimodules we may consider X as OQ−OQ−bimodule with the left action induced by resriction through σ to Q, and then X(Q) becomes naturally an A(Q)−module. Observe that if Y is another A − OQ−bimodule, the space Hom_{A⊗1}(X, Y) is an OQ − OQ−bimodule via the right actions of OQ on X and Y. This gives rise to two bifunctors on the category _{A}Mod_{OQ} of finitely generated A−OQ−bimodules, which turn out to be naturally isomorphic on a suitable subcategory; this is the content of the next proposition (which is crucial for the proof of 1.1 and which is mainly a straightforward generalization of Rickard’s lemma [14, 4.2]):

Proposition 2.4. Let P be a finite p−group, A an interior P −algebra and Q a subgroup of P. Suppose that A has a P × P −stable O−basis such that A is projective as left and right OP −module.

Let M be the full additive subcategory of _{A}Mod_{OQ} generated by the direct summands of the A−OQ−bimodules A ⊗_{OR}(_{φ}OQ) , where R runs over the set of subgroups

of P and φ runs over the set of injective group homomorphisms from R to Q,with the additional condition that R = Q and φ = Id_{Q} if φ is an isomorphism.

Then, on M, the bifunctors defined by

(X, Y) ⟶ (Hom_{A⊗1}(X, Y))(Q)

and

(X, Y) ⟶ Hom_{A(Q)}(X(Q), Y (Q))

are naturally isomorphic.

Proof. For any two A − OQ−bimodules X, Y we have an obvious natural map

(Hom_{A⊗1}(X, Y))^{Q} = Hom_{A⊗OQ}(X, Y) ⟶ Hom_{A(Q)}(X(Q), Y (Q))

which induces a natural map

Hom_{A⊗1}(X, Y)(Q) ⟶ Hom_{A(Q)}(X(Q), Y (Q)),

and we have to show that this map is an isomorphism, if X and Y belong to the subcategory M. We clearly may assume that X = A ⊗_{OR}(_{φ}OQ) and Y = A ⊗_{OS} (_{ψ}OQ),

where R, S are subgroups of P and φ : R ⟶ Q , ψ : S ⟶ Q are injective group homomorphisms.

Note that the standard adjunctions give isomorphisms of OQ − OQ−bimodules

Hom_{A⊗1}(X, Y) ≅ Hom_{OR⊗1}(_{φ}(OQ), Y) ≅ (OQ)_{φ} ⊗_{OR} Y ≅ (OQ)_{φ} ⊗_{OR} A ⊗_{OS} _{ψ}(OQ).

Using 2.2 above and the hypotheses on A, if R (resp. S) is not isomorphic to Q we have X(Q) = 0 (resp. Y (Q) = 0) and also (Hom_{A⊗1}(X, Y))(Q) = 0 , so both bifunctors map (X, Y) to zero.

It remains to treat the case X = Y = A ⊗_{OQ} OQ ≅ A. Since Hom_{A⊗1}(A, A) ≅ A^{0}

we have natural isomorphisms (Hom_{A⊗1}(A, A))(Q) ≅ A(Q)^{0} ≅ Hom_{A(Q)}(A(Q), A(Q)), and the result follows.

The Brauer construction gives rise to different characterizations of defect groups which we recall without proof (see e.g. [16, 18.5] ):

Proposition 2.5. Let G be a finite group, b a block of G and P a subgroup of G. The following are equivalent:

(i) P is a defect group of b.

(ii) P is a minimal subgroup of G such that b ∈ (OG)_{P}^{G}.

(iii) P is a maximal p−subgroup of G such that Br_{P}^{G}(b) ≠ 0.

In that case, for any idempotent e ∈ (OGb)^{P} satisfying Br_{P} (e) ≠ 0, the bimodules OGe and eOG induce a Morita equivalence between the algebras OGb and eOGe.

We finally recall a technical result on relative projectivity (which has various generalizations, but we state it in the form we need it):

Lemma 2.6. Let P be a finite p−group, A an interior P −algebra, Q a subgroup of P and j a primitive idempotent in A^{Q}. If Br_{Q}^{A}(j) = 0 there is a proper subgroup R of Q such that the homomorphism of A − OQ−bimodules

Aj ⊗_{OR} OQ ⟶ Aj

mapping aj ⊗ u to aju = auj , where a ∈ A, u ∈ Q, has a section.

Proof. If Br_{Q}^{A}(j) = 0, by Rosenberg’s lemma [16, 4.9] there is a proper subgroup R of P such that j ∈ A_{R}^{Q}. Then if c ∈ A^{R} such that j = ∑_{u∈[Q/R]}ucu^{−1}, clearly the map

sending a ∈ Aj to ∑_{u∈[Q/R]}aucj ⊗ u^{−1} is a section as required.

3 On fusion in block algebras

A systematical treatment of fusion in block algebras and block source algebras can be found in various works of Alperin-Broué [1], Broué-Puig [5] and Puig [11], for instance. See also [16, chapter 6 and 7].

We give here a very short ad hoc treatment of this subject strictly limited to what we need in this paper.

3.1 Let G be a finite group, b a block of G and P a defect group of b. Recall that for any p−subgroup Q of G the map

Br_{Q} : (OG)^{Q} ⟶ kC_{G}(Q).

is a surjective algebra homomorphism. Hence Br_{Q}(b) is either zero or an idempotent in Z(kC_{G}(Q)), thus a sum of blocks of kC_{G}(Q). Following [1], a b-Brauer pair is a pair (Q, e) consisting of a p−subgroup Q of G and a block e of kC_{G}(Q) satisfying Br_{Q}(b)e = e. Note that then in particular Br_{Q}(b) ≠ 0, hence Q is contained in some defect group of b (or, equivalently, some conjugate of Q is contained in P).

Following [5], there is an inclusion of b-Brauerpairs (Q, e), (R, f) : we define (R, f) ⊂ (Q, e) and say that (R, f) is contained in (Q, e), if R ⊂ Q and there is a primitive idempotent j ∈ (OGb)^{Q} such that Br_{Q}(j)e ≠ 0 and Br_{R}(j)f ≠ 0. In that case, since j is primitive in (OGb)^{Q}, its image Br_{Q}(j) is primitive in kC_{G}(Q), and hence Br_{Q}(j)e = Br_{Q}(j). Even though j need no longer be primitive in (OGb)^{R}, it is still true that Br_{R}(j)f = Br_{R}(j) and that f does not depend on the choice of j. Thus f is uniquely determined by e (cf. [5, 1.8]). We collect some standard properties of Brauer pairs we need (see [1] and [5]):

3.1.1. If P is a defect group of b, there is a block e_{P} of kC_{G}(P) such that Br_{P} (b)e_{P} = e_{P}.

3.1.2. For any b−Brauer pair (Q, e) there is x ∈ G such that ^{x}(Q, e) ⊂ (P, e_{P}); in particular, all maximal b−Brauer pairs are conjugate to (P, e_{P}).

3.1.3. For any subgroup Q of P there is a unique block e_{Q} of kC_{G}(Q) such that (Q, e_{Q}) ⊂ (P, e_{P}).

For any two b−Brauerpairs (Q, e), (R, f), we denote by { \atop Hom}(Q, R) the set of equivalence classes of group homomorphisms from Q to R with respect to the equivalence relation declaring two group homomorphisms φ, ψ from Q to R to be equivalent if there is an inner automorphism τ of R such that φ = τψ. We denote then by φ̃ the image of φ in { \atop Hom}(Q, R) and by E_{G}((Q, e), (R, f)) the image in { \atop Hom}(Q, R) of of the set of all group homomorphisms φ : Q ⟶ R for which there is x ∈ G satisfying φ(u) = ^{x}u for all u ∈ Q and ^{x}(Q, e) ⊂ (R, f).

3.2 Following [10], for any subgroup H of G, a point of H on OGb is an ((OGb)^{H})^{×}−conjugacy class β of primitive idempotents in (OGb)^{H}, and we call then H_{β} a pointed group on OGb.

For any p−subgroup Q of G, a local point of Q on OGb is a point δ of Q on OGb such that Br_{Q}(δ) ≠ 0, and we call then Q_{δ} a local pointed group on OGb. Clearly G acts by conjugation on the set of (local) pointed groups on OGb.

If Q_{δ} is a local pointed group on OGb then since Br_{Q} is a surjective algebra homomorphism, Br_{Q}(δ) is a conjugacy class of primitive idempotents in kC_{G}(Q)Br_{Q}(b) (cf. [16, 3.2]) , hence determines a unique b−Brauer pair (Q, e_{δ}) via the condition Br_{Q}(δ)e_{δ} ≠ 0.

If Q_{δ}, R_{ϵ} are pointed groups on OGb we write R_{ϵ} ⊂ Q_{δ} if R ⊂ Q and for some (and then necessarily any) j ∈ δ there is k ∈ ϵ such that k = jk = kj.

Note that an inclusion of local pointed groups R_{ϵ} ⊂ Q_{δ} induces an inclusion of the corresponding Brauer pairs (R, e_{ϵ}) ⊂ (Q, e_{δ}). We again recall two basic properties of local pointed groups (cf. [10]):

3.2.1. If P is a defect group of b , there is a local point γ of P on OGb.

3.2.2. For any local pointed group Q_{δ} on OGb there is x ∈ G such that ^{x}(Q_{δ}) ⊂ P_{γ}. In particular, the maximal local pointed groups on OGb are all conjugate to P_{γ}.

We have no longer a uniqueness of the inclusion as in 3.1.3, since for a subgroup Q of P there may be different local points δ, δ′ of Q on OGb such that Q_{δ} ⊂ P_{γ}, Q_{δ′} ⊂ P_{γ}. Yet, statement 3.1.3 implies then e_{δ} = e_{δ′}.

The next lemmas contain the technical facts about Brauer pairs and local pointed groups required for the proofs of the statements in section 1 (statement 3.3 (ii) is a reformulation of [1, 4.5] and statement 3.3 (v) is based on Puig’s idea of relating the bimodule structure of a block algebra to its fusion ; see for instance the proof of [11, 3.1]):

Lemma 3.3. Let G be a finite group, b a block of G , P_{γ} a maximal local pointed group on OGb and choose i ∈ γ. For any subgroup Q of P let e_{Q} be the unique block of kC_{Q}(Q) such that Br_{Q}(i)e_{Q} = Br_{Q}(i).

(i) For any subgroup Q of P, C_{P} (Q) is contained in a defect group of e_{Q}.

(ii) For any subgroup Q of P there is x ∈ G such that ^{x}(Q, e_{Q}) ⊂ (P, e_{P}) and such that C_{P} (^{x}Q) is a defect group of ^{x}e_{Q} = e_{^{x}Q}.

(iii) Let Q be a subgroup of P such that C_{P} (Q) is a defect group of e_{Q}. Then kC_{G}(Q)e_{Q} and Br_{Q}(i)kC_{G}(Q)Br_{Q}(i) are Morita equivalent, and for any local point δ of Q on OGb such that Br_{Q}(δ)e_{Q} ≠ 0 we have Q_{δ} ⊂ P_{γ}.

(iv) Let Q, R be isomorphic subgroups of P and x ∈ G such that ^{x}(R, e_{R}) = (Q, e_{Q}) and such that C_{P} (Q) is a defect group of e_{Q}. Then for any local point ϵ of R on OGb satisfying R_{ϵ} ⊂ P_{γ} we have ^{x}(R_{ϵ}) ⊂ P_{γ}.

(v) Let Q, R be isomorphic subgroups of P and φ : R ⟶ Q a group isomorphism. If _{φ}(OQ) is isomorphic to a direct summand of iOGi as OR − OQ−bimodule, then φ̃ ∈ E_{G}((R, e_{R}), (Q, e_{Q})). The converse holds if moreover C_{P} (Q) is a defect group of e_{Q}.

Proof. (i) Since Br_{P} (i) ≠ 0 we have Br_{C_{P} (Q)}(Br_{Q}(i)) = Br_{QC_{P} (Q)}(i) ≠ 0, hence Br_{C_{P} (Q)}(e_{Q}) ≠ 0, and so (i) follows from 2.5.

(ii) Let (R, f) be a maximal e−Brauer pair; that is, R is a defect group of e and f is a block of k_{C_{G}(Q)}(R) = kC_{G}(QR) such that Br_{R}(e_{Q})f = f. Clearly (QR, f) is then a b−Brauer pair such that (Q, e_{Q}) ⊂ (QR, f). By 3.1.2 there is x ∈ G such that ^{x}(Q, e_{Q}) ⊂ ^{x}(QR, f) ⊂ (P, e_{P}). Since ^{x}R is a defect group of ^{x}e_{Q} contained in C_{P} (^{x}(Q)), the statement follows from (i).

(iii) Again, as Br_{C_{P} (Q)}(Br_{Q}(i)) = Br_{QC_{P} (Q)}(i) ≠ 0, it follows from the last statement of 2.5 that the algebras Br_{Q}(i)kC_{G}(Q)Br_{Q}(i) and kC_{G}(Q)e_{Q} are Morita equivalent. Thus for every local point δ of Q on OGb satisfying Br_{Q}(δ)e_{Q} = Br_{Q}(δ) some element of Br_{Q}(δ) lies in Br_{Q}(i)kC_{G}(Q)Br_{Q}(i), which is only possible if Q_{δ} ⊂ P_{γ}.

(iv) This follows from (iii) applied to to ^{x}(R_{ϵ}).

(v) Since _{φ}(OQ) is isomorphic to an indecomposable direct summand of iOGi as OR − OQ−bimodule, there are primitive idempotents j ∈ (OGb)^{Q}, l ∈ (OGb)^{R}, such that _{φ}(OQ) is isomorphic to a direct summand of lOGj as OR − OQ−bimodule. Since iOGi is a direct summand of OG = ⊕_{x∈[R\G/Q]}O[RxQ] as OR − OQ−bimodule, there is x ∈ G such that _{φ}(OQ) ≅ O[Rx^{−1}] = O[x^{−1}Q] and φ(u) = ^{x}u for all u ∈ R. Hence OQ is isomorphic to a direct summand of ^{x}lOGj as OQ − OQ−bimodule. Applying the Brauer construction shows that kZ(Q) is isomorphic to a direct summand of Br_{Q}(^{x}l)kC_{G}(Q)Br_{Q}(j); in particular, the latter expression is non zero. Thus Br_{Q}(^{x}l), Br_{Q}(j) are non zero primitive idempotents in kC_{G}(Q) belonging to the same block e_{Q}. As Br_{R}(l) belongs to e_{R} and ^{x}Br_{R}(l) = Br_{Q}(^{x}l) belongs to e_{Q}, it follows that ^{x}e_{R} = e_{Q} , or , equivalently, ^{x}(R, e_{R}) = (Q, e_{Q}).

In order to prove the converse, assume now that C_{P} (Q) is a defect group of e_{Q}. Let x ∈ G such that ^{x}(R, e_{R}) = (Q, e_{Q}). Denote by φ : R ⟶ Q the group isomorphism mapping u ∈ R to ^{x}u. Choose any local point ϵ of R on OGb such that R_{ϵ} ⊂ P_{γ}. By (iii) we have ^{x}(R_{ϵ}) ⊂ P_{γ}. Thus, if we pick any l ∈ ϵ, as an OR − OQ−bimodule, lOG^{x}l is isomorphic to a direct summand of iOGi. Since Br_{R}(l) ≠ 0, the OR − OR−bimodule lOGl has a direct summand isomorphic to OR. Therefore kOG^{x}l has a direct summand isomorphic to O[Rx^{−1}] = O[x^{−1}Q] ≅ _{φ}(OQ), which concludes the proof.

Lemma 3.4. Let G, H be finite groups, b, c be blocks of G, H, respectively, having a common defect group P. Let i ∈ (OGb)^{P}, j ∈ (OHc)^{P} be primitive idempotents such that Br_{P} (i) ≠ 0, Br_{P} (j) ≠ 0. For any subgroup Q of P denote by e_{Q} and f_{Q} the unique blocks of kC_{G}(Q) and kC_{H}(Q) satisfying Br_{Q}(i)e_{Q} ≠ 0 and Br_{Q}(j)f_{Q} ≠ 0, respectively. Assume that for any two subgroups Q, R of P we have E_{G}((Q, e_{Q}), (R, e_{R})) = E_{H}((Q, f_{Q}), (R, f_{R})).

Then for any subgroup Q of P there are elements x ∈ G and y ∈ H such that x(Q, e_{Q}) ⊂ (P, e_{P}), ^{y}(Q, f_{Q}) ⊂ (P, f_{P}), ^{x}u = ^{y}u for all u ∈ Q, and C_{P} (^{x}Q) = C_{P} (^{y}Q) is a defect group of both ^{x}e_{Q}, ^{y}f_{Q}.

Proof. Let Q be a subgroup of P. By 3.3(ii) there is x ∈ G such that ^{x}(Q, e_{Q}) ⊂ (P, e_{P}) and C_{P} (^{x}Q) is a defect group of ^{x}e_{Q}. By the assumptions, there is y ∈ H such that ^{y}(Q, f_{Q}) ⊂ (P, f_{P}) and ^{y}u = ^{x}u for all u ∈ Q. It therefore suffices to show that assuming that C_{P} (Q) is a defect group of e_{Q}, it is a defect group of f_{Q}, too. For this, choose a defect group R of f_{Q} in C_{H}(Q) and a block f of kC_{C_{H}(Q)}(R) = kC_{H}(QR) such that (R, f) is an f_{Q}−Brauer-pair. Then (RQ, f) is a b−Brauer pair such that (Q, f_{Q}) ⊂ (QR, f). Thus there is y ∈ H satisfying ^{y}(Q, f_{Q}) ⊂ ^{y}(QR, f) ⊂ (P, f_{P}) by 3.1.2. Note that then ^{y}f = f_{^{y}(QR)} . The assumptions imply that there is x ∈ G such that ^{x}(Q, e_{Q}) ⊂ (P, e_{P}) and ^{x}u = ^{y}u for all u ∈ Q. In particular, we have ^{x}Q = ^{y}Q and hence ^{x}(Q, e_{Q}) ⊂ (^{y}(QR), e_{^{y}(QR)}). This in turn means that (^{y}R, e_{^{y}(QR)}) is an xe_{Q}−Brauer pair, or, equivalently, (^{x^{−1}}(^{y}R), e_{Q(^{x^{−1}} (^{y}R))}) is an e-Brauer pair. But then, again by 3.1.2, R is conjugate to a subgroup of C_{P} (Q). Since R contains C_{P} (Q) by 3.3 (i), we have in fact the equality R = C_{P} (Q), and the lemma follows.

4 Proofs

We keep the notation of 1.1.

Proof of 1.1. Set A = iOGi and B = jOHj. Since OGb ⊗_{O} (OHc)^{0} and A ⊗_{O} B^{0}

are Morita equivalent (cf. 1.4) clearly T = iXj is a Rickard tilting complex of A − B−modules. Let Q be a subgroup of P. We choose first Q in such a way that C_{P} (Q) is a defect group of both e_{Q}, f_{Q}. Let M be the subcategory of _{A}Mod_{OQ} as defined in 2.4. The main step of the proof is to show, that the terms of the complex T, when restricted to A − OQ, all belong to the category M. By the hypotheses, it suffices to show, that

4.1 for any subgroup R of P, the restriction to A − OQ of the A − B−bimodule A ⊗_{OR} B belongs to the category M.

In order to prove 4.1, it suffices to show, that for any indecomposable direct summand W of B as OR − OQ−bimodule and any primitive idempotent l ∈ A^{R} the A − OQ−bimodule Al ⊗_{OR} W belongs to the category M. Since W is a direct

summand of B, hence of OH as OR − OQ−bimodule, there is y ∈ H such that

W ≅ O[RyQ] ≅ OR ⊗_{OS} _{φ}(OQ), where S = ^{y}Q ∩ R and φ(u) = y^{−1}uy for all u ∈ S.

Thus Al ⊗_{OR} W ≅ Al ⊗_{OS} _{φ}(OQ). Therefore, we may assume that R = S. By 2.6, we

may also assume that l belongs to a local point ϵ of R on OGb such that R_{ϵ} ⊂ P_{γ}, where γ is the local point of P on OGb containing i. If φ is not an isomorphism, then Al ⊗_{OR} _{φ}(OQ) belongs to M by definition. If φ is an isomorphism, we have

φ̃ ∈ E_{G}((R, e_{R}), (Q, e_{Q})) by 3.3 (v). Let x ∈ G such that ^{x}(R, e_{R}) ⊂ (Q, e_{Q}) and φ(u) = ^{x}u for all u ∈ R. Then by 3.3 (iii) we have ^{x}(R_{ϵ}) ⊂ P_{γ}.

Thus Al ⊗_{OR} _{φ}(OQ) ≅ (Al)_{φ^{−1}} ≅ iOG^{x}l as A − OQ−bimodules. Since ^{x}(R_{ϵ}) ⊂ P_{γ},

iOG^{x}l is isomorphic to a direct summand of iOGi = A and lies therefore in M. This shows 4.1.

The proof of 1.1 concludes now as follows: since, by 4.1, all terms of T are in the category M, we may apply 2.4 and get an isomorphism of complexes

4.2 (Hom_{A⊗1}(T, T))(Q) ≅ Hom_{A(Q)}(T (Q), T (Q)).

Since the terms of T are left and right projective and since A is symmetric (i.e. A ≅ A^{∗} as A − A−bimodules) we have Hom_{A⊗1}(T, T) ≅ T^{∗} ⊗_{A} T , and similarly, we

have Hom_{A(Q)}(T (Q), T (Q)) ≅ T (Q)^{∗} ⊗_{A(Q)} T (Q) as complexes; thus 4.2 translates to

an isomorphism of complexes

4.3 T (Q)^{∗} ⊗_{A(Q)} T (Q) ≅ (T^{∗} ⊗_{A} T)(Q).

Now T^{∗} ⊗_{A} T is homotopic to B, and therefore, (T^{∗} ⊗_{A} T)(Q) is homotopic to B(Q).

This shows, still under the assumtion that C_{P} (Q) is a defect group of both e_{Q}, f_{Q}, that

4.4 the complex T (Q) is a Rickard tilting complex of A(Q) − B(Q)−bimodules.

From the Morita equivalences in 3.3(iii) follows therefore that

4.5 the complex e_{Q}X(Q)f_{Q} is a Rickard tilting complex of kC_{G}(Q)e_{Q} − kC_{H}(Q)f_{Q}−bimodules.

Clearly the property of e_{Q}X(Q)f_{Q} being a Rickard tilting complex is invariant under “simultaneous conjugation” of Q in P by elements of G and H, thus lemma 3.4 implies immediately, that in fact e_{Q}X(Q)f_{Q} is a Rickard tilting complex for all subgroups Q of P.

Proof of 1.2. Since Br_{P} (b) = e_{P} there is a unique local point γ of P on OGb (cf. [16, (40.13)(b)]). Similarly, there is a unique local point δ of P on OHc. Choose i ∈ γ and j ∈ δ. Let M be an indecomposable OGb − OHc−bimodule such that M is isomorphic to OGb ⊗_{OQ} OHc for some subgroup Q of P, and choose Q to be minimal

with this property. By the assumptions, in order to apply 1.1, it suffices to show that in fact M is then isomorphic to a direct summand of OGi ⊗_{OR} jOH for some subgroup

R of P.

Note first that if x ∈ G, y ∈ H such that ^{x}Q ⊂ P, ^{y}Q ⊂ P and ^{x}u = ^{y}u for all u ∈ Q, there is an isomorphism of OGb−OHc−bimodules OGb ⊗_{OQ}OHc ≅ OGb ⊗_{O^{x}Q}OHc

mapping a ⊗ d to ax^{−1} ⊗ yd for any a ∈ OGb, d ∈ OHc. Therefore, by lemma 3.4, we may assume that C_{P} (Q) is a defect group of e_{Q} and f_{Q}.

Observe next that since M is indecomposable, there are primitive idempotents m ∈ (OGb)^{Q} and n ∈ (OHc)^{Q} such that M is isomorphic to a direct summand of OGm ⊗_{OQ} nOH. By the minimality of Q, the idempotents m and n belong to local

points μ and ν of Q on OGb and OHc, respectively (cf. 2.6). By 3.3(iii) we have Q_{μ} ⊂ P_{γ} and Q_{ν} ⊂ P_{δ}. Consequently, the bimodule OGm ⊗_{OQ} nOH and hence M is

isomorphic to a direct summand of OGi ⊗_{OQ} jOH.

Thus all terms of X are indeed isomorphic to sums of direct summands of the bimodules OGi ⊗_{OQ} jOH, where Q runs over the set of subgroups of P, hence 1.1

applies and the proof of 1.2 is complete.

Proof of 1.3. Set Y = OGi ⊗_{OP} Ind_{ΔP}^{P ×P} (X_{V}) ⊗_{OP} jOH and

U = OGi ⊗_{OP} Ind_{ΔP}^{P ×P} (V) ⊗_{OP} jOH. Since OGb is symmetric and all terms of Y are

projective as left OGb−modules, we have an isomorphism of the total complexes of OHc − OHc−bimodules

4.6 Hom_{OGb⊗1}(Y, Y) ≅ Y^{∗} ⊗_{OGb} Y

and our first step is to show that

4.7 the complex Y^{∗} ⊗_{OGb} Y is split.

As Y^{∗} ⊗_{OGb} Y is isomorphic to the complex

OHj ⊗_{OP} Ind_{ΔP}^{P ×P} (X_{V}^{∗}) ⊗_{OP} iOGi ⊗_{OP} Ind_{ΔP}^{P ×P} (X_{V}) ⊗_{OP} jOH, it suffices to show that

for any direct summand W of iOGi as OP − OP −bimodule

4.8 the complex of OP − OP −bimodules Ind_{ΔP}^{P ×P} (X_{V}^{∗}) ⊗_{OP} W ⊗_{OP} Ind_{ΔP}^{P ×P} (X_{V}) is

split.

By 3.3 (v) we have W ≅ OP ⊗_{OQ}_{φ}(OP) for some subgroup Q of P and an injective

group homomorphism φ : Q ⟶ P such that φ̃ ∈ E_{G}((R, e_{R}), (Q, e_{Q})). Thus the complex in 4.8 is isomorphic to

4.9 Ind_{ΔP}^{P ×P} (X_{V}^{∗}) ⊗_{OQ} _{φ}Ind_{ΔP}^{P ×P} (X_{V}) ≅ Ind_{ΔQ}^{P ×P} (X_{V}^{∗} ⊗_{O} _{φ}X_{V})

and the latter is split since _{φ}X_{V} ≅ Res_{Q}^{P}(X_{V}) and X_{V}^{∗} ⊗_{O} X_{V} is split by the

assumptions.

We use now an argument of Rickard (occurring in the proof of [14, 7.5]) to show that an isomorphism between the degree zero homology of X_{V} and V induces an algebra isomorphism

4.10 End_{K^{b}(OGb⊗_{O}(OHc)^{0})}(Y) ≅ End_{OGb⊗_{O}(OHc)^{0}}(U).

Indeed, since the complex Hom_{OGb⊗1}(Y, Y) is split by 4.6 and 4.7, taking homology commutes to taking H−fixpoints. If we take first H−fixpoints, we obtain the complex Hom_{OGb⊗_{O}(OHc)^{0}}(Y, Y), and its degree zero homology is well-known to be isomorphic

to End_{K^{b}(OGb⊗_{O}(OHc)^{0})}(Y). As X_{V} has homology concentrated in degree zero isomor-

phic to V, the complex Y has homology concentrated in degree zero isomorphic to U. It follows that the complex Hom_{OGb⊗1}(Y, Y) has homology concentrated in degree zero isomorphic to End_{OGb⊗1}(U). Taking H−fixpoints yields End_{OGb⊗_{O}(OHc)^{0}}(U),

which implies 4.10.

Now M is isomorphic to a direct summand of U, thus corresponds to a primitive idempotent in End_{OGb⊗_{O}(OHc)^{0}}(U). This corresponds to a primitive idempotent in

End_{K^{b}(OGb⊗_{O}(OHc)^{0})}(Y) through the algebra isomorphism 4.10, and yields hence a

direct summand X of Y. Clearly X has the properties stated in 1.3. The last statement is a corollary to Puig’s work [12, 5.3], [11, section 3] on fusion.

Acknowledgements. The author wishes to thank K.W Roggenkamp and the members of the Institute of Mathematics at the University of Stuttgart for the invitation to Stuttgart and their hospitality, as well as the Deutsche Forschungsgemeinschaft for their support during the writing of this paper.

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

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