TRIVIAL SOURCE BIMODULE RINGS FOR

BLOCKS AND p-PERMUTATION EQUIVALENCES

Markus Linckelmann

September 2006

Abstract. We associate with any p-block of a finite group a Grothendieck ring of certain p-permutation bimodules. We extend the notion of p-permutation equivalences introduced by Boltje and Xu [4] to source algebras of p-blocks of finite groups. We show that a p-permutation equivalence between two source algebras A, B of blocks with a common defect group and same local structure induces an isotypy.

1 Introduction

Let p be a prime, let O be a complete discrete valuation ring having a quotient field K of characteristic zero and a algebraically closed residue field k of prime characteristic p. For any finite p-group P and any interior P -algebra A we denote by T (A) the Grothendieck group, with respect to split exact sequences, of finite direct sums of summands of the A-A-bimodules A ⊗_{OQ} A, with Q running

over the subgroups of P. If A is a source algebra of a block of a finite group algebra OG then the group T (A) has a ring structure:

Theorem 1.1. Let A be a source algebra of a block of a finite group with defect group P. The tensorproduct over A induces an associative bilinear multiplication on T (A) through which T (A) becomes a unitary ring having an augmentation and an antipode.

We will restate and prove the various parts of Theorem 1.1 in greater generality in 8.2, 8.6 and 12.3. We show in 8.4 that the ring T (A) does not depend on the choice of a source algebra of the considered block b ∈ Z(OG) of a finite group G, and is hence an invariant of the block b. One may think of T (A) as a bimodule version for source algebras of the trivial source ring of a finite group and call T (A) the trivial source bimodule ring of the block b. If A = OP then T (A) is isomorphic to the Burnside ring of P (cf. 12.1), which is also the trivial source ring in that case. Results of Alperin-Broué [1] and Broué-Puig [9] imply that a source algebra A of a block with defect group P determines a fusion system on P. For P a finite p-group, A an interior P -algebra and Q a subgroup of P, the Brauer construction with respect to the conjugation action of Q on A yields an interior C_{P} (Q)-algebra A(ΔQ). In certain cases, this is compatible with the ring structure on T (A):

Theorem 1.2. Let A be a source algebra of a block with defect group P and fusion system F. For any fully F-centralised subgroup Q of P the tensorproduct over A(ΔQ) induces an associative unitary ring structure on T (A(ΔQ)) and the Brauer construction induces a unitary ring homomorphism T (A) ⟶ T (A(ΔQ)).

Typeset by AMS-TEX

This is proved in 9.1 and 9.2. The difficult part is to show that the Brauer construction does actually induce a map from T (A) to T (A(ΔQ)); the fact that this map is multiplicative follows then from a result in [15]. This Theorem does not hold, in general, for the block algebra OGb instead of A. For A, B two source algebras of blocks of finite groups having a common defect group P we denote more generally by T (A, B) the Grothendieck group, with respect to split exact sequences, of finite direct sums of summands of the A-B-bimodules A ⊗_{OQ} B, with Q running over the subgroups

of P. Since A, B are symmetric O-algebras, the O-dual of an A-B-bimodule of the form A ⊗_{OQ} B

is isomorphic to the B-A-module B ⊗_{OQ} A. In other words, O-duality induces a group isomorphism

T (A, B) ≅ T (B, A). For X an element in T (A, B) we denote by X^{∗} its image in T (B, A) under this isomorphism. The map sending a pair (M, N) consisting of an A-B-bimodule M and a B-A-bimodule N to the A-A-bimodule M ⊗_{B} N induces a bilinear map − ·_{B}− from T (A, B) × T (B, A)

to the Grothendieck group of A-A-bimodules, again taken with respect to split exact sequences. The following definition is the source algebra version of the concept of a p-permutation equivalence between blocks of finite groups, as introduced by Boltje and Xu [4].

Definition 1.3. Let A, B be source algebras of blocks of finite groups with a common defect group P. A p-permutation equivalence between A and B is an element X ∈ T (A, B) such that X ·_{B}X^{∗} = [A]

in the Grothendieck group of A-A-bimodules and X^{∗} ·_{A}X = [B] in the Grothendieck group of B-B-bimodules.

A p-permutation equivalence between source algebras A, B induces a p-permutation equivalence in the sense of Boltje and Xu [4]. The converse is true for principal blocks, but not for arbitrary blocks in general (cf. 7.5, 7.6). What one gains by adopting the slightly more restrictive definition 1.3 is a generalisation of [4, 1.11] to arbitrary blocks:

Theorem 1.4. Let A, B be source algebras of blocks b, c of finite groups G, H, respectively, having a common defect group P and the same fusion system on P. If there is a p-permutation equivalence between the source algebras A and B then the blocks b and c are isotypic.

This is proved in 10.1; as in [4], we will show that the isotypy between b and c is obtained from applying the Brauer construction to the p-permutation equivalence. Following [15, 1.10] or [17, 1.1] a splendid derived equivalence between source algebras A, B of blocks of finite groups with a common defect group P is a derived equivalence given by a bounded two-sided tilting complex X of A-B-bimodules whose components in each degree are direct sums of summands of the A-B-bimodules A ⊗_{OQ} B, where Q runs over the subgroups of P. A two-sided tilting complex X of this form is then

called a splendid tilting complex of A-B-bimodules. This is a variation of Rickard’s version [25] of a splendid equivalence - our definition is again slightly more restrictive in that it is relative to a choice of source idempotents. The motivation then as here is to make the notion “splendid” work for arbitrary blocks. If X is a bounded complex of A-B-bimodules, we set [X] = ∑_{i∈ℤ} (−1)^{i}[X_{i}], where

[X_{i}] is the isomorphism class of the degree i component of X viewed as element in the Grothendieck group of A-B-bimodules. We obtain the result corresponding to [4, 1.5] in this setting:

Theorem 1.5. Let A, B be source algebras of blocks of finite groups having a common defect group P and let X be a splendid tilting complex of A-B-bimodules. The image [X] of X in T (A, B) is a p-permutation equivalence.

TRIVIAL SOURCE BIMODULE RINGS AND p-PERMUTATION EQUIVALENCES 3

This is an immediate consequence of [14, 11.4.2]. Combining the present material with a result of Puig [24] yields the following invariance statement:

Theorem 1.6. Let A, B be source algebras of blocks b, c of finite groups G, H, respectively, having a common defect group P. Suppose that OGb and OHc are Morita equivalent via a bimodule with endo-permutation source. Then T (A) ≅ T (B).

The structure theory of nilpotent blocks [23] in conjunction with 1.6 implies:

Corollary 1.7. Let A be a source algebra of a nilpotent block with defect group P. Then T (A) ≅ T (OP).

Further remarks in §13 include the observation that the Hochschild cohomology and block cohomology of a block are modules over the trivial source bimodule ring of the block (cf. 13.2), as well as considerations on how far off the trivial source bimodule ring is from being commutative (cf. 13.3).

Convention: all O-algebras are assumed to be finitely generated as O-modules, and all modules are finitely generated left modules, unless stated otherwise. Given two O-algebras A, B, we assume that for any A-B-bimodule M the left and right O-module structure on M concident; in other words, M can be viewed as A ⊗_{O} B^{op}-module. Given finite groups G, H, an OG-OH-bimodule M can be viewed

as O(G × H)-module, with (x, y) ∈ G × H acting on M by sending m ∈ M to xmy^{−1}. In particular, if M is an indecomposable OG-OH-bimodule, it makes sense to consider its vertices and sources as O(G × H)-module.

§2 Fusion systems

As a consequence of results in [1], [9], any block of a finite group gives rise to a fusion system on any of its defect groups (see 4.4 below). When dealing with fusion systems - a concept introduced by Puig in the early 1990’s - we follow the terminology in [18, §2]. Given a fusion system F on a finite p-group P, a subgroup Q is called fully F-centralised if |C_{P} (Q)| ≥ |C_{P} (φ(Q))| for any φ ∈ Hom_{F}(Q, P). By a result of Puig, any fully F-centralised subgroup Q of P gives rise to a fusion system, denoted by C_{F}(Q), on C_{P} (Q). See e.g. [19] for a more detailed introduction to fusion systems. We collect in this section some elementary facts on centralisers in fusion systems.

Proposition 2.1. Let F be a fusion system on a finite p-group P, let Q be a fully F-centralised subgroup of P and let S be a subgroup of C_{P} (Q). Then S is fully C_{F}(Q)-centralised if and only if QS is fully F-centralised.

Proof. Suppose that S is fully C_{F}(Q)-centralised. Let ψ : QS → P be a morphism in F. Denote by τ : ψ(Q) → Q the inverse of the isomorphism ψ|_{Q} : Q ≅ ψ(Q) induced by ψ. Since Q is fully F-centralised, τ extends to a morphism σ : ψ(Q)C_{P} (ψ(Q)) → QC_{P} (Q) in F (cf. [18]). Note that ψ(S) ⊆ C_{P} (ψ(Q)). By construction, we have σ ∘ ψ|_{Q} = Id_{Q}. Thus σ ∘ ψ|_{S} is a morphism in C_{F}(Q). Since S is fully C_{F}(Q)-centralised, we have

|C_{P} (QS)| = |C_{C_{P} (Q)}(S)| ≥ |C_{C_{P} (Q)}(σ(ψ(S)))| = |C_{P} (Qσ(ψ(S))|

Now σ sends C_{P} (ψ(QS)) to C_{P} (Qσ(ψ(S))), and hence |C_{P} (Qσ(ψ(S)))| ≥ |C_{P} (ψ(QS)|, which proves that QS is fully F-centralised. Conversely, suppose that QS is fully F-centralised. Let τ : S → C_{P} (Q) be a morphism in C_{F}(Q). That is, τ extends to a morphism σ : QS → P in F such that σ|_{Q} = Id_{Q}. Since QS is fully F-centralised, we have

|C_{C_{P} (Q)}(S)| = |C_{P} (QS)| ≥ |C_{P} (σ(QS))| = |C_{P} (Qτ(S))| = |C_{C_{P} (Q)}(τ(S))|

which proves that S is fully C_{F}(Q)-centralised. □

Let P, Q be finite p-groups, let F be a fusion system on P and let G be a fusion system on Q. By [6], the category on P × Q generated by F and G is a fusion system on P × Q, denoted by F × G. In particular, F determines a fusion system F × F on P × P.

Proposition 2.2. Let F be a fusion system on a finite p-group P, let Q be a subgroup of P, let τ ∈ Hom_{F}(Q, P) and set Δ_{τ} Q = {(u, τ(u)) | u ∈ Q}. The subgroup Δ_{τ} Q of P × P is fully F × Fcentralised if and only if Q and τ(Q) are fully F-centralised.

Proof. Let σ : Δ_{τ} Q → P be a morphism in F × F. By the definition of F × F, this means there are morphisms φ ∈ Hom_{F}(Q, P) and ψ ∈ Hom_{F}(τ(Q), P) such that σ(u, τ(u)) = (φ(u), ψ(τ(u))) for all u ∈ Q. Clearly C_{P ×P} (Δ_{τ} Q) = C_{P} (Q) × C_{P} (τ(Q)) and C_{P ×P} (σ(Δ_{τ} Q)) = C_{P} (φ(Q)) × C_{P} (ψ(τ(Q))). Thus the inequality |C_{P ×P} (σ(Δ_{τ} Q))| ≤ |C_{P ×P} (Δ_{τ} Q)| holds for all morphisms σ : Δ_{τ} Q → P × P in F × F if and only if the inequalities |C_{P} (φ(Q))| ≤ |C_{P} (Q)| and |C_{P} (ψ(τ(Q)))| ≤ |C_{P} (τ(Q))| hold for all φ ∈ Hom_{F}(Q, P) and all ψ ∈ Hom_{F}(τ(Q), P). The result follows. □

Proposition 2.3. Let F be a fusion system on a finite p-group P and let Q be a fully F-centralised subgroup of P. Then the fusion systems C_{F×F}(ΔQ) and C_{F}(Q) × C_{F}(Q) on C_{P ×P} (ΔQ) = C_{P} (Q) × C_{P} (Q) are equal.

Proof. Since Q is fully F-centralised it follows from 2.2 that ΔQ is fully F × F-centralised. Thus both C_{F×F}(ΔQ) and C_{F}(Q) × C_{F}(Q) are fusion systems on C_{P ×P} (ΔQ) = C_{P} (Q) × C_{P} (Q). Let R be a subgroup of C_{P} (Q) × C_{P} (Q). Denote by R_{1}, R_{2} the images of R in C_{P} (Q) under the two projections of the direct product C_{P} (Q) × C_{P} (Q) onto its first and second factor, respectively. Let φ : R → C_{P} (Q) × C_{P} (Q) be a morphism in C_{F×F}(ΔQ). That is, φ extends to a morphism ψ : RΔQ → P × P in F × F such that ψ|_{ΔQ} = Id_{ΔQ}. The images in P of RΔQ under the two projections of P × P onto its factors are equal to R_{1}Q and R_{2}Q. Thus ψ being a morphism in F × F is equivalent to the existence of morphisms ψ_{1} : R_{1}Q → P and ψ_{2} : R_{2}Q → P which are the identity on Q and which induce ψ. The result follows. □

§3 The generalised Brauer construction

3.1. The Brauer construction has its origins in Brauer’s observation that the canonical linear projection kG → kC_{G}(P) induces an algebra homomorphism Z(kG) → Z(kC_{G}(P)), for any finite group G and any p-subgroup P of G. Broué [7] observed that this extends to a surjective algebra homomorphism (kG)^{ΔP} → kC_{G}(P). This in turn has been extended to G-algebras in [9], to modules in [11], and generalised further by Boltje and Külshammer in [3]. We briefly recall the definition and some of the basic properties of this construction, following [3, §2]. Let P be a finite p-group and let θ : P → O^{×} be a group homomorphism. For any OP -module M we set

M^{(P,θ)} = {m ∈ M | ym = θ(y)m (∀y ∈ P)} .

For θ = 1 (the trivial group homomorphism) this is the usual O-submodule of P -fixpoints in M; we write M^{P} instead of M^{(P,1)}. For Q a subgroup of P and τ = θ|_{Q} : Q → O^{×} we denote by [P/Q] a system of representatives in P of the cosets P/Q, and for m ∈ M^{(Q,τ)} we set Tr_{(Q,τ)}^{(P,θ)}(m) = ∑_{y∈[P/Q]} θ(y)^{−1}ym . This expression does not depend on the choice of [P/Q] and hence defines an

O-linear map Tr_{(Q,τ)}^{(P,θ)} : M^{(Q,τ)} → M^{(P,θ)}. We set M_{(Q,τ)}^{(P,θ)} = Im(Tr_{(Q,τ)}^{(P,θ)}). If θ = 1 we write Tr_{Q}^{P}

and M_{Q}^{P} instead of Tr_{(Q,τ)}^{(P,θ)} and M_{(Q,τ)}^{(P,θ)}, respectively. The generalised Brauer construction [3, 2.2] is defined by

M(P, θ) = M^{(P,θ)}/(∑_{Q} M_{(Q,θ|_{Q})}^{(P,θ)} + J(O)M^{(P,θ)})

where Q runs over the set of all proper subgroups of P. The Brauer homomorphism is defined to be the canonical surjection

Br_{(P,θ)}^{M} : M^{(P,θ)} ⟶ M(P, θ)

If no confusion arises we will write Br_{(P,θ)} instead of Br_{(P,θ)}^{M}. If moreover θ is the trivial homomorphism we write M(P) instead of M(P, θ) and Br_{P} instead of Br_{(P,θ)}. One can play back this construction to the usual Brauer construction as follows: for θ : P → O^{×} a group homomorphism, denote by θ̄ the O-algebra automorphism of OP sending u ∈ P to uθ(u^{−1}). For M an OP -module denote by θ̄M the OP -module obtained from restricting the action of OP along θ̄; that is, θ̄M = M as O-module, and y ∈ P acts on m ∈ M as multiplication by yθ(y)^{−1}. Then M^{(P,θ)} = (θ̄M)^{P} and we have

M(P, θ) = (θ̄M)(P) .

Note that if M is a permutation OP -module and if θ : P → O^{×} is a non-trivial group homomorphism then M(P, θ) = {0}.

3.2. If G is a finite group having P as subgroup, then OG is an O(P ×P)-module with (u, v) ∈ P ×P acting on a ∈ OG as uav^{−1}. For Q a subgroup of P we set ΔQ = {(u, u) | u ∈ Q}. This is a subgroup of P × P, hence acts on OG (this is the conjugation action of Q on OG). The corresponding Brauer construction (OG)(ΔQ) is well-known to be canonically isomorphic to kC_{G}(Q), and under the canonical identification (OG)(ΔQ) = kC_{G}(Q) the Brauer homomorphism

Br_{ΔQ} : (OG)^{ΔQ} ⟶ kC_{G}(Q)

is a surjective algebra homomorphism induced by the linear map sending x ∈ G to its image in kC_{G}(Q) if x ∈ C_{G}(Q) and to zero otherwise.

3.3. Let A be an interior P -algebra; that is, A is an O-algebra endowed with a group homomorphism σ : P → A^{×} (or equivalently, with a unitary algebra homomorphism OP → A). Then A becomes an O(P × P)-module with (u, v) ∈ P × P acting on a ∈ A by σ(u)aσ(v^{−1}). For Q a subgroup of P, the Brauer construction A(ΔQ) becomes then an interior C_{P} (Q)-algebra, with multiplication induced by that in A^{ΔQ} via the Brauer homomorphism Br_{ΔQ}^{A}, and with structural homomorphism

C_{P} (Q) ⟶^{σ|_{C_{P} (Q)}} (A^{×})^{ΔQ} = (A^{ΔQ})^{×} ⟶^{Br_{ΔQ}} A(ΔQ)^{×}

If M is an A-module, then M can be viewed as OP -module via σ : P → A^{×}. The A-module structure on M induces then an A(ΔQ)-module structure on the corresponding Brauer construction M(Q), for Q a subgroup of P. If A and B are two interior P -algebras with structural homomorphisms σ : P → A^{×} and τ : P → B^{×}, and if M is an A-B-bimodule, then M can be viewed as O(P × P)- module with (u, v) ∈ P × P acting on m ∈ M as σ(u)mτ(v^{−1}). For Q a subgroup of P, the A-B-bimodule structure on M restricts to an A^{ΔQ}-B^{ΔQ}-bimodule structure on M^{ΔQ} which in turn induces an A(ΔQ)-B(ΔQ)-bimodule structure on M(ΔQ). In particular, if G, H are two finite groups having P as common p-subgroup and if M is an OG-OH-bimodule, we regard M(ΔQ) as kC_{G}(Q)-kC_{H}(Q)-bimodule through the canonical identifications.

The Brauer construction is functorial, sends p-permutation modules to p-permutation modules, commutes with duality, and in some cases, commutes with tensor products. See [28, §11, §27] for more details. Calculating the Brauer construction for p-permutation modules is particularly easy:

Lemma 3.4. ([28, (27.6)]) Let P be a finite p-group and let M be an OP -module having a P -stable O-basis X. Then the image in M(P) of the fixpoint set X^{P} under Br_{P}^{M} is a k-basis of M(P).

Lemma 3.5. ([15, 6.1.(iii)]) Let P be a finite p-group and let M be an indecomposable OP -OP - bimodule having a P × P -stable O-basis such that M is projective as left and right OP -module. Then M(ΔP) is non zero if and only if M ≅ OP as OP -OP -bimodule.

3.6 Let P be a finite p-group, let θ : P → O^{×} be a group homomorphism, let A be an interior P -algebra with structural homomorphism σ : P → A^{×} and let M be an A-module. Denote by θ̄A the interior P -algebra which is equal to A as O-algebra, with structural homomorphism sending y ∈ P to σ(y)θ(y^{−1}). Note that σ(y) and σ(y)θ(y^{−1}) act in the same way on A. Thus (θ̄A)(ΔP) = A(ΔP) as k-algebras. More precisely, for Q a subgroup of P we get that

(θ̄A)(ΔQ) = A(ΔQ)

as interior C_{P} (Q)-algebras. The A-module M is still a θ̄A-module, and its restriction to OP via the structural homomorphism is now θ̄M. Thus, using 3.3, we get that M(P, θ) is an A(ΔP)-module with a module structure which is induced by the A^{ΔP} -module structure on M^{(P,θ)}. In particular, if A = OGb for some finite group G containing P and some idempotent b ∈ Z(OG) it follows that M(P, θ) is a kC_{G}(P)Br_{ΔP} (b)-module.

3.7 By [4, Theorem 2.4], the generalised Brauer construction can be used to “lift” generalised decomposition maps. We describe this briefly. Let G be a finite group and let b be an idempotent in Z(OG). Denote by ℤIrr_{K}(G, b) the group of generalised characters of G over K associated with b and by ℤIBr_{K}(G, b) the corresponding group of generalised Brauer characters. If b = 1 we write ℤIrr_{K}(G) instead of ℤIrr_{K}(G, b). Denote by b̄ the image of b in kG. For a kGb̄-module N denote by [[N]] its Brauer character viewed as element in the group ℤIBr_{K}(G, b) associated with b (which can be identified with the Grothendieck group of kGb̄-modules taken with respect to all short exact sequences). For u a p-element in G and e an idempotent in Z(kC_{G}(u)) the generalised decomposition map

d_{(G,b)}^{(u,e)} : ℤIrr_{K}(G, b) ⟶ K ⊗_{ℤ} ℤIBr_{K}(C_{G}(u), e)

is defined by d_{(G,b)}^{(u,e)}(χ)(s) = χ(eus) for all p′-elements s in C_{G}(u). An OG-module M is called a linear source module if M is a finite direct sum of summands of OG-modules of the form OG ⊗_{OQ} _{θ}O, where

Q runs over the p-subgroups of G and θ over the group homomorphisms from Q to O^{×} and where θO is the rank one module determined by θ.

Theorem 3.8. ([4, Theorem 2.4]) Let G be a finite group, let u be a p-element in G, let b be an idempotent in Z(OG) and let e be an idempotent in Z(kC_{G}(u)). Let M be a linear source OGb-module and let χ be the character of M. We have

d_{(G,b)}^{(u,e)}(χ) = ∑_{θ∈Hom(⟨u⟩,O^{×})} θ(u) ⊗ [[M(⟨u⟩, θ)]] .

The generalised Brauer construction “commutes” with “cutting by idempotents”:

Lemma 3.9. Let G be a finite group, let P be a p-subgroup of G, let θ : P → O^{×} be a group homomorphism and let i be an idempotent in (OG)^{ΔP} such that Br_{ΔP} (i) ≠ 0. For any OG-module M there is a natural isomorphism of Br_{ΔP} (i)kC_{G}(P)Br_{ΔP} (i)-modules

(iM)(P, θ) ≅ Br_{ΔP} (i)M(P, θ) .

Proof. The kC_{G}(P)-module structure on M(P, θ) is induced by the (OG)^{ΔP} -module structure on M^{(P,θ)}. That is, for a ∈ (OG)^{ΔP} and m ∈ M^{(P,θ)} we have

am + ker(Br_{(P,θ)}^{M}) = Br_{ΔP} (a)(m + ker(Br_{(P,θ)}^{M})) .

Since i commutes with P we have (iM)^{(P,θ)} = i(M^{(P,θ)}). It follows that the map sending im + ker(Br_{(P,θ)}^{M}) to Br_{ΔP} (i)(m+ker(Br_{(P,θ)}^{M}) is in fact an equality (iM)(P, θ) = Br_{ΔP} (i)M(P, θ) of subsets of M(P, θ). □

3.10. Let P be a finite p-group and let A be an interior P -algebra. Following [21] a local point of a subgroup Q of P on A is an (A^{ΔQ})^{×}-conjugacy class δ of primitive idempotents in A^{ΔQ} such that Br_{ΔQ}(δ) ≠ 0. Since Br_{ΔQ} : A^{ΔQ} → A(ΔQ) is a surjective algebra homomorphism, the standard lifting theorems for idempotents imply that then Br_{ΔQ}(δ) is a conjugacy class of primitive idempotents in A(ΔQ). If Q, R are subgroups of P, δ a local point of Q on A and ϵ a local point of R on A, we write Q_{δ} ⊆ R_{ϵ} if Q ⊆ R and if there are i ∈ δ, j ∈ ϵ satisfying ij = ji = i. We refer to [28] for a detailed account on local pointed groups.

§4 Almost source algebras

Let G be a finite group and let b be a block of OG; that is b is a primitive idempotent in Z(OG). Let P be a defect group of b; that is, P is a maximal p-subgroup of G with the property that Br_{ΔP} (b) ≠ 0, where Br_{ΔP} : (OG)^{ΔP} → kC_{G}(P) is the Brauer homomorphism as described in the previous section. Equivalently, P is a minimal subgroup of G such that b ∈ (OG)_{P}^{G}. Since Br_{ΔP} (b) ≠ 0 there must be a primitive idempotent i ∈ (OGb)^{ΔP} satisfying Br_{ΔP} (i) ≠ 0. Such an idempotent i is called a source idempotent and the algebra A = iOGi is called a source algebra of b. This concept is due to Puig [21]. A source algebra is always considered as interior P -algebra; that is, together with the group homomorphism P → A^{×} sending u ∈ P to ui = iu = iui ∈ A^{×}. If P′ is another defect group of b and i′ ∈ (OGb)^{ΔP′} a source idempotent then there is an O-algebra isomorphism α : iOGi ≅ i′OGi′ and an element x ∈ G such that ^{x}P = P′ and such that α(ui) = ^{x}ui′ for all u ∈ P. Source algebras of a block b are Morita equivalent to the block algebra OGb; more generally:

Proposition 4.1. ([21, 3.5]) Let G be a finite group, let b be a block of OG, let P be a defect group of b and let i be an idempotent in (OGb)^{ΔP} such that Br_{ΔP} (i) ≠ 0. The block algebra OGb and the algebra iOGi are Morita equivalent via the OGb-iOGi-bimodule OGi and its O-dual iOG.

Proof. We sketch a proof for the convenience of the reader. The obvious map iOG ⊗_{OG} OGi → iOGi

given by multiplication in OG is clearly an isomorphism. For any idempotent j ∈ (OGb)^{ΔP}, (OGj)_{ΔP}^{ΔG} is a left ideal, hence an ideal in the commutative algebra Z(OGb). Since b ∈ (OGb)_{ΔP}^{ΔG}, Rosenberg’s Lemma [28, (4.9)] implies that b ∈ (OGj)_{ΔP}^{ΔG} for some primitive idempotent j ∈ (OGb)^{ΔP}. Since Br_{ΔP} (b) ≠ 0 this implies Br_{ΔP} (j) ≠ 0 and hence that j belongs to a local point of P on OGb. Since all local points of P on OGb are N_{G}(P)-conjugate it follws that b ∈ (OGi)_{ΔP}^{ΔG}. Thus there is c ∈ (OG)^{ΔP} such that Tr_{ΔP}^{ΔG}(ci) = b. Now Tr_{ΔP}^{ΔG}(ci) belongs to the two-sided ideal OGiOG generated by i, and hence b is in the image of the the map OGi ⊗_{iOGi} iOG → OG given by multiplication in OG.

Therefore this map is surjective and the result follows. □

Source algebras are known to be “relatively separable” with respect to their defect groups. As before, this is true in slightly greater generality:

Proposition 4.2. Let G be a finite group, let b be a block of OG, let P be a defect group of b and let i be an idempotent in (OGb)^{ΔP} such that Br_{ΔP} (i) ≠ 0. Set A = iOGi. The canonical map A ⊗_{OP} A → A induced by multiplication in A splits as homomorphism of A-A-bimodules; in particular,

A is isomorphic to a direct summand of A ⊗_{OP} A as A-A-bimodule.

Proof. As in the proof of 4.1, write b = Tr_{ΔP}^{ΔG}(ci) for some c ∈ (OG)^{ΔP}. One checks that the map sending a ∈ OG to ∑_{x∈[G/P]} axci⊗ix^{−1} in OGi ⊗_{OP} iOG is a section for the canonical OG-OG-bimodule

homomorphism OGi ⊗_{OG}iOG → OGb given by multiplication in OG. Multiplying this homomorphism

by i on the left and right yields the required section as stated. □

Note that the idempotent i ∈ (OGb)^{ΔP} in 4.1 and 4.2 need not be primitive (if it is, then A is a source algebra of b). A crucial property of a source idempotent i ∈ (OGb)^{ΔP} is that, by [9, 1.8], for any subgroup Q of P the idempotent Br_{ΔQ}(i) in kC_{G}(Q) belongs in fact to a unique block algebra of kC_{G}(Q). In the following definition we keep this key property but require no longer that i is primitive:

Definition 4.3 Let G be a finite group, let b be a block of OG, let P be a defect group of b. An idempotent i in (OGb)^{ΔP} is called an almost source idempotent if Br_{ΔP} (i) ≠ 0 and for every subgroup Q of P there is a unique block e_{Q} of kC_{G}(Q) such that Br_{ΔQ}(i) ∈ kC_{G}(Q)e_{Q}. The interior P -algebra iOGi is then called an almost source algebra of the block b.

Remark 4.4. As pointed out before, any source idempotent is an almost source idempotent, by [9, 1.8]. The choice of an almost source idempotent i ∈ (OGb)^{ΔP} determines a fusion system F as follows. For any subgroup Q of P denote by e_{Q} the unique block of kC_{G}(Q) such that Br_{ΔQ}(i)e_{Q} ≠ 0. For any two subgroups Q, R of P, the set Hom_{F}(Q, R) is the set of all group homomorphisms φ : Q → R for which there is an element x ∈ G satisfying φ(u) = xux^{−1} for all u ∈ Q and satisfying xe_{Q}x^{−1} = e_{xQx^{−1}}, which is equivalent to the inclusion of Brauer pairs ^{x}(Q, e_{Q}) ⊆ (R, e_{R}) as defined in [1]. It follows from the results in [1] that F is indeed a fusion system; see e.g. [18, §2] or [19] for some more details. Moreover, a subgroup Q of P is fully F-centralised if and only if C_{P} (Q) is a defect group of the block e_{Q} of kC_{G}(Q). Given a subgroup Q of P it is always possible to find a subgroup R of P such that Q ≅ R in F and such that R is fully F-centralised. The following observation explains why we will need to work with fully centralised subgroups and almost source idempotents rather than source idempotents:

Proposition 4.5. Let G be a finite group, let b be a block of OG, let P be a defect group of b and let i be an almost source idempotent in (OGb)^{ΔP}. Let F be the fusion system determined by i on P, let Q be a fully F-centralised subgroup of P and let e be the unique block of kC_{G}(Q) such that Br_{ΔQ}(i)e ≠ 0. The idempotent Br_{ΔQ}(i) in (kC_{G}(Q)e)^{ΔC_{P} (Q)} is an almost source idempotent of the block e of kC_{G}(Q).

Proof. Since Q is fully F-centralised, the group C_{P} (Q) is a defect group of e (cf. [18, §2]). The idempotent Br_{ΔQ}(i) belongs to kC_{G}(Q)^{C_{P} (Q)} and satisfies Br_{ΔC_{P} (Q)}(Br_{ΔQ}(i)) = Br_{ΔQC_{P} (Q)}(i) which is non zero because even Br_{ΔP} (i) is non zero. Moreover, for every subgroup S of C_{P} (Q) we have C_{C_{P} (Q)}(S) = C_{P} (QS) and C_{C_{G}(Q)}(S) = C_{G}(QS). Thus the unique block f of C_{G}(QS) satisfying Br_{ΔQS}(i)f ≠ 0 is also the unique block of kC_{C_{G}(Q)}(S) satisfying Br_{ΔS}(Br_{ΔQ}(i))f ≠ 0, and so Br_{ΔQ}(i) is an almost source idempotent. □

Proposition 4.6. Let G be a finite group, let b be a block of OG, let P be a defect group of b and let i be an almost source idempotent in (OGb)^{ΔP}. Set A = iOGi and let F be the fusion system of A on P. Let Q be a fully F-centralised subgroup of P and let e be the unique block of kC_{G}(Q) such that Br_{ΔQ}(i)e ≠ 0.

(i) The block algebra kC_{G}(Q)e_{Q} is Morita equivalent to the algebra A(ΔQ) via the bimodule kC_{G}(Q)Br_{ΔQ}(i) and its dual. (ii) For any local point δ of Q on OGb satisfying Br_{ΔQ}(δ)e_{Q} ≠ {0} we have δ ∩ A ≠ ∅. (iii) The canonical map A(ΔQ) ⊗_{kC_{P} (Q)} A(ΔQ) → A(ΔQ) induced by multiplication in A(ΔQ) splits

as homomorphism of A(ΔQ)-A(ΔQ)-bimodules.

Proof. By 4.5, Br_{ΔQ}(i) is an almost source idempotent, and thus 4.1.(i) applies to kC_{G}(Q)e and Br_{ΔQ}(i) which implies statement (i). If δ is a local point of Q on OGb satisfying Br_{ΔQ}(e_{Q}) ≠ {0} then Br_{ΔQ}(δ) is a conjugacy class of primitive idempotents in kC_{G}(Q)e. Since kC_{G}(Q)e is Morita equivalent to A(ΔQ) = Br_{ΔQ}(i)kC_{G}(Q)Br_{ΔQ}(i) there is j ∈ δ such that Br_{ΔQ}(j) ∈ Br_{ΔQ}(i)kC_{G}(Q)Br_{ΔQ}(i). The usual lifting theorems for idempotents imply that j can be chosen in A^{ΔQ} = i(kG)^{ΔQ}i, whence statement (ii). Statement (iii) follows from 4.1.(ii) applied to kC_{G}(Q), e_{Q}, Br_{ΔQ}(i) and C_{P} (Q) instead of OG, b, i and P, respectively. □

This explains why it is technically easier to work at the source algebra level: while (OGb)(ΔQ) = kC_{G}(Q)Br_{ΔQ}(b) may decompose as product of more than one block algebra, the algebra A(ΔQ) remains indecomposable so long as one chooses Q to be fully centralised with respect to the fusion system of A on P. The following result is a tool to switch back and forth between block and source algebras:

Proposition 4.7. Let G be a finite group, let b be a block of OG, let P be a defect group of b and let i be an almost source idempotent in (OGb)^{ΔP}. Set A = iOGi and let F be the fusion system of A on P. Let Q be a fully F-centralised subgroup of P and let e be the unique block of kC_{G}(Q) such that Br_{ΔQ}(i)e ≠ 0. Let θ : Q → O^{×} be a group homomorphism. For any A-module M there is a natural isomorphism of kC_{G}(Q)e-modules

e((OGi ⊗_{A} M)(Q, θ)) ≅ kC_{G}(Q)Br_{ΔQ}(i) ⊗_{A(ΔQ)} M(Q, θ) .

Proof. Since Br_{ΔQ}(i) ∈ kC_{G}(Q)e, both sides in the statement are indeed kC_{G}(Q)e-modules. By 4.6.(i), multiplication by Br_{ΔQ}(i) induces a Morita equivalence between kC_{G}(Q)e and A(ΔQ). Thus, in order to show that there is a natural isomorphism as stated, it suffices to show this after multiplying both sides with Br_{ΔQ}(i). Using 3.9, the left side becomes

Br_{ΔQ}(i)(kGi ⊗_{A} M)(Q, θ)) = (ikGi ⊗_{A} M)(Q, θ) = (A ⊗_{A} M)(Q, θ) ≅ M(Q, θ)

and the right side becomes

Br_{ΔQ}(i)kC_{G}(Q)Br_{ΔQ}(i) ⊗_{A(ΔQ)} M(Q, θ) ≅ M(Q, θ) ;

both sides are naturally isomorphic as A(ΔQ)-modules. □

Proposition 4.8. Let G be a finite group, b a block of OG, P a defect group of b and i ∈ (OGb)^{ΔP} an almost source idempotent. Let Q be a subgroup of P and let e, e′ be two different blocks of kC_{G}(Q). For any subgroup R of P and any direct summand Y of OGi ⊗_{OR} iOG we have eY (ΔQ)e′ = {0}.

Proof. We may assume O = k. Since i is an almost source idempotent, for any subgroup S of P there is a unique block e_{S} of kC_{G}(S) satisfying Br_{ΔS}(i)e_{S} ≠ 0. The image of the set G × G in kG ⊗_{kR} kG is

a k-basis which is stable under the action of ΔQ. In order to compute Y (ΔQ) we have to determine the ΔQ-fixpoints in this basis. For x, y ∈ G the image x ⊗ y in kG ⊗_{kR} kG is fixed under the action

of ΔQ if for every u ∈ Q there is r_{u} ∈ R such that ux = xr_{u} and yu^{−1} = (r_{u})^{−1}y, or equivalently, if and only if xy ∈ C_{G}(Q) and Q^{x} = ^{y}Q ⊆ R. Thus we have

(kG ⊗_{kR} kG)^{ΔQ} = ∑_{(x,y)} k(x ⊗ y) ⊕ ker(Br_{ΔQ})

where (x, y) runs over the set of pairs in G × G satisfying xy ∈ C_{G}(Q) and Q^{x} ⊆ R. Consequently, we get that

(ekGi ⊗_{kR} ikGe′)^{ΔQ} = ∑_{(x,y)} k(exi ⊗ iye′) + ker(Br_{ΔQ})

with (x, y) as before. Write exi = e(^{x}i)x. Since i is an almost source idempotent we have e(^{x}i) ∈ ker(Br_{ΔQ}) unless e = ^{x}(e_{Q^{x}}). Similarly, we have (^{y}i)e′ ∈ ker(Br_{ΔQ}) unless e′ = (e_{^{y}Q})^{y}. But these two equalities would imply the contradiction e = e′ since xy ∈ C_{G}(Q). Thus at least one of e(^{x}i), (^{y}i)e′ is contained in ker(Br_{ΔQ}), whence the result. □

Proposition 4.9. Let G, H be finite groups, let b, c be blocks of OG, OH with defect group P, Q, respectively. Let i ∈ (OGb)^{ΔP} and j ∈ (OHc)^{ΔQ} be almost source idempotents, let F be the fusion system on P determined by i and let G be the fusion system on Q determined by j. Identify OG⊗_{O}OH

and O(G × H) through the canonical isomorphism.

(i) b ⊗ c is a block of O(G × H) and P × Q is a defect group of b ⊗ c. (ii) i ⊗ j is an almost source idempotent of b ⊗ c. (iii) The fusion system determined by i ⊗ j on P × Q is equal to F × G.

Proof. Statement (i) is well-known (and easy to prove). For R a subgroup of P ×Q, denote by R_{1} the image of R under the projection P ×Q → P and by R_{2} the image of R under the projection P ×Q → Q. Then R ⊆ R_{1} × R_{2} and C_{G×H}(R) = C_{G}(R_{1}) × C_{H}(R_{2}). Thus Br_{ΔR}(i ⊗ j) = Br_{ΔR_{1}}(i) ⊗ Br_{ΔR_{2}}(j). Therefore, if e_{1}, f_{2} are the unique blocks of kC_{G}(R_{1}), kC_{H}(R_{2}), respectively, satisfying Br_{ΔR_{1}}(i)e_{1} ≠ 0 and Br_{ΔR_{2}}(j)f_{2} ≠ 0 then e = e_{1} ⊗ f_{2} is the unique block of C_{G×H}(R) satisfying Br_{ΔR}(i ⊗ j)e ≠ 0. This shows (ii), and (iii) is clear. □

Proposition 4.10. Let G be a finite group, let b be a block of OG, let P be a defect group of b and let i ∈ (OGb)^{ΔP} be an almost source idempotent. Let α be the anti-automorphism of OG sending x ∈ G to x^{−1}. Set b^{0} = α(b) and i^{0} = α(i). Then b^{0} is a block of OG having P as defect group and i^{0} ∈ (OGb^{0})^{ΔP} is an almost source idempotent of b^{0}. Moreover, the fusion systems on P determined by i and by i^{0} are equal.

Proof. Straightforward verification. □

Combining 4.7 and 4.9, 4.10 yields another statement about switching between block algebras and source algebras for bimodules which will be needed in the proof of 1.4:

Proposition 4.11. Let G, H be finite groups, let b, c be blocks of OG, OH, respectively, having a common defect group P. Let i ∈ (OGb)^{ΔP} and j ∈ (OHc)^{ΔP} be almost source idempotents. Suppose that i and j determine the same fusion system F on P. Let Q be a fully F-centralised subgroup of P and let e and f be the unique blocks of kC_{G}(Q) and kC_{H}(Q), respectively, satisfying Br_{ΔQ}(i)e ≠ 0 and Br_{ΔQ}(j)f ≠ 0. Let X be an A-B-bimodule and set Y = e((OGi ⊗_{A} X ⊗_{B} jOH)(ΔQ))f. Then

Y ≅ kC_{G}(Q)Br_{ΔQ}(i) ⊗_{A(ΔQ)} X(ΔQ) ⊗_{B(ΔQ)} Br_{ΔQ}(j)kC_{H}(Q)

as kC_{G}(Q)e-kC_{H}(Q)f-bimodules and

Br_{ΔQ}(i)Y Br_{ΔQ}(j) ≅ X(ΔQ)

as A(ΔQ)-B(ΔQ)-bimodules.

Proof. By 2.2, ΔQ is fully F ×F-centralised. By 4.7 applied to G×H, the almost source idempotent i ⊗ j^{0}, the subgroup ΔQ of P × P and the block e ⊗ f^{0} of C_{G×H}(ΔQ) we get the first of the two isomorphisms, and multiplying this on the left and on the right by Br_{ΔQ}(i) and Br_{ΔQ}(j), respectively, yields the second isomorphism. □

§5 Fusion in almost source algebras

By [22, 3.1], the fusion system F of a block b of OG can be read off the OP -OP -bimodule structure of the source algebra A = iOGi of b. The following Propositions collect some technical details around this theme, similar to material in [15, 17], extended here to almost source idempotents.

Proposition 5.1. Let G be a finite group, let b be a block of OG with defect group P, let i ∈ (OGb)^{ΔP} be an almost source idempotent and set A = iOGi. Denote by F the fusion system of A on P. Let Q be a fully F-centralised subgroup of P and let φ : Q → P be a morphism in F. Set R = φ(Q). Denote by e_{Q}, e_{R} the unique blocks of kC_{G}(Q), kC_{G}(R) satisfying Br_{ΔQ}(i)e_{Q} ≠ 0 and Br_{ΔR}(i)e_{R} ≠ 0.

(i) For any primitive idempotent n in (OGb)^{ΔR} satisfying Br_{ΔR}(n)e_{R} ≠ 0 there is a primitive idempotent m in A^{ΔQ} satisfying Br_{ΔQ}(m) ≠ 0 such that mOG ≅ _{φ}(nOG) as OQ-OGb-bimodules and such that OGm ≅ (OGn)_{φ} as OGb-OQ-bimodules. (ii) For any primitive idempotent n in A^{ΔR} satisfying Br_{ΔR}(n) ≠ 0 there is a primitive idempotent m in A^{ΔQ} satisfying Br_{ΔQ}(m) ≠ 0 such that mA ≅ _{φ}(nA) as OQ-A-bimodules and such that Am ≅ (An)_{φ} as A-OQ-bimodules.

Proof. Since φ belongs to the fusion system F there is an element x ∈ G such that φ(u) = xux^{−1} for all u ∈ Q and such that ^{x}(e_{Q}) = e_{R}. Let n be a primitive idempotent in (OGb)^{ΔR} satisfying Br_{ΔR}(n)e_{R} ≠ 0. Denote by ν be the local point of R on OGb containing n. Since ^{x}Q = R there is a local point μ of Q on OGb such that ν = ^{x}μ. Since Br_{ΔR}(ν)e_{R} ≠ 0 we have Br_{ΔQ}(μ)e_{Q} ≠ 0. This implies that Br_{ΔQ}(μ) is in fact a conjugacy class of primitive idempotents in kC_{G}(Q)e_{Q}. Now Q is fully F-centralised, and hence by 4.6.(ii), μ must contain an element m such that m ∈ A. Moreover, n^{x} and m belong both to the same point μ of Q on OGb, and hence there is an element c ∈ ((OGb)^{ΔQ})^{×} such that ^{xc}m = n. The map sending ma ∈ mOG to xcma = nxca is the required isomorphism mOG ≅ _{φ}(nOG). The map sending am to amc^{−1}x^{−1} = ac^{−1}x^{−1}n is the required isomorphism OGm ≅ (OGn)_{φ}. This proves (i). Statement (ii) is played back to (i) via the Morita equivalence between OGb and A. Let n be a primitive idempotent in A^{ΔR} satisfying Br_{ΔR}(n) ≠ 0. The unit element i of A satisfies Br_{ΔR}(i)e_{R} = Br_{ΔR}(i) by the uniqueness of the inclusion of Brauer pairs. Thus Br_{ΔR}(n)e_{R} ≠ 0. Therefore statement (i) applies and yields a primitive idempotent m ∈ A^{ΔQ} satisfying Br_{ΔQ}(m) ≠ 0 such that there are isomorphisms mOG ≅ _{φ}(nOG) and OGm ≅ (OGn)_{φ}. Multiplying these isomorphisms by i on the right and on the left, respectively, yields the isomorphisms as stated in (ii). □

The following is [17, 7.7, 7.8] generalised to almost source idempotents.

Proposition 5.2. Let G be a finite group, let b be a block of OG with defect group P, let i be an almost source idempotent in (OGb)^{ΔP} and set A = iOGi. Denote by F be the fusion system of A on P. Let Q, R be subgroups of P.

(i) Every indecomposable direct summand of A as OQ-OR-bimodule is isomorphic to OQ ⊗_{OS} _{φ}OR

for some subgroup S of Q and some morphism φ : S → R belonging to F. (ii) If φ : Q → R is an isomorphism in F such that R is fully F-centralised then _{φ}OR is isomorphic to a direct summand of A as OQ-OR-bimodule.

Proof. (i) Let Y be an indecomposable direct summand of A as OQ-OR-bimodule. Then Y has a Q × R-stable O-basis on which Q and R act freely on the left and on the right, respectively. Thus

Y ≅ OQ ⊗_{OS} _{φ}OR

for some subgroup S of Q and some injective group homomorphism φ : S → R. Set T = φ(S). Restricting Y to S × T shows that _{φ}OT is isomorphic to a direct summand of Y, hence of A, as OS-OT -bimodule. Now A is a direct summand of OG as OP -OP -bimodule. In particular, _{φ}OT is isomorphic to a direct summand of OG as OS-OT -bimodule, hence isomorphic to OSy^{−1} = y^{−1}OT for some element y ∈ G such that^{y}S = T and such that^{y}s = φ(s) for all s ∈ S. Then OS is isomorphic to a direct summand of iOGiy = iOGy^{−1}iy as OS-OS-bimodule. Thus Br_{ΔS}(iOGy^{−1}iy) ≠ 0 by 3.5. Since Br_{ΔS}(i) ∈ kC_{G}(S)e_{S} this forces also that Br_{ΔS}(y^{−1}iy)e_{S} ≠ 0. Conjugating by y yields that Br_{ΔT} (i)^{y}e_{S} ≠ 0. But then necessarily ^{y}e_{S} = e_{T} because e_{T} is the unique block of kC_{G}(T) with the property Br_{ΔT} (i)e_{T} ≠ 0. This shows that φ is a morphism in the fusion system F, whence (i).

(ii) Since φ belongs to F there is an element x ∈ G such that φ(u) = ^{x}u for all u ∈ Q and such that xe_{Q} = e_{R}. Let μ be a local point of Q on OGb such that μ ∩ A ≠ ∅. Set ν = ^{x}μ; that is, ν is the local point of R on OGb such that ^{x}(Q_{δ}) = R_{ν}. Since μ ∩ A ≠ ∅ we have Br_{ΔQ}(μ)e_{Q} ≠ 0. Conjugating by x implies that Br_{ΔR}(ν)e_{R} ≠ 0. Since R is fully F-centralised, it follows from 4.6.(ii) that also ν ∩A ≠ ∅. Let m ∈ μ∩A and let n ∈ ν ∩A. Note that n and ^{x}m belong both to ν, hence are conjugate in (A^{R})^{×}. Since Br_{ΔQ}(m) ≠ 0 we get (mOGm)(ΔQ) ≠ {0}, hence mOGm has a direct summand isomorphic to OQ as OQ-OQ-bimodule. Therefore, mOGmx^{−1} = mOGxmx^{−1} ≅ mOGn = mAn has a direct summand isomorphic to OQ_{φ^{−1}} ≅ _{φ}OR as OQ-OR-bimodule. □

Corollary 5.3. With the notation of 5.2, the fusion system F is generated by the set of inclusions between subgroups of P and automorphisms φ of any subgroup Q of P for which _{φ}OQ is isomorphic to a direct summand of A as OQ-OQ-bimodule.

Proof. This follows from 5.2 and Alperin’s fusion theorem. □

6 Vertices of p-permutation modules for almost source algebras

The Brauer construction sends p-permutation modules to p-permutation modules. Combining material of the previous sections we can be more precise regarding vertices:

Proposition 6.1. Let A be an almost source algebra of a block with defect group P and fusion system F. Let R be a subgroup of P and let X be an indecomposable direct summand of the A-module A ⊗_{OR}O.

Then X is isomorphic to a direct summand of Am ⊗_{OQ} O for some fully F-centralised subgroup Q of

P which is isomorphic, in F, to a subgroup of R, and some primitive idempotent m ∈ A^{ΔQ} satisfying Br_{ΔQ}(m) ≠ 0.

Proof. Since X is indecomposable, X is isomorphic to a direct summand of An ⊗_{OR} O for some

primitive idempotent n ∈ A^{ΔR}. Up to replacing R by a proper subgroup we may assume by [15, 2.6] that Br_{ΔR}(n) ≠ 0. Let Q be a fully F-entralised subgroup of P such that there is an isomorphism φ : Q ≅ R in F. By 5.1.(ii) there is a primitive idempotent m ∈ A^{ΔQ} such that Br_{ΔQ}(m) ≠ 0 and such that Am ≅ (An)_{φ} as A-OQ-bimodules. Then An ⊗_{OR} O ≅ Am ⊗_{OQ} O, and hence in particular, X

is isomorphic to a direct summand of A ⊗_{OQ} O as stated. □

We use 6.1 to track to vertices of indecomposable summands of p-permutation modules after applying the Brauer construction:

Theorem 6.2. Let A be an almost source algebra of a block with defect group P and fusion system F. Let Q, R be subgroups of P such that Q is fully F-centralised. Let Y be an indecomposable direct summand of the A(ΔQ)-module (A ⊗_{OR} O)(Q). Then Y is isomorphic to a direct summand

of A(ΔQ)m ⊗_{kS} k for some subgroup S of C_{P} (Q) containing Z(Q) and some primitive idempotent

m ∈ A(ΔQ)^{ΔS} satisfying Br_{ΔS}(m) ≠ 0 such that S is fully C_{F}(Q)-centralised and QS is isomorphic in F to a subgroup of R.

Proof. By 4.6, A(ΔQ) is isomorphic to a direct summand of A(ΔQ) ⊗_{kC_{P} (Q)} A(ΔQ). Tensoring with

− ⊗_{A(ΔQ)}Y implies that Y is isomorphic to a direct summand of A(ΔQ) ⊗_{kC_{P} (Q)}Y. Since Y is indecom-

posable, Y is in fact isomorphic to a direct summand of A(ΔQ) ⊗_{kC_{P} (Q)} W for some indecomposable

direct summand of Y as kC_{P} (Q)-module. Thus we need to determine the kC_{P} (Q)-module structure of Y. Let X be a P -P -stable O-basis of A. We use the notation X ⊗ 1 for the image of the set X in A ⊗_{OR} O. The set X ⊗ 1 is a P -stable O-basis of A ⊗_{OR} O. Thus, in order to compute (A ⊗_{OR} O)(Q) we

need to determine the Q-fixpoints in the set X ⊗ 1. For x, y ∈ X the images x ⊗ 1 and y ⊗ 1 in X ⊗ 1 are equal if and only if there is an element r ∈ R such that y = xr. Therefore, x ⊗ 1 ∈ (X ⊗ 1)^{Q} if and only if for every u ∈ Q there is r_{u} ∈ R such that ux = xr_{u}. In that case, r_{u} is then uniquely determined by u because P acts freely on the right of X. Since P acts also freely on the left of X, the map sending u ∈ Q to r_{u} ∈ R is an injective group homomorphism φ : Q → R. Set

U_{φ} = {x ∈ X | ux = xφ(u) (∀ u ∈ Q)}

Note that by 5.2.(i) any φ arising in this way belongs to the fusion system F because OQx = xOφ(Q) ≅ _{φ}Oφ(Q) is a direct summand of A as OQ-Oφ(Q)-bimodule. It follows from the above that

(X ⊗ 1)^{Q} = ⋃_{φ} U_{φ} ⊗ 1

where φ runs over the set Hom_{F}(Q, R). Let φ, ψ ∈ Hom_{F}(Q, R) and denote by φ̃, ψ̃ their images in the orbit space Inn(R)\Hom_{F}(Q, R), where the group of inner automorphisms Inn(R) of R acts by composition of group homomorphisms on the set Hom_{F}(Q, R). Suppose there is an element x ∈ U_{φ} and y ∈ U_{ψ} such that x ⊗ 1 = y ⊗ 1. Then there is an element r ∈ R such that y = xr, and for all u ∈ U, we have ux = xφ(u) and uy = yψ(u). Thus uxr = xφ(u)r = xrψ(u), hence φ(u)r = ψ(u)r, or equivalently, φ(u) = ^{r}ψ(u) for all u ∈ Q. This means that φ̃ = ψ̃. It follows that U_{φ} ⊗ 1 = U_{ψ} ⊗ 1 if φ̃ = ψ̃ and that U_{φ} ⊗ 1 ∩ U_{ψ} ⊗ 1 = ∅ if φ̃ ≠ ψ̃. In other words, (X ⊗ 1)^{Q} is in fact the disjoint union

(X ⊗ 1)^{Q} = ⋃_{φ} U_{φ} ⊗ 1

with φ running over a set of representatives in Hom_{F}(Q, R) of Inn(R)\Hom_{F}(Q, R). Therefore, by 3.4, we have

(A ⊗_{OR} O)(Q) ≅ k((X ⊗ 1)^{Q}) = ⊕_{φ} k(U_{φ} ⊗ 1)

with φ running as before over a set of representatives in Hom_{F}(Q, R) of Inn(R)\Hom_{F}(Q, R). This is a decomposition of (A ⊗_{OR} O)(Q) as kC_{P} (Q)-module because the subsets U_{φ} of X are invariant

under the left action of C_{P} (Q). Thus the direct summand W of Y as kC_{P} (Q)-module is isomorphic to a direct summand of k(U_{φ} ⊗ 1) for some φ ∈ Hom_{F}(Q, R). Now W is a permutation kC_{P} (Q)- module, and hence, in order to determine its structure, it suffices to determine the stabilisers in C_{P} (Q) of basis elements x ⊗ 1 ∈ U_{φ} ⊗ 1. Let x, y ∈ U_{φ} such that x ⊗ 1 = y ⊗ 1. Then, on one hand, y = xr for some r ∈ R, and on the other hand, ux = xφ(u) and uy = yφ(u) for all u ∈ Q. Thus uxr = xφ(u)r = xrφ(u), which implies r ∈ C_{P} (φ(Q)). It follows that the stabiliser S in C_{P} (Q) of an element x⊗1 ∈ U_{φ}⊗1 consists of all elements z ∈ C_{P} (Q) such that zx = xr_{z} for some r_{z} ∈ C_{P} (φ(Q)). In particular Z(Q) ⊆ S because zx = xφ(z) for z ∈ Z(Q). It follows that there is a well-defined group homomorphism ψ : QS ⟶ R given by ψ(uz) = φ(u)r_{z} for all u ∈ Q and z ∈ S. Note that ψ has the property uzx = xψ(uz), so O(QS)x = xO(ψ(QS)) ≅ _{ψ}Oψ(QS) is a direct summand of A as O(QS)-Oψ(QS)-bimodule. Hence ψ belongs to the fusion system F by 5.2.(i). This shows that Y is isomorphic to a direct summand of A(ΔQ) ⊗_{kS} k for some subgroup S of C_{P} (Q) containing Z(Q) for

which there exists a morphism ψ : QS → R in F. We need to show that S can furthermore be chosen to be fully C_{F}(Q)-centralised. Applying 6.1 to A(ΔQ) and its fusion system C_{F}(Q) shows that Y is isomorphic to a direct summand of A(ΔQ)m ⊗_{kT} k for some fully C_{F}(Q)-centralised subgroup T

of C_{P} (Q) for which there exists a morphism τ : T → S in C_{F}(Q) and some primitive idempotent m ∈ A(ΔQ)^{ΔT} satisfying Br_{ΔT} (m) ≠ 0. By the definition of C_{F}(Q), the morphism τ extends to a morphism σ : QT → QS in F satisfying σ|_{Q} = Id_{Q}. Thus replacing S by T and ψ by ψ ∘ σ concludes the proof. □

Corollary 6.3. Let A be an almost source algebra of a block with defect group P and fusion system F. Let Q, R be subgroups of P such that Q is fully F-centralised. If Hom_{F}(Q, R) = ∅ then (A ⊗_{OR}O)(Q) =

{0}.

Corollay 6.4. Let A, B be almost source algebras of blocks of finite groups having a common defect group P and the same fusion system F. Let Q, R be subgroups of P such that Q is fully F-centralised. If Hom_{F}(Q, R) = ∅ then (A ⊗_{OR} B)(ΔQ) = {0}.

Proof. This follows from 6.3 in conjunction with 4.9 and 2.2. □

Corollary 6.5. Let A be an almost source algebra of a block with defect group P and fusion system F. Let Q be a fully F-centralised subgroup of P. Every indecomposable direct summand of (A ⊗_{OQ} O)(Q)

is isomorphic to A(ΔQ)m ⊗_{kZ(Q)} k for some primitive idempotent m ∈ A(ΔQ).

Proof. By 6.2, an indecomposable direct summand Y of (A ⊗_{OQ} O)(Q) is isomorphic to a direct

summand of A(ΔQ)m ⊗_{kS} k for some subgroup S of C_{P} (Q) containing Z(Q) and some primitive

idempotent m ∈ A(ΔQ)^{ΔS} satisfying Br_{ΔS}(m) ≠ 0 such that QS is isomorphic to a subgroup of Q in F. This forces S ⊆ Q, hence S = Z(Q). Since the image of Z(Q) in A(ΔQ)^{×} belongs to Z(A(ΔQ)), a primitive idempotent in A(ΔQ)^{Z(Q)} is primitive in A(ΔQ) and hence the module A(ΔQ)m ⊗_{kZ(Q)} k

is indecomposable since it is an indecomposable projective module for the algebra A(ΔQ) ⊗_{kZ(Q)} k. □

7 Grothendieck groups of trivial and linear source modules for almost source algebras

We translate in this section some of the concepts and terminology related to trivial source and linear source modules for block algebras to the source algebra level.

Definition 7.1. Let A, B be O-algebras. We denote by R(A, B) the Grothendieck group of A-B-bimodules which are finitely generated as left A-modules and as right B-modules, with respect to split exact sequences. We denote by P(A, B) the subgroup of R(A, B) generated by the A-B-bimodules which are finitely generated projective as left A-modules and as right B-modules. For A, B, C three O-algebras we denote by

− ·_{B} − : R(A, B) × R(B, C) ⟶ R(A, C)

the group homomorphism induced by the tensorproduct M ⊗_{B} N over B of an A-B-bimodule M and

a B-C-bimodule N. We consider R(A) = R(A, A) as unitary associative ring with multiplication − ·_{A} − and unit element [A], the isomorphism class of A as A-A-bimodule in R(A), and we consider

R(A, B) as R(A)-R(B)-bimodule via − ·_{A} − and − ·_{B} −. Clearly the bilinear map − ·_{B} − induces a bilinear map

− ·_{B} − : P(A, B) × P(B, C) ⟶ P(A, C) ;

in particular, P(A) = P(A, A) is a unitary subring of R(A).

Following [21], given a finite group P, an interior P -algebra is an O-algebra A endowed with a group homomorphism P → A^{×}, or equivalently, endowed with a unitary algebra homomorphism OP → A.

Definition 7.2. Let P be a finite group and let A, B be interior P -algebras. We denote by T (A, B) the subgroup of R(A, B) generated by the finite direct sums of summands of the A-B-bimodules A ⊗_{OQ} B, where Q runs over the subgroups of P.

We will also need a source algebra version of linear source modules. Let Q be a finite p-group. Any group homomorphism θ : Q → O^{×} induces an O-algebra automorphism sending u ∈ Q to θ(u)u.

Given an OQ-module U we denote by _{θ}U the OQ-module which is equal to U as O-module, with u ∈ Q acting as θ(u)u on U. We use a similar notation for right modules and bimodules.

Definition 7.3 Let P be a finite p-group and let A, B be interior P -algebras. We denote by L(A, B) the Grothendieck group of finite direct sums of summands of A-B-bimodules of the form A ⊗_{OQ} _{θ}B,

with Q running over the subgroups of P and θ over the set Hom(Q, O^{×}) of group homomorphisms from Q to O^{×}.

Clearly T (A, B) is a subgroup of L(A, B). If A, B are finitely generated projective over OP then T (A, B) and L(A, B) are subgroups of P(A, B). If A is isomorphic to a direct summand of A ⊗_{OP} A as

A-A-bimodule then [A] belongs to T (A) = T (A, A). By 4.2 this case occurs whenever A is an almost source algebra of a block with P as defect group. The groups T (A, B) and L(A, B) are independent of the choice of almost source idempotents in the following sense:

Theorem 7.4. Let G, H be finite groups and let b, c be blocks of OG, OH with a common defect group P, respectively. Let i ∈ (OGb)^{ΔP}, j ∈ (OHc)^{ΔP} be almost source idempotents. Set A = iOGi and B = jOHj. Let i′ ∈ (OGb)^{ΔP}, j′ ∈ (OHc)^{ΔP} be idempotents satisfying i = ii′ = i′i and j = jj′ = j′j. Set A′ = i′OGi′ and B′ = j′OHj′. The map sending an A-B-bimodule M to the A′-B′-bimodule A′i ⊗_{A} M ⊗_{B} jB′ induces injective group homomorphisms

T (A, B) ⟶ T (A′, B′),
L(A, B) ⟶ L(A′, B′) .

If the idempotents i′, j′ are also almost source idempotents and if i, j determine the same fusion system on P then these maps are isomorphisms whose inverses are induced by the map sending an A′-B′-bimodule N to the A-B-bimodule iNj.

Proof. The condition i = ii′ = i′i is equivalent to i ∈ A′, so the statement makes sense. Let M be a direct summand of A ⊗_{OR} _{θ}B for some subgroup R of P and some group homomorphism θ : R → O.

Then A′i ⊗_{A} M ⊗_{B} jB′ is a direct summand of A′i ⊗_{A} A ⊗_{OR} _{θ}B ⊗_{B} jB′ ≅ A′i ⊗_{OR} _{θ}jB′, which is a direct

summand of A′ ⊗_{OR} _{θ}B′, Thus the map sending M to A′i ⊗_{A} M ⊗_{B} jB′ induces group homomorphisms

T (A, B) → T (A′, B′) and L(A, B) → L(A′, B′). These group homomorphisms are injective because A, A′ and B, B′ are Morita equivalent via the bimodules A′i, B′j and their duals, respectively, as a consequence of 4.1. Suppose now that i′ and j′ are almost source idempotents. In order to show the surjectivity of the above maps we may assume that i, j are source idempotents. Let N be an indecomposable direct summand of A′ ⊗_{OR}_{θ}B′ for some group homomorphism θ : Q → O^{×}. Thus N is

an indecomposable direct summand of A′n ⊗_{OR} _{θ}sB′ for some primitive idempotents n ∈ (A′)^{ΔR} and

s ∈ (B′)^{ΔR}. By choosing R of minimal order we may assume that Br_{ΔR}(n) ≠ 0 and Br_{ΔR}(s) ≠ 0. In other words, n belongs to a local point ν of Q on A′, and s belongs to a local point σ of R on B′. Since i′, j′ are almost source idempotents, the defect group P has unique local points γ, δ on A′, B′, respectively, and so i ∈ γ and j ∈ δ. Denote by F the common fusion system of G and H on P. Let φ : Q ≅ R be an isomorphism in F such that Q is fully F-centralised. By 5.1 there is a primitive idempotent m ∈ A^{ΔQ} satisfying Br_{ΔQ}(m) ≠ 0 such that OGm ≅ (OGn)_{φ} as OGb-OQ-bimodules. Multiplying by i′ yields an isomorphism A′m ≅ (A′n)_{φ} as A′-OQ-bimodules. Similarly, there is a primitive idempotent r ∈ B^{ΔQ} satisfying Br_{ΔQ}(r) ≠ 0 such that rB′ ≅ _{φ}sB′ as OQ-B′- bimodules. Then _{θ∘φ}rB′ ≅ _{φθ}sB′ as OQ-B′- bimodules. Thus N is isomorphic to a direct

summand of A′n ⊗_{OR} _{θ}sB′ ≅ A′m ⊗_{OQ} _{τ} sB′, where τ = θ ∘ φ : Q → O^{×}. Since m ∈ A, r ∈ B the

latter is isomorphic to a direct summand of A′i ⊗_{OQ} _{τ} jB′. Thus N is in fact a direct summand of

A′i ⊗_{OQ} _{τ} jB′. Multiplication by i on the left and by j on the right is a Morita equivalence between

the category of A′-B′-bimodules and the category of A-B-bimodules. Thus, setting M = iNj we get that N ≅ A′i ⊗_{A} M ⊗_{B} jB′. Since by construction M is a direct summand of A ⊗_{OQ} _{τ} B the surjectivity

of the given maps follows. □

Following the notation in [4, 1.2], if G, H are finite groups, b a block of OG and c a block of OH with a common defect group P, then T^{ΔP} (OGb, OHc) denotes the Grothendieck group of OGb-OHc-bimodules (with respect to split exact sequences) whose indecomposable direct summands have trivial source and a vertex contained in ΔP, when viewed as O(G × H)-modules. Similarly, L^{ΔP} (OGb, OHc) denotes the Grothendieck group of linear source OGb-OHc-bimodules (with respect to split exact sequences) whose indecomposable direct summands have a linear source and a vertex contained in ΔP. The following consequence of Theorem 7.4 implies in particular that the notion of p-permutation equivalence as introduced by Boltje and Xu is equivalent to the source algebra version 1.3 above in case the involved blocks are of principal type. Recall that b is said to be of principal type if Br_{ΔQ}(b) is a block of kC_{G}(Q) for every subgroup Q of a defect group P of b. Equivalently, b is of principal type if and only if b itself is an almost source idempotent. If b is the principal block of OG then by Brauer’s Third Main Theorem, Br_{ΔQ}(b) is the principal block of kC_{G}(Q) for any subgroup Q of a Sylow-p-subgroup P of G, and hence in particular and not surprisingly, the principal block b is of principal type.

Corollary 7.5. Let G, H be finite groups and let b, c be blocks of OG, OH with a common defect group P, respectively. Let i ∈ (OGb)^{ΔP}, j ∈ (OHc)^{ΔP} be almost source idempotents. Set A = iOGi and B = jOHj. The map sending an A-B-bimodule M to the OGb-OHc-bimodule OGi ⊗_{A} M ⊗_{B} jOH

induces injective group homomorphisms

T (A, B) ⟶ T^{ΔP} (OGb, OHc) ,
L(A, B) ⟶ L^{ΔP} (OGb, OHc) .

If the blocks b, c are of principal type and have the same fusion system these maps are isomorphisms.

Proof. For any subgroup R of P and any group homomorphism θ : R → O^{×} we have an isomorphism of O(G × H)-modules

OG ⊗_{OR} _{θ}OH ≅ Ind_{ΔR}^{G×H}(_{θ}O)

sending x ⊗ y^{−1} to (x, y) ⊗ 1 for x ∈ G, y ∈ H. Thus T (OGb, OHc) ≅ T^{ΔP} (OGb, OHc) and L(OGb, OHc) ≅ L^{ΔP} (OGb, OHc). Thus 7.4 applied to i′ = b and j′ = c yields the injective group homomorphisms as stated. If b and c are of principal type then b and c are almost source idempotents, and hence, if they determine the same fusion system it follows again from 7.4 that the maps in the statement are isomorphisms. □

Remark 7.6. For non principal blocks the maps in 7.5 need not be isomorphisms, in general. If OGb is a block algebra whose defect group P has two different local points γ, γ′ on OGb then OGi ⊗_{OP} i′OG

is a p-permutation O(G × G)-module, where i ∈ γ, i′ ∈ γ′, but not all indecomposable summands of iOGi ⊗_{OP} i′OGi as iOGi-iOGi-bimodules will be summands of bimodules of the form iOGi ⊗_{OQ} iOGi.

This is because γ′ = ^{x}γ for some x ∈ N_{G}(P) not contained in N_{G}(P_{γ}), so conjugation by x is an automorphism of P which does not belong to the A-fusion on P.

For the sake of completeness we include some further observations relating the notation of the present paper to that in [4]. Let A be an almost source algebra of a block b of a finite group G with defect group P. Consider O as trivial interior P -algebra with structural homomorphism P → O^{×} sending any element u ∈ P to 1_{O}. Then T (A, O) is the Grothendieck group of sums of summands of the left A-modules A ⊗_{OQ} O, where Q runs over the subgroups of P. Similarly,

L(A, O) is the Grothendieck group of sums of summands of the left A-modules A ⊗_{OQ} _{θ}O, where

Q ⊆ P and θ ∈ Hom(Q, O^{×}). Following the notation of Boltje and Xu [4], we denote by T (OGb) the Grothendieck group of p-permutation OGb-modules; that is, T (OGb) is generated by the isomorphism classes of summands of OGb ⊗_{OQ} O, with Q ⊆ P. Similarly, we denote by L(OGb) the Grothendieck

group generated by the linear source OGb-modules; that is, L(OGb) is generated by the isomorphism classes of summands of OGb ⊗_{OQ} _{θ}O, with Q ⊆ P and θ ∈ Hom(Q, O^{×}).

Proposition 7.7. Let G be a finite group, let b be a block of OG, let P be a defect group of b and let i ∈ (OGb)^{ΔP} be an idempotent satisfying Br_{ΔP} (i) ≠ 0 and set A = iOGi. The functor OGi ⊗_{A} − :

mod(A) → mod(OGb) induces group isomorphisms T (A, O) ≅ T (OGb) and L(A, O) ≅ L(OGb).

Proof. Let Q be a subgroup of P, let θ : Q → O^{×} be a group homomorphism and let Y be a direct summand of A ⊗_{OQ}_{θ}O. Then OGi ⊗_{A} Y is a direct summand of OGi ⊗_{OQ}_{θ}O, hence of OGb ⊗_{OQ}_{θ}O. Thus

the functor OGi ⊗_{A} − induces group homomorphisms T (A, O) → T (OGb) and L(A, O) → L(OGb).

These group homomorphisms are injective because the functor OGi ⊗_{A} − is an equivalence, by 4.1. To

see that they are also surjective, let X be an indecomposable linear source OGb-module. It follows from [13, 6.3] that one can choose a vertex R of X such that X is isomorphic to a direct summand of OGi ⊗_{OR} _{ρ}O for some group homomorphism ρ : R → O^{×}. Thus the functor OGi ⊗_{A} − induces the

isomorphisms as stated. □

With the notation of 7.5, the map given by − ·_{OHc} − sends T^{ΔP} (OGb, OHc) × L(OHc) to L(OGb)

(cf. [2, 2.3]). Clearly the map − ·_{B}− sends T (A, B)×L(B, O) to L(A, O). These two maps correspond to each other via the maps from 7.5 and 7.7:

Proposition 7.8. Let G, H be finite groups and let b, c be blocks of OG, OH with a common defect group P, respectively. Let i ∈ (OGb)^{ΔP}, j ∈ (OHc)^{ΔP} be almost source idempotents. Set A = iOGi and B = jOHj. We have a commutative diagram

\xymatrix{T (A, B) × L(B, O) \ar[r]^{− ·_{B}−} \ar[d] & L(A, O) \ar[d]_{≅} \\ T^{ΔP} (OGb, OHc) × L(OHc) \ar[r]_{− ·_{OHc}−} & L(OGb)}

where the vertical arrows are induced by those in 7.5 and 7.7.

Proof. Let M be a direct summand of an A-B-bimodule of the form A ⊗_{OQ} B for some subgroup Q of

P and let Y be a direct summand of a left B-module B ⊗_{OR} _{θ}O for some subgroup R of P and some

group homomorphism θ : R → O^{×}. Then M ⊗_{B} Y is a direct summand of A ⊗_{OQ} B ⊗_{OR} _{θ}O. Since B is a permutation O(Q × R)-module it follows that M ⊗_{B} Y is a direct sum of linear source A-modules.

The composition of maps T (A, B) × L(B, O) → T^{ΔP} (OGb, OHc) × L(OHc) → L(OGb) is induced by the map

(M, Y) ↦ (OGi ⊗_{A} M ⊗_{B} jOH, OHj ⊗_{B} Y) ↦ OGi ⊗_{A} M ⊗_{B} jOH ⊗_{OHc} OHj ⊗_{B} Y ≅ OGi ⊗_{A} M ⊗_{B} Y

and the composition of maps T (A, B) × L(B, O) → L(B, O) ≅ L(OGb) is induced by the map

(M, Y) ↦ M ⊗_{B} Y ↦ OGi ⊗_{A} M ⊗_{B} Y

and hence both maps coincide. □

§8 Ring and module structures of T (A, B) and L(A, B)

As a consequence of [17, §2] we get some extra structure on T (A, B) in case A, B are source algebras. The following theorem extends [17, 2.3.(i)] to linear source modules over almost source algebras.

Theorem 8.1. Let A, B, C be almost source algebras of p-blocks of finite groups with a common defect group P all having the same fusion system F on P. The tensorproduct over B induces bilinear maps

− ·_{B}− : L(A, B) × L(B, C) ⟶ L(A, C) ,

− ·_{B}− : T (A, B) × T (B, C) ⟶ T (A, C) .

Proof. Let M be an indecomposable direct summand of A ⊗_{OQ} _{θ}B and let N be an indecomposable

direct summand of B ⊗_{OR}_{η}C for some subgroups Q, R of P and some group homomorphisms θ : Q →

O^{×} and η : R → O^{×}. Let X be an indecomposable direct summand of M ⊗_{B} N. We have to show

that X is isomorphic to a direct summand of A ⊗_{OS} _{τ} C for some subgroup S of P and some group

homomorphism τ : S → O^{×}, such that τ is trivial whenever both θ, η are trivial. The bimodule X is isomorphic to a direct summand of

A ⊗_{OQ} _{θ}B ⊗_{OR} _{η}C .

Since X is indecomposable, there is an indecomposable direct summand W of B as OQ-OR-bimodule such that X is isomorphic to a direct summand of

A ⊗_{OQ} _{θ}W ⊗_{OR} _{η}C .

By 5.2.(i), the OQ-OR-bimodule W is isomorphic to OQ ⊗_{OS}_{φ}OR for some subgroup S of Q and some

morphism φ : S → R belonging to the fusion system F. Thus X is isomorphic to a direct summand of

A ⊗_{OS} _{φ}(_{σ}C) ,

where σ : φ(S) → O^{×} is the group homomorphism sending φ(y) to θ(y)η(φ(y)), for every y ∈ S. Since X is indecomposable, X is in fact isomorphic to a direct summand of An ⊗_{OS}_{φ}(_{σ}sC) for some primitive idempotent n ∈ A^{ΔS} and some primitive idempotent s ∈ C^{Δφ(S)}. By [15, 2.6] we may assume that n, s belong to local points of S, φ(S) on A, C, respectively. Let T be a fully F-centralised subgroup of P such that there is an isomorphism ψ : T → S in F. By 5.1 there are primitive idempotents m ∈ A^{ΔT} and r ∈ C^{ΔT} such that Am ≅ An_{ψ} as A-OT -bimodules, and such that rC ≅ _{φψ}sC as OT -C-bimodules. Thus X is a direct summand of

Am ⊗_{OT} _{τ} rC

where τ = σ ∘ ψ : T → O^{×}. This shows that the isomorphism class of X belongs to L(A, C). By construction, if θ, η are trivial, so is τ, and hence in that case, the isomorphism class of X belongs to T (A, C). □

Corollary 8.2. Let A be an almost source algebra of a p-block of a finite group with defect group P. The abelian groups T (A) and L(A) are unitary subrings of P(A).

Proof. Applying 8.1 to A = B = C shows that T (A) and L(A) are rings; the fact that they are unitary follows from 4.2. □

Corollary 8.3. Let A, B be almost source algebras of p-blocks of finite groups with a common defect group P and the same fusion system. Then L(A, B) is an L(A)-L(B)-bimodule and T (A, B) is an T (A)-T (B)-bimodule, with left and right module structure induced by the tensorproducts over A and B, respectively.

Proof. This follows from 8.1 applied to the cases A = B and B = C. □

Corollary 8.4. Let A, B be almost source algebras of p-blocks of finite groups with a common defect group P and the same fusion system. Let X be a p-permutation equivalence between A and B. The map sending Y ∈ T (A) to X^{∗} ·_{A}Y ·_{A}X is a ring isomorphism

T (A) ≅ T (B) .

Proof. For Y, Z in T (A) we have (X^{∗} ·_{A}Y ·_{A}X) ·_{B}(X^{∗} ·_{A}Z ·_{A}X) = X^{∗} ·_{A}Y ·_{A}(X ·_{B}X^{∗}) ·_{A}Z ·_{A}X =

X^{∗} ·_{A}Y ·_{A}[A] ·_{A}Z ·_{A}X = X^{∗} ·_{A}(Y ·_{A}Z) ·_{A}X, which shows that the given map is a ring homomorphism. Exchanging the roles of A, B and X, X^{∗} yields the inverse. □

The ring structure of T (A) does not depend on the choice of the almost source idempotents:

Proposition 8.5. Let G be a finite group, let b be a block of OG, let P, P′ be defect groups of b, let i ∈ (OGb)^{ΔP} and i′ ∈ (OGb)^{ΔP′} be almost source idempotents. Set A = iOGi and A′ = i′OGi′. There is a canonical ring isomorphism T (A) ≅ T (A′).

Proof. As a consequence of 7.4 we may assume that i, i′ are source idempotents. Let γ, γ′ be the local points of P, P′ on OGb containing i, i′, respectively. By [21, 1.2] there is x ∈ G such that P′_{γ′} = ^{x}(P_{γ}). We may assume that i′ = ^{x}i. Let α : A ≅ A′ be the O-algebra isomorphism sending a ∈ A to ^{x}a ∈ A′. Then α induces an isomorphism of A-A-bimodules

A ⊗_{OQ} A ≅ _{α}A′ ⊗_{Oα(Q)} A′_{α}

where we have identified Q to its image in A. Thus the map sending an A-A-bimodule M to the A′-A′-bimodule _{α^{−1}}M_{α^{−1}} induces a ring isomorphism T (A) ≅ T (A′). We need to show that this isomorphism does not depend on the choice of x. If y is another element such that P′_{γ′} = ^{y}(P_{γ}) then y = xn for some n ∈ N_{G}(P_{γ}), so it suffices to show that the automorphism of T (A) induced by n is the identity. By [22, 2.12] there is an element c ∈ A^{×} which normalises the image of P in A such that conjugation by c induces the same group automorphism P as conjugation by n. Denote by β the inner automorphism of A sending a ∈ A to ^{c}a. Then β induces for any A-A-bimodule M an isomorphism of A-A-bimodules M ≅ _{β}M_{β}. The result follows. □

Given two O-algebras A, B, the the O-dual M^{∗} = Hom_{O}(M, O) of an A-B-bimodule is a B-A-bimodule via (b.μ.a)(m) = μ(amb) for a ∈ A, b ∈ B, m ∈ M and μ ∈ M^{∗}. This, taking O-duality induces a group homomorphism R(A, B) ≅ R(B, A). Recall that an O-algebra A is called symmmetric if A is free of finite rank as O-module and if A ≅ A^{∗} as A-A-bimodule. Group algebras of finite groups, their block algebras and almost source algebras are symmetric.

Proposition 8.6. Let A, B be symmetric O-algebras. Taking O-duals of bimodules induces a group isomorphism P(A, B) ≅ P(B, A) and an anti-automorphism on the ring P(A).

Proof. Since A, B are symmetric, the O-dual of an A-B-bimodule which is finitely generated projective as left A-module and as right B-module is again finitely generated projective as left B-module and as right A-module. This shows the first statememt. If M, N are A-A-bimodules which are finitely generated projective as left and right A-modules then (M ⊗_{A} N)^{∗} ≅ N^{∗} ⊗_{A} M^{∗} which implies

the second statement. □

Proposition 8.7. Let A, B be almost source algebras of blocks of finite groups with a common defect group P. Taking O-duals of bimodules induces group isomorphisms L(A, B) ≅ L(B, A) and T (A, B) ≅ T (B, A), as well as anti-automorphisms of the rings L(A) and T (A).

Proof. Since A, B, OP are symmetric, the O-dual of an A-B-bimodule of the form A ⊗_{OQ} _{θ}B for

some subgroup Q of P and a group homomorphism θ : Q → O^{×} is isomorphic to the B-A-bimodule B_{θ} ⊗_{OQ} A ≅ B ⊗_{OQ} _{(θ^{−1})}A. The first statement follows. The second statement follows from 8.6. □

Given a finite group G, any p-permutation kG-module lifts uniquely, up to isomorphism, to a p-permutation OG-module. Conversely, a linear source OG-module reduces modulo J(O) to a trivial source kG-module because there are no non trivial group homomorphisms from a finite p-group to k^{×}. At the source algebra level, this translates to the following statement:

Proposition 8.8. Let A, B be almost source algebras of blocks with a common defect group P. The map sending an A-B-bimodule M to k ⊗_{O} M induces a surjective group homomorphism L(A, B) →

T (k ⊗_{O} A, k ⊗_{O} B) and a group isomorphism T (A, B) ≅ T (k ⊗_{O} A, k ⊗_{O} B). For A = B this is a ring

isomorphism T (A) ≅ T (k ⊗_{O} A).

Proof. If A and B are almost source algebras of blocks b and c of finite groups G and H, respectively, with a common defect group P then A⊗_{O}B is an almost source algebra of the block b⊗c of G×H. Via

the Morita equivalence between a block algebra and its almost source algebra any direct summand of Ā ⊗_{kQ} B̄ corresponds to a ΔP -projective p-permutation k(G × H)-module, and hence lifts uniquely, up to isomorphism, to a direct summand of A ⊗_{OQ} B, where Q is a subgroup of P and where Ā = k ⊗_{O} A,

B̄ = k ⊗_{O} B. The result follows. □

.

§9 Compatibility with the Brauer construction

The first result in this section says that under suitable hypotheses, the Brauer construction sends T (A, B) to T (A(ΔQ), B(ΔQ)). For P abelian this is [4, 3.3].

Theorem 9.1. Let A, B be almost source algebras of p-blocks b, c of finite groups G, H, respectively, with a common defect group P having the same fusion system F on P. Let Q be a fully F-centralised subgroup of P. For any X ∈ T (A, B) we have X(ΔQ) ∈ T (A(ΔQ), B(ΔQ)).

Proof. Let i ∈ (OGb)^{ΔP} and j ∈ (OHc)^{ΔP} be almost source idempotents such that A = iOGi and B = jOHj. Denote by c^{0} and j^{0} the images in OH of c and j, respectively, of the anti-automorphism of OH sending y ∈ H to y^{−1}. Then B^{op} ≅ j^{0}OHj^{0} is an almost source algebra of the block c^{0} of OH, by 4.10. Let R be a subgroup of P. We need to show that (A ⊗_{OR} B)(ΔQ) is a direct sum of

summands of the A(ΔQ)-B(ΔQ)- bimodules A(ΔQ) ⊗_{kS} B(ΔQ), with S running over the subgroups

of C_{P} (Q). Let Y be an indecomposable direct summand of (A ⊗_{OR} B)(ΔQ). As A ⊗_{O} B^{op}-modules we

have an isomorphism

A ⊗_{OR} B ≅ (A ⊗_{O} B^{op}) ⊗_{OΔR} O

sending x ⊗ y to (x ⊗ y) ⊗ 1_{O}, for x ∈ A and y ∈ B. By 4.9, the algebra A ⊗_{O} B^{op} is an almost source

algebra of the block b ⊗ c^{0} of G × H. By the assumptions, A, B and hence B^{op} determine the same fusion system F on P (cf. 4.10). Since Q is fully F-centralised, ΔQ is fully F × F-centralised by 2.2. We apply now 6.2 to A ⊗_{O} B^{op}, P × P, F × F, ΔQ, ΔR instead of A, P, F Q, R, respectively. The

conclusion of 6.2 yields a fully C_{F×F}(ΔQ)-centralised subgroup S of C_{P} (Q)×C_{P} (Q) and a morphism ψ : ΔQ · S → ΔR in F × F such that Y is isomorphic to a direct summand of (A ⊗_{O} B^{op})(ΔQ) ⊗_{kS} k

as (A ⊗_{O} B^{op})(ΔQ)-modules. Note that we have obvious isomorphisms

(A ⊗_{O} B^{op})(ΔQ) ≅ A(ΔQ) ⊗_{k} B^{op}(ΔQ) ≅ A(ΔQ) ⊗_{k} B(ΔQ)^{op} .

Since ψ : ΔQ · S → ΔR is an injective group homomorphism, the group S must be of the form

S = {(t, τ(t)) | t ∈ T }

for some subgroup T of C_{P} (Q) and some injective group homomorphism τ : T → C_{P} (Q). Since ψ is a morphism in F × F there are morphisms ψ_{1} : QT → P and ψ_{2} : Qτ(T) → P such that

ψ(ut, vτ(s)) = (ψ_{1}(ut), ψ_{2}(vτ(t)))

for all u, v ∈ Q and t, s ∈ T. Since ψ(ΔQ) ⊆ ΔR we have ψ_{1}(u) = ψ_{2}(u) for all u ∈ Q. Since also ψ(S) ⊆ ΔR we have ψ_{1}(t) = ψ_{2}(τ(t)) for all t ∈ T. Thus ψ_{2}^{−1} ∘ ψ_{1}|_{Q} = Id_{Q} and ψ_{2}^{−1} ∘ ψ_{1}|_{T} = τ. This proves that τ is a morphism in C_{F}(Q). Thus we have an isomorphism

Δ(QT) ≅ ΔQ · S

in F × F mapping (ut, ut) to (u, u) · (t, τ(t)) for all u ∈ Q and t ∈ T. Now S is fully C_{F×F}(ΔQ)- centralised and C_{F×F}(ΔQ) = C_{F}(Q) × C_{F}(Q) by 2.3. Thus, by 2.2, in particular T is fully C_{F}(Q)- centralised. But then, by 2.2 again, ΔT is fully C_{F}(Q) × C_{F}(Q)-centralised. Thus, by combining 5.1 and 6.2 we may replace S by ΔT. This, however, shows that Y is isomorphic to a direct summand of

(A(ΔQ) ⊗_{k} B(ΔQ)^{op}) ⊗_{kΔT} k ≅ A(ΔQ) ⊗_{kT} B(ΔQ) .

Thus the isomorphism class of Y belongs to T (A(ΔQ), B(ΔQ)). The result follows. □

The second theorem in this section extends similar statements in [15, 2.4], [17, 2.3.(ii)], [4, 3.6] and includes in particular a proof of Theorem 1.2.

Theorem 9.2. Let A, B be almost source algebras of p-blocks of finite groups with a common defect group P having the same fusion system F on P and let C be an interior P -algebra. Let Q be a fully F-centralised subgroup of P and let θ : Q → O^{×} be a group homomorphism. For any X ∈ T (A, B) and any Y ∈ R(B, C) we have

(X ·_{B}Y)(ΔQ, Δθ) = X(ΔQ) ·_{B(ΔQ)}Y (ΔQ, Δθ)

where Δθ : ΔQ → O^{×} is the group homomorphism sending (u, u) to θ(u) for all u ∈ Q. In particular, for any fully F-centralised subgroup Q of P the map sending X ∈ T (A) to X(ΔQ) is a ring homomorphism ρ_{Q} : T (A) → T (A(ΔQ)).

Proof. Let M be an A-B-bimodule and let N be a B-C-bimodule. Consider the obvious maps

M^{ΔQ} ⊗_{O} N^{(ΔQ,Δθ)} → (M ⊗_{O} N)^{(ΔQ,Δθ)} → (M ⊗_{B} N)^{(ΔQ,Δθ)} → (M ⊗_{B} N)(ΔQ, Δθ) .

The composition of these maps induces a homomorphism of A(ΔQ)-C(ΔQ)-bimodules

M(ΔQ) ⊗_{B(ΔQ)} N(ΔQ, Δθ) ⟶ (M ⊗_{B} N)(ΔQ, Δθ)

If we fix N and let M vary, this map is additive functorial in M. Thus, in order to show that this is an isomorphism if M is a finite direct sum of summands of bimodules of the form A ⊗_{OR} B, we may

assume that M is a direct summand of A ⊗_{OR} B as OQ-B-bimodule for some subgroup R of P. It

suffices to show that this map is a k-linear isomorphism, and so we may ignore the left A(ΔQ)-module structure. Thus we may in fact assume that M = W ⊗_{OR} B for some indecomposable direct summand

W of A as OQ-OR-bimodule, for some subgroup R of P. By 5.2 we have W ≅ OQ ⊗_{OT} _{φ}OR for some

subgroup T of Q and some group homomorphism φ : T → R belonging to the common fusion system of A and B on P. If T is a proper subgroup of Q then both sides in the above map are zero. Assume that Q = T, hence M ≅ _{φ}B as OQ-B-bimodule. By decomposing M further we may in fact assume that M = _{φ}nB for some primitive idempotent n ∈ B^{Δφ(Q)}. Then again, if Br_{Δφ(Q)}(n) = 0, both sides in the above map are easily seen to be zero. Thus we may assume that Br_{Δφ(Q)}(n) ≠ 0. As Q is fully F-centralised, we can apply 5.1, which shows that there is a primitive idempotent m ∈ B^{ΔQ} such that _{φ}nB ≅ mB as OQ-B-bimodules. Thus we may assume M = mB. But then M is a direct summand of B itself, so we may assume that M = B as OQ-B-bimodule. But in that case, both sides in the above map are canonically isomorphic to N(ΔQ). The last statement follows from applying this to A = B = C and θ = 1. □

If one translates 9.2 back to modules over block algebras, one gets in particular the following statement (needed in the proof of 1.4):

Corollary 9.3. Let G, H be finite groups, let b be a block of OG, and let c be a block of OH. Suppose that b and c have a common defect group P and that there are source idempotents i ∈ (OGb)^{ΔP} and j ∈ (OHc)^{ΔP} such that the fusion systems on P determined by i and j coincide. For any subgroup Q of P denote by e_{Q} the unique block of kC_{G}(Q) satisfying Br_{ΔQ}(i)e_{Q} ≠ 0 and by f_{Q} the unique block of kC_{H}(Q) satisfying Br_{ΔQ}(j)f_{Q} ≠ 0. Let M be an OGb-OHc-bimodule which is a finite direct sum of summands of the bimodules OGi ⊗_{OQ} jOH, with Q running over the subgroups of P. For

any OHc-module U, any subgroup Q of P and any group homomorphism θ : Q → O^{×} we have an isomorphism of kC_{G}(Q)e_{Q}-modules

e_{Q}(M(ΔQ))f_{Q} ⊗_{kC_{H}(Q)f_{Q}} f_{Q}(U(Q, θ)) ≅ e_{Q}((M ⊗_{kHc} U)(Q, θ)) .

Proof. By conjugating Q simultaneously we may assume that Q is fully centralised with respect to the common fusion system of A = iOGi and B = jOHj. Then, by making use of the Morita equivalences between A(ΔQ), kC_{G}(e_{Q}) and between B(ΔQ), kC_{H}(Q)f_{Q} from 4.6 it suffices to show that there is an isomorphism of A(ΔQ)-modules

Br_{ΔQ}(i)M(ΔQ)Br_{ΔQ}(j) ⊗_{B(ΔQ)} Br_{ΔQ}(j)U(Q, θ) ≅ Br_{ΔQ}(i)((Mj ⊗_{B} jU)(Q, θ)) .

Now the left side is isomorphic to (iMj)(ΔQ) ⊗_{B(ΔQ)}(jU)(Q, θ) by 3.9, and the right side is isomorphic

to (iMj ⊗_{B} jU)(Q, θ). Both are isomorphic thanks to 9.2 applied to A, B, C = O, the isomorphism

class of iMj instead of X and the isomorphism class of jU instead of Y. □

Remark 9.4. Theorem 9.1 adds to the results in [15, 17] that if X is a splendid tiliting complex of A-B-bimodules then for any F-centric subgroup Q of P the Rickard complex X(ΔQ) of A(ΔQ)- B(ΔQ)-bimodules is splendid (in the slightly more restrictive sense of the definition given in [15, 1.10] or [17, 1.1]).

Remark 9.5. With the notation of 9.2, if Q is a fully F-centralised subgroup of P and S a fully C_{F}(Q)-centralised subgroup of C_{P} (Q), then R = QS is fully F-centralised by 2.1. By the usual transitivity properties of the Brauer construction we have A(ΔQ)(ΔS) = A(ΔR) and the following diagram of ring homomorphisms is commutative:

\xymatrix{T (A) \ar[rr]^{ρ_{Q}} \ar[dr]_{ρ_{R}} &  & T (A(ΔQ)) \ar[dl]^{ρ_{S}} \\  & T (A(ΔR))}

§10 Isotypies

Let G, H be finite groups, let b be a block of OG and let c be a block of OH. Suppose that the quotient field K of O is large enough for b, c and their Brauer pairs. As before, denote by ℤIrr_{K}(G, b) the group of generalised characters of G over K associated with the block b, and denote by ℤIBr_{K}(G, b) the corresponding group of generalised Brauer characters. Following Broué, a perfect isometry between b and c is a group isomorphism

Ψ : ℤIrr_{K}(H, c) ≅ ℤIrr_{K}(G, b)

sending χ ∈ Irr_{K}(H, c) to ϵ(χ)η_{χ} for some ϵ(χ) ∈ {1, −1} and some η_{χ} ∈ Irr_{K}(G, b) such that the character μ ∈ ℤIrr(H × G) defined by

μ(y, x) = ∑_{χ∈Irr_{K}(H,c)} ϵ(χ)χ(y)η_{χ}(x^{−1})

for all (y, x) ∈ H × G has the following properties: the character value μ(y, x) is divisible in O by the orders of C_{G}(x), C_{H}(y), for all x ∈ G, all y ∈ H, and μ(x, y) = 0 if exactly one of x, y is a p′-element. Since the decomposition map ℤIrr_{K}(G, b) → ℤIBr_{K}(G, b) given by restricting a generalised character to the set of p′-elements in G is surjective, the arithmetic properties of a perfect isometry Ψ imply that Ψ induces an isomorphism

Ψ̄ : ℤIBr_{K}(H, c) ≅ ℤIBr_{K}(G, b) .

Since μ determines the map Ψ by the formula Ψ(χ)(x) = \frac{1}{|H|} ∑_{y∈H} μ(y, x)χ(y^{−1}) for all x ∈ G and all

χ ∈ ℤIrr_{K}(H, c), we will sometimes use abusively the same notation for Ψ and μ.

Suppose now that b and c have a common defect group P, let i ∈ (OGb)^{ΔP} and j ∈ (OHc)^{ΔP} be source idempotents. Suppose that the fusion systems of the source algebras iOGi and jOHj on P are equal. For any subgroup Q of P denote by e_{Q} the unique block of kC_{G}(Q) satisfying Br_{ΔQ}(i)e_{Q} ≠ 0 and by f_{Q} the unique block of kC_{H}(Q) satisfying Br_{ΔQ}(j)f_{Q} ≠ 0. Denote by ê_{Q} and f̂_{Q} the blocks of OC_{G}(Q) and OC_{H}(Q) which lift e_{Q} and f_{Q}, respectively. Again following Broué, an isotypy between b and c is a family of perfect isometries

Ψ_{Q} : ℤIrr_{K}(C_{H}(Q), f̂_{Q}) ≅ ℤIrr_{K}(C_{G}(Q), ê_{Q})

for every subgroup Q of P, with the following properties:

(1) for any isomorphism φ : Q ≅ R in the common fusion system F we have ^{φ}Ψ_{Q} = Ψ_{R}, where ^{φ}Ψ_{Q} is obtained from composing Ψ_{Q} with the isomorphisms ℤIrr_{K}(C_{G}(Q), ê_{Q}) ≅ ℤIrr_{K}(C_{G}(R), ê_{R}) and ℤIrr_{K}(C_{H}(Q), f̂_{Q}) ≅ ℤIrr_{K}(C_{H}(R), f̂_{R}) given by conjugation with elements x ∈ G, y ∈ H satisfying φ(u) = xux^{−1} = yuy^{−1} for all u ∈ Q;

(2) for any subgroup Q of P, any element u ∈ C_{P} (Q), setting R = Q⟨u⟩, we have an equality of maps

d_{(C_{G}(Q),e_{Q})}^{(u,e_{R})} ∘ Ψ_{Q} = Ψ̄_{R} ∘ d_{(C_{H}(Q),f_{Q})}^{(u,e_{R})}

from ℤIrr_{K}(C_{H}(Q), f̂_{Q}) to K⊗_{ℤ}ℤIBr_{K}(C_{G}(R), ê_{R}), where d_{(C_{G}(Q),e_{Q})}^{(u,e_{R})} is the generalised decomposition

map from ℤIrr_{K}(C_{G}(Q), ê_{Q}) to K ⊗_{ℤ}ℤIBr_{K}(C_{G}(R), ê_{R}) as mentioned in 3.7 and d_{(C_{H}(Q),f_{Q})}^{(u,f_{R})} is defined

analogously.

Theorem 10.1. Let G, H be finite groups, let b be a block of OG, and let c be a block of OH. Suppose that b and c have a common defect group P and that there are source idempotents i ∈ (OGb)^{ΔP} and j ∈ (OHc)^{ΔP} such that the fusion systems on P of the source algebras A = iOGi and B = jOHj coincide. Suppose there is a p-permutation equivalence X between A and B. Then for any subgroup Q of P the character Ψ_{Q} of C_{G}(Q) × C_{H}(Q) determined by e_{Q}(OGi ⊗_{A} X ⊗_{B} jOH)(ΔQ)f_{Q} is a perfect

isometry between the unique blocks e_{Q}, f_{Q} of C_{G}(Q), C_{H}(Q) determined by i, j, respectively, and the family {Ψ_{Q} | Q ⊆ P } is an isotypy between b and c.

Proof. Observe the abuse of notation in the statement: we mean by e_{Q}((OGi ⊗_{A} X ⊗_{B} jOH)(ΔQ))f_{Q}

the image in T (kC_{G}(Q)e_{Q}), kC_{H}(Q)f_{Q}) of X under the map which is induced by the map sending an A-B-bimodule M to the kC_{G}(Q)e_{Q}-kC_{H}(Q)f_{Q}-bimodule e_{Q}((OGi ⊗_{A} M ⊗_{B} jOH)(ΔQ))f_{Q}. This

could have been denoted more accurately by e_{Q}(([OGi] ·_{A}X ·_{B}[jOH])(ΔQ))f_{Q}. Condition (1) holds clearly. Thus, in order to show that Ψ_{Q} induces a perfect isometry, we may assume that Q is fully F-centralised. Set

Y_{Q} = e_{Q}((OGi ⊗_{A} X ⊗_{B} jOH)(ΔQ))f_{Q} .

By 4.11 we have

Y_{Q} = kC_{G}(Q)Br_{ΔQ}(i) ⊗_{A(ΔQ)} X(ΔQ) ⊗_{B(ΔQ)} Br_{ΔQ}(j)kC_{H}(Q) .

By 9.1, X(ΔQ) belongs to T (A(ΔQ), B(ΔQ) and hence, by 7.5, Y_{Q} belongs to T^{ΔC_{P} (Q)}(kC_{G}(Q)e_{Q}, kC_{H}(Q)f_{Q}). Since p-permutation modules over k lift uniquely, up to isomorphism, to p-permutation modules over O preserving vertices, Y_{Q} lifts to a unique element Ŷ_{Q} in T^{ΔC_{P} (Q)}(OC_{G}(Q)ê_{Q}, OC_{H}(Q)f̂_{Q}). Moreover, Y_{Q} is a virtual module which is projective on the left and on the right. Therefore, in order to show that this induces a perfect isometry it suffices to show that it induces an isometry. The fact that p-permutation modules lift uniquely as mentioned above implies that it suffices to show that

Y_{Q} ·_{kC_{H}(Q)f_{Q}} Y_{Q}^{∗} = [kC_{G}(Q)e_{Q}] .

Since Q is fully F-centralised, cutting with Br_{ΔQ}(i) and Br_{ΔQ}(j) yields Morita equivalences. Thus, it suffices to show that

Br_{ΔQ}(i)Y_{Q}Br_{ΔQ}(i) ·_{B(ΔQ)} Br_{ΔQ}(i)Y_{Q}^{∗}Br_{ΔQ}(i) = [A(ΔQ)]

Using 4.11 we get

Br_{ΔQ}(i)Y_{Q}Br_{ΔQ}(i) = X(ΔQ)

and hence this is simply equivalent to showing that

X(ΔQ) ·_{B(ΔQ)} X(ΔQ)^{∗} = [A(ΔQ)]

Thanks to 9.2 we have

X(ΔQ) ·_{B(ΔQ)} X(ΔQ)^{∗} = (X ·_{B}X^{∗})(ΔQ) = [A](ΔQ) = [A(ΔQ)]

as required, and thus Ψ_{Q} is a perfect isometry. The commutativity (2) follows from translating the argument given by Boltje and Xu in [4, 3.8. (e)] with Br_{P} (e), Br_{Q}(e), Br_{P} (f), Br_{Q}(f) replaced by e_{Q}, e_{R}, f_{Q}, f_{R}, respectively, where the notation is now as in (2). Note that f_{R} is a block of C_{C_{G}(Q)}(u) = C_{G}(R). In what follows we write d^{(u,e_{R})} instead of d_{(C_{G}(Q),ê_{Q})}^{(u,e_{R})} and smilarly we write

d^{(u,f_{R})} instead of d_{(C_{H}(Q),f̂_{Q})}^{(u,f_{R})} We need to show the equality

(d^{(u,e_{R})} ∘ Ψ_{Q})(χ) = (Ψ̄_{R} ∘ d^{(u,f_{R})})(χ)

for any χ ∈ ℤIrr_{K}(C_{H}(Q), f̂_{Q}). By a theorem of Brauer, every generalised character is a ℤ-linear combination of characters of linear source modules. Thus it suffices to prove the above equality in the case where χ is the character of a linear source OC_{H}(Q)f̂_{Q}-module U. Then Ψ_{Q}(χ) is the character of Ŷ_{Q} ⊗_{OC_{H}(Q)f̂_{Q}} U (which in line with our convention on abuse of notation means Ŷ_{Q} ·_{OC_{H}(Q)f̂_{Q}} [U]).

This is again a virtual linear source OC_{G}(Q)ê_{Q}-module (cf. 7.8). Thus, by 3.8, we get that

d^{(u,e_{R})}(Ψ_{Q}(χ)) = ∑_{θ∈Hom(⟨u⟩,O^{×})} θ(u) ⊗ [[f_{R}((Ŷ_{Q} ⊗_{OC_{H}(Q)f̂_{Q}} U)(⟨u⟩, θ))]]

Similarly, again by 3.8, we have d^{(u,f_{R})}(χ) = ∑_{θ∈Hom(⟨u⟩,O^{×})} θ(u) ⊗ [[f_{R}(U(⟨u⟩, θ))]]. Applying Ψ̄_{R} to

this expression yields

Ψ̄_{R}(d^{(u,f_{R})}(χ)) = ∑_{θ∈Hom(⟨u⟩,O^{×})} θ(u) ⊗ [[Y_{R} ⊗_{kC_{H}(R)f_{R}} f_{R}(U(⟨u⟩, θ))]] .

Since Y_{R} = e_{R}(Ŷ_{Q}(Δ⟨u⟩))f_{R}, the required equality of these two expressions follows now from 9.3. □

§11 Invariance under Morita equivalences with endo-permutation source

If b, c are blocks of finite groups G, H such that the block algebras OGb, OHc are Morita equivalent via an OGb-OHc-bimodule M with endo-permutation source V, then by a result of Puig [24], b and c have isomorphic defect groups, and there is an identification of defect groups of b and c such that for some choice of source idempotents the corresponding fusion systems are equal and such that V is stable with respect to this fusion system (see e.g [12] for the terminology regarding fusion stable endo-permutation modules). Thus Theorem 1.6 is a consequence of the following Theorem:

Theorem 11.1. Let A, B be source algebras of blocks b, c of finite groups G, H with a common defect group P and the same fusion system F on P. Suppose that there is an indecomposable F-stable endo-permutation OP -module V and an indecomposable direct summand M of the A-B-bimodule A ⊗_{OP} Ind_{ΔP}^{P ×P} (V) ⊗_{OP} B satisfying M ⊗_{B} M^{∗} ≅ A and M^{∗} ⊗_{A} M ≅ B. Then the map sending an

A-A-bimodule X to the B-B-bimodule M^{∗} ⊗_{A} X ⊗_{A} M induces a ring isomorphism T (A) ≅ T (B).

Proof. We need to show that for Q a subgroup of P, the isomorphism class of the B-B-bimodule M^{∗} ⊗_{A} A ⊗_{OQ} A ⊗_{A} M belongs to T (B). By the assumptions on M, the bimodule M^{∗} ⊗_{A} A ⊗_{OQ} A ⊗_{A} M

is isomorphic to a direct summand of B ⊗_{OP} Ind_{ΔP}^{P ×P} (V^{∗}) ⊗_{OP} A ⊗_{OQ} A ⊗_{OP} Ind_{ΔP}^{P ×P} (V) ⊗_{OP} B. It follows

from 5.2 that every indecomposable direct summand of A ⊗_{OQ} A as OP -OP -bimodule is isomorphic to

OP ⊗_{OR}_{φ}OP for some subgroup R of P and some φ ∈ Hom_{F}(R, P). Thus, if Y is an indecomposable

direct summand of the bimodule M^{∗} ⊗_{A} A ⊗_{OQ} A ⊗_{A} M then Y is isomorphic to a direct summand of

B ⊗_{OP} Ind_{ΔP}^{P ×P} (V^{∗}) ⊗_{OR} _{φ}Ind_{ΔP}^{P ×P} (V) ⊗_{OP} B. One checks that

Ind_{ΔP}^{P ×P} (V^{∗}) ⊗_{OR} _{φ}Ind_{ΔP}^{P ×P} (V) ≅ Ind_{ΔR}^{P ×P} (V^{∗} ⊗_{O} _{φ}V)

and since V is fusion stable, it follows that V^{∗} ⊗_{O} _{φ}V is a permutation OR-module. Thus the iso-

morphism class of Y belongs to T (B). The fact that the map sending an A-A-bimodule X to M^{∗} ⊗_{A} X ⊗_{A} M induces a ring homomorphism follows from the isomorphism M ⊗_{B} M^{∗} ≅ B. Finally,

this is an isomorphism - its inverse is obtained by exchanging the roles of A, B and of M, M^{∗}. □

Corollary 11.2. Let A be a source algebra of a nilpotent block with defect group P. Then T (A) ≅ T (OP).

Proof. By the main result in [23] we can identify A = S ⊗_{O} OP, where S = End_{O}(V) for some

indecomposable endo-permutation OP -module V with vertex P. Set

M = V ⊗_{O} OP ,

viewed as A-OP -bimodule via the canonical action of S on V and the regular actions of OP on the left and the right of OP. Using the canonical isomorphism S ≅ V ⊗_{O} V^{∗} one shows that M induces

a Morita equivalence. As A-A-bimodule, A is isomorphic to a direct summand of A ⊗_{OP} A. Thus,

as A-OP -bimodule, M is isomorphic to a direct summand of A ⊗_{OP} M = A ⊗_{OP} (V ⊗_{O} OP). When

considered as A ⊗_{O} OP -module, this is isomorphic to (A ⊗_{O} OP) ⊗_{OΔP} V via the map sending a ⊗ (v ⊗ y)

to (a⊗y^{−1})⊗v, for all a ∈ A, v ∈ V and y ∈ P. This in turn is isomorphic to A ⊗_{OP} Ind_{ΔP}^{P ×P} (V) ⊗_{OP} OP

via the map sending (a ⊗ y^{−1}) ⊗ v to a ⊗ ((1, 1) ⊗ v) ⊗ y, so the isomorphism T (A) ≅ T (OP) is a particular case of 11.1. □

Remark 11.3. A nilpotent block with defect group P is known to be isotypic to the defect group algebra OP. Does any such isotypy lift to a p-permutation equivalence? The answer is positive whenever a source V of the unique simple module has an endo-split p-permutation resolution in the sense of Rickard [25, §7], because then the isotypy is in fact induced by a splendid derived equivalence (cf. [15, 1.3] and [25, 6.3]).

§12 Augmentation homomorphisms

Given a finite p-group P we denote by B(P) the Burnside ring of P.

Proposition 12.1. Let P be a finite p-group. The map sending for any subgroup Q of P the P -set P/Q to the OP -OP -bimodule OP ⊗_{OQ} OP induces a ring isomorphism

δ_{P} : B(P) ≅ T (OP) .

In particular, the ring T (OP) is commutative and the ℤ-rank of T (OP) is equal to the number of conjugacy classes of subgroups of P.

Proof. One can verify this directly; alternatively, by [14, 11.4.6], the map sending an OP -module V to the OP -OP -bimodule Ind_{ΔP}^{P ×P} (V) induces an injective ring homomorphism from the Green ring of OP -modules to the ring P(OP), where here V is considered as module over OΔP via the obvious isomorphism ΔP ≅ P. This map sends the isomorphism class of OP/Q to that of OP ⊗_{OQ} OP, for

any subgroup Q of P, and hence induces an isomorphism B(P) ≅ T (OP) because any bimodule of the form OP ⊗_{OQ} OP is indecomposable. □

The Burnside ring of a finite p-group P admits for any subgroup Q an augmentation homomorphism σ_{Q} : B(P) → ℤ, induced by the map sending a finite P -set X to the number of Q-fixpoints |X^{Q}|. Composed with δ_{P} this yields an augmentation of T (OP) which can be described explicitly as follows:

Proposition 12.2. Let P be a finite p-group and let Q be a subgroup of P. The map sending an OP - OP -bimodule M to the rational number \frac{dim_{k}(M(ΔQ)}{|C_{P} (Q)|} induces a ring homomorphism ρ_{Q} : T (OP) → ℤ, and we have σ_{Q} = ρ_{Q} ∘ δ_{P}.

Proof. Let R be a subgroup of P. The image P ×_{R} P of P × P in OP ⊗_{OR} OP is a P -P -stable O-basis

of OP ⊗_{OR} OP. In order to calculate (OP ⊗_{OR} OP)(ΔQ) we need to determine the set of Q-fixpoints in

this basis. Let y, z ∈ P and denote by y ⊗ z the image of (y, z) in OP ⊗_{OR} OP. This is a ΔQ-fixpoint

if and only if for every u ∈ Q there is r_{u} ∈ R such that uy ⊗ z = yr_{u} ⊗ z = y ⊗ r_{u}z = y ⊗ zu, hence if and only if Q^{y} ⊆ R and yz ∈ C_{P} (Q). The condition Q^{y} ⊆ R is also equivalent to yR being in the Q-fixpoint set (P/R)^{Q}. Note that C_{P} (Q) acts freely on the left on the set of ΔQ-fixpoints in the basis P ×_{R} P of OP ⊗_{OR} OP. Thus the map sending y ∈ P satisfying Q^{y} ⊆ R to y ⊗ y^{−1} induces a

bijection between the Q-fixpoints in P/R and the C_{P} (Q)-orbits of ΔQ-fixpoints in P ×_{R} P. The result

follows. □

We get in particular an augmentation for the ring T (A):

Proposition 12.3. Let A be a source algebra of a block of a finite group with defect group P. The map sending an A-A-bimodule M to to the rational number \frac{dim_{k}(M(ΔP))}{|Z(P)|} induces a unitary ring homomorphism α_{P} : T (A) → ℤ.

Proof. By [28, 38.10] we have A(ΔP) ≅ kZ(P). Thus, by 9.2, the Brauer construction with respect to ΔP induces a unitary ring homomorphism T (A) → T (kZ(P)). The result follows from 12.2 applied to kZ(P) and 1 instead of OP and Q, respectively. □

As in the case of Burnside rings of finite groups, there are other homomorphisms from T (A) to ℤ. In order to describe them, we need the following terminology.

Definition 12.4. Let F be a fusion system on a finite p-group P. A subgroup Q of P is called F-nilpotent if Q is fully F-centralised and if C_{F}(Q) is the trivial fusion system F_{C_{P} (Q)}(C_{P} (Q)) of the group C_{P} (Q) on itself.

In particular, if Q is F-centric then Q is F-nilpotent. It is easy to check that if Q, R are fully F-centralised subgroups of P such that Q is F-nilpotent and Hom_{F}(Q, R) ≠ ∅ then R is F-nilpotent as well.

Remark 12.5. Let G be a finite group, b a block of OG, P a defect group of b and let i ∈ (OGb)^{ΔP} be an almost source idempotent of b. Set A = iOGi, denote by F the fusion system of A on P and for any subgroup Q of P denote by e_{Q} the unique block of kC_{G}(Q) satisfying Br_{ΔP} (i)e_{Q} ≠ 0. Then a subgroup Q of P is F-nilpotent if and only if e_{Q} is a nilpotent block of kC_{G}(Q) having C_{P} (Q) as defect group (this is well-known and follows, for instance, from [18, §2]). Therefore, if Q is Fnilpotent we have T (A(ΔQ)) ≅ T (kC_{P} (Q)) by 11.2. Thus the composition of ring homomorphisms T (A) → T (A(ΔQ)) ≅ T (kC_{P} (Q)) → ℤ yields a unitary ring homomorphism T (A) ⟶ ℤ for any F-nilpotent subgroup Q of P. We can describe these ring homomorphisms explicitly:

Theorem 12.6. Let A be an almost source algebra of a block with defect group P and fusion system F. Let Q be an F-nilpotent subgroup of P and let j ∈ A^{ΔQ} be a primitive idempotent such that Br_{ΔQ}(j) ≠ 0. The map sending an A-A-bimodule M to the rational number \frac{dim_{k}((jMj)(ΔQ))}{|C_{P} (Q)|} induces a unitary ring homomorphism α_{Q} : T (A) → ℤ. Moreover, α_{Q} is independent of the choice of j.

Proof. The ring homomorphism ρ_{Q} : T (A) → T (A(ΔQ)) is induced by the map sending an A-A-bimodule M to M(ΔQ). The block e_{Q} of C_{G}(Q) determined by A is nilpotent because Q is F-nilpotent. Thus e_{Q} has C_{P} (Q) as defect group, there is a unique local point μ of C_{P} (Q) on A(ΔQ), and for any m ∈ μ we have an isomorphism of interior C_{P} (Q)-algebra

mA(ΔQ)m ≅ S ⊗_{k} kC_{P} (Q)

with S = End_{k}(V) for some indecomposable endo-permutation kC_{P} (Q)-module V. Every primitive idempotent in S remains primitive in mA(ΔQ)m and hence lifts to a primitive idempotent j in A(ΔQ)^{ΔQ}, uniquely up to conjugation, and hence

(jAj)(ΔQ) ≅ kC_{P} (Q)

Thus the map sending M to (jMj)(ΔQ) induces a ring homomorphism T (A) → T (kC_{P} (Q)). Composing this with the augmentation T (kC_{P} (Q)) → ℤ obtained from 12.2 (with C_{P} (Q) and 1 instead of P and Q) yields the ring homomorphism α_{Q} as stated. □

The ring homomorphisms α_{Q} depend only on the isomorphism class of Q in the fusion system. More precisely:

Theorem 12.7. Let A be a source algebra of a block of a finite group G with defect group P and fusion system F. Let Q, R be F-nilpotent subgroups of P. We have α_{Q} = α_{R} if and only if Q ≅ R in F.

Proof. Suppose that α_{Q} = α_{R}. Let j ∈ A^{ΔQ} be a primitive idempotent satisfying Br_{ΔQ}(j) ≠ 0. Then (jA ⊗_{OQ} Aj)(ΔQ) is non zero, hence the image of [A ⊗_{OQ} A] under α_{Q} is non zero. But then its

image under α_{R} is non zero, which forces (A ⊗_{OQ} A)(ΔR) ≠ {0}. It follows from 6.4 that Hom_{F}(R, Q)

is non-empty. Exchanging the roles of Q and R yields an isomorphism Q ≅ R in F. Suppose conversely that Q ≅ R in F. By [22, 3.1], the existence of an isomorphism Q ≅ R in F is equivalent to the existence of an invertible element a ∈ A^{×} and primitive idempotents m ∈ A^{ΔQ}, n ∈ A^{ΔR} not contained in ker(Br_{ΔQ}), ker(Br_{ΔR}), respectively, such that ^{a}(Qm) = Rn. Thus conjugation by a induces an isomorphism (mAm)(ΔQ) ≅ (nAn)(ΔR), and since the blocks e_{Q}, e_{R} of kC_{G}(Q), kC_{G}(R), respectively, determined by A are nilpotent, we have kC_{P} (Q) ≅ (mAm)(ΔQ) and kC_{P} (R) ≅ (nAn)(ΔR). Thus we get a commutative diagram of ring homomorphisms of the form

\xymatrix{ & T (A(ΔQ)) \ar[r]^{≅} & T ((mAm)(ΔQ)) \ar[dd]_{≅} \ar[dr] \\ T (A) \ar[ur] \ar[dr] &  &  & ℤ \\  & T (A(ΔR)) \ar[r]_{≅} & T ((nAn)(ΔR)) \ar[ur]}

where the vertical arrow is induced by conjugation with a. The result follows. □

§13 Further remarks

13.1. One advantage of the concept of a p-permutation equivalence introduced by Boltje and Xu [4] is that it admits the obvious source algebra formulation 1.3 above. The concept of isotypies, however, is formulated at the block algebra level, involving character values of group elements. Some of the technical issues in this paper are caused by the need to switch between block algebras and their source algebras; a version of isotypies in terms of source algebras would simplify some parts. The other advantage of p-permutation equivalences versus isotypies is that the rather technical compatibility conditions of an isotypy get replaced by a condition which is simpler in that it does not refer to the local structure of the blocks. It does not seem to be known whether every isotypy between two blocks can be “lifted” to a p-permutation equivalence.

13.2. Let A, B be symmetric O-algebras. Following [16, 2.9] any A-B-bimodule M which is finitely generated projective as left A-module and as right B-module gives rise to a transfer map on Hochschild cohomology

t_{M} : HH^{∗}(B) ⟶ HH^{∗}(A) .

This depends only on the isomorphism class of M. Thus, any element in X ∈ P(A, B) determines a transfer map t_{X} : HH^{∗}(B) ⟶ HH^{∗}(A). When applied to A = B this yields a graded P(A)-module structure on HH^{∗}(A) given by

X · ζ = t_{X}(ζ)

for any X ∈ P(A) and any ζ ∈ HH^{∗}(A). If A is a source algebra of a block b of some finite group with defect group P then this module structure restricts to a T (A)-module structure on HH^{∗}(A). By [16, 5.6.(iii)], the Hochschild cohomology algebra HH^{∗}(A) contains a canonical copy of the block cohomology H^{∗}(G, b). By [16, 6.7] this is a T (A)-submodule of HH^{∗}(A).

13.3. Unlike the Burnside ring of a finite group, the trivial source bimodule ring of a block need not be commutative. Consider p = 2 and G = A_{4} = P ⋊E, where P ≅ C_{2} ×C_{2} and |E| = 3. Then OG has a unique block. Write Hom(E, O^{×}) = {μ_{0}, μ_{1}, μ_{2}} with μ_{0} = 1. Every μ_{i} extends to an automorphism α_{i} of OG defined by α_{i}(eu) = μ_{i}(e)ue, for u ∈ P and e ∈ E. These automorphisms fix P and hence the three OG-OG-bimodules (OG)_{α_{i}} have vertex ΔP and trivial source. Thus they determine three elements in T (OG). Setting e_{i} = \frac{1}{3} ∑_{e∈E} μ_{i}(e)e we get a primitive decomposition \{e_{1}, e_{2}, e_{3}\}

of 1 in OG. One checks that α_{i}(e_{0}) = e_{i} and that α_{2} = (α_{1})^{−1}. Any algebra automorphism α of OG induces an isomorphism of bimodules _{α}OG ≅ OG_{α−1} and isomorphisms of left OG-modules αOGj ≅ OGα^{−1}(j) for any idempotent j in OG. Therefore

[(OG)_{α_{1}}] ·_{OG} [OGe_{0} ⊗_{O} e_{0}OG] = [OGe_{1} ⊗_{O} e_{0}OG] ,

[OGe_{0} ⊗_{O} e_{0}OG] ·_{OG} [(OG)_{α_{1}}] = [OGe_{0} ⊗_{O} e_{2}OG] .

In particular T (OG) is not commutative.

Appendix: On stable equivalences and blocks of defect 2

The technology of this paper can be used to show that the proofs of two results due to R. Rouquier [27, 5.6, 6.3] for principal blocks carry over nearly verbatim to arbitrary blocks, giving in particular proofs of some of the results announced by Rouquier in [27, 6.4, Appendix]. For A, B two symmetric O-algebras, a bounded complex X of A-B-bimodules which are projective as left A-modules and as right B-modules is said to induce a stable equivalence if there are isomorphisms of complexes of bimodules X ⊗_{B} X^{∗} ≅ A ⊕ Y and X^{∗} ⊗_{A} X ≅ B ⊕ Z with Y and Z homotopy equivalent to bounded

complexes of projective A-A-bimodules and B-B-bimodules, respectively. If Y and Z are homotopic to zero then X is called a Rickard complex. The following two theorems are [27, 5.6] and [27, 6.3] for arbitrary blocks:

Theorem A.1. Let G, H be finite groups, b, c blocks of OG, OH, respectively, having a common defect group P, let i ∈ (OGb)^{ΔP} and j ∈ (OHc)^{ΔP} be source idempotents. Suppose that i and j determine the same fusion system F on P. 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_{ΔQ}(i)e_{Q} ≠ 0 and Br_{ΔQ}(j)f_{Q} ≠ 0. Let X be a bounded complex of OGb-OHc bimodules whose terms are finite direct sums of summands of the OGb-OHc-bimodules OGi ⊗_{OQ} jOH, where Q runs over the subgroups of P. The following are equivalent:

(i) The complex X induces a stable equivalence. (ii) For every non-trivial subgroup Q of P the complex e_{Q}X(ΔQ)f_{Q} is a Rickard complex of kC_{G}(Q)e_{Q}-kC_{H}(Q)f_{Q}-bimodules.

Proof. Suppose that (i) holds. Write X ⊗_{OHc} X^{∗} ≅ OGb ⊕ Y for some bounded complex Y of OGb-

OGb-bimodules which is homotopy equivalent to a complex of projective OGb-OGb-bimodules. Since the Brauer construction with respect to a non-trivial p-subgroup sends any projective module to zero and preserves homotopies, we get that Y (ΔQ) ≃ {0} for any non-trivial subgroup Q of P. Thus, by [17, 2.3] (or by 9.2 translated back to block algebras via 4.7) we get that

(e_{Q}X(ΔQ)f_{Q}) ⊗_{kC_{H}(Q)f_{Q}} (f_{Q}X^{∗}(ΔQ)e_{Q}) ≅ e_{Q}(X ⊗_{OHc} X^{∗})(ΔQ)e_{Q}

≃ e_{Q}(OGb)(ΔQ)e_{Q} = kC_{G}(Q)e_{Q}

for any non-trivial fully F-centralised subgroup Q of P. Since any subgroup of P is isomorphic, in the fusion system F, to a fully F-centralised subgroup we get this isomorphism for any non-trivial subgroup Q of P. In particular, e_{Q}X(ΔQ)f_{Q} is a Rickard complex for any non-trivial subgroup Q of P. Suppose conversely that (ii) holds. Let Y be the mapping cone of the adjunction unit OGb → X ⊗_{OHc} X^{∗}. By the assumptions, the induced map

kC_{G}(Q)e_{Q} → (e_{Q}X(ΔQ)f_{Q}) ⊗_{kC_{H}(Q)f_{Q}} (f_{Q}X^{∗}(ΔQ)e_{Q})

is a homotopy equivalence, and so its mapping cone is contractible, for every non-trivial subgroup Q of P. But as before, the right side is canonically isomorphic to e_{Q}(X ⊗_{OHc} X^{∗})(ΔQ)e_{Q}, and so the

corresponding mapping cone is e_{Q}Y (ΔQ)f_{Q}. This is contractible, for every non-trivial Q. But then in fact Y (ΔQ) ≃ 0 for every non-trivial subgroup Q of P, thanks to 4.8. A theorem of Bouc [5, 7.9] implies that Y is homotopy equivalent to a bounded complex of projective bimodules, whence (i). □

Theorem A.2. Let G be a finite group, let b be a block of OG with a defect group of order p^{2}, set H = N_{G}(P) and denote by c the block of OH satisfying Br_{ΔP} (b) = Br_{ΔP} (c). Let i ∈ (OGb)^{ΔP} and j ∈ (OHc)^{ΔP} be source idempotents such that Br_{ΔP} (i) = Br_{ΔP} (j). There is a bounded complex of OGb-OHc-bimodules whose components are finite direct sums of summands of the bimodules OGi ⊗_{OQ}

jOH, with Q running over the subgroups of P, such that X induces a stable equivalence.

Proof. If P is cyclic there is a Rickard complex of OGb-OHc-bimodules explicitly constructed by Rouquier [26]; it is observed in [17, §7] that the components of this complex are as stated. Thus we may assume that P is elementary abelian of rank 2. The proof follows the lines of [27, 6.3]. Denote by e_{Q} and f_{Q} the unique blocks of kC_{G}(Q) and kC_{H}(Q), respectively, satisfying Br_{ΔQ}(i)e_{Q} ≠ 0 and Br_{ΔQ}(j)f_{Q} ≠ 0. Since P is abelian, the fusion systems on P determined by i and by j are equal to that of kN(P, e_{P})e_{P}. Hence, for any subgroup Q of P, we have

N_{G}(Q, e_{Q})/C_{G}(Q) ≅ N_{H}(Q, f_{Q})/C_{H}(Q)

and both sides are p′-groups. The blocks e_{Q}, f_{Q} lift to unique blocks ê_{Q}, f̂_{Q} of OC_{G}(Q), OC_{H}(Q), respectively. The images of ê_{Q}, f̂_{Q} in OC_{G}(Q)/Q, OC_{H}(Q)/Q are blocks, denoted by ē_{Q}, f̄_{Q}, respectively. If Q has order p then ē_{Q}, f̄_{Q} have the cyclic group P/Q as defect group, and C_{H}(Q)/Q is the normaliser in C_{G}(Q)/Q of P/Q. Thus f̄_{Q} is in fact the Brauer correspondent of ē_{Q}. By Rouquier’s construction of Rickard complexes for blocks with cyclic defect groups in [26, 10.3], there is a Rickard complex of OC_{G}(Q)/Qē_{Q}-OC_{H}(Q)f̄_{Q}-bimodules C̄_{Q} of the form

C̄_{Q} = ⋯ → 0 → N̄_{Q} ⟶^{Φ̄_{Q}} ē_{Q}OC_{G}(Q)/Qf̄_{Q} → 0 → ⋯

for some projective bimodule N̄_{Q}. This complex lifts to a Rickard complex of OC_{G}(Q)ê_{Q}-OC_{H}(Q)f̂_{Q}- bimodules of the form

C_{Q} = ⋯ → 0 → N_{Q} ⟶^{Φ_{Q}} e_{Q}OC_{G}(Q)f_{Q} → 0 → ⋯

where N_{Q} is a projective O(C_{G}(Q)×OC_{H}(Q))/ΔQ-module lifting N̄_{Q}, inflated to O(C_{G}(Q)×C_{H}(Q)), and where Φ_{Q} lifts the map Φ̄_{Q}. By adapting arguments of Marcuş [20, 5.5] this complex extends to the group

T = N_{G×H}(ΔQ) ∩ (N_{G}(Q, e_{Q}) × N_{H}(Q, f_{Q}))

and T contains C_{G}(Q) × C_{H}(Q) as normal subgroup of p′-index by the above remarks. Thus N_{Q} remains projective when considered as OT/ΔQ-module. We set

V_{Q} = Ind_{T}^{G×H}(N_{Q}) .

The inclusion C_{G}(Q) ⊆ G induces an OT -homomorphism e_{Q}OC_{G}(Q)f_{Q} ⟶ bOGc, which, by adjunction, yields a homomorphism of O(G × H)-modules

α_{Q} : Ind_{T}^{G×H}(e_{Q}OC_{G}(Q)f_{Q}) ⟶ bOGc .

Set ψ_{Q} = α_{Q} ∘ Ind_{T}^{G×H}(Φ_{Q}) : V_{Q} → bOGc and define the complex X by

X = ⋯ → 0 → ⊕_{Q} V_{Q} ⟶^{∑_{Q}ψ_{Q}} bOGc → 0 → ⋯

with bOGc in degree zero, where Q runs over a set of representatives of the N_{G}(P, e_{P})-conjugacy classes of subgroups of order p of P. If Q, R are two subgroups of order p which are not N_{G}(P, e_{P})- conjugate then V_{R}(ΔQ) = {0}. This implies that e_{Q}X(ΔQ)f_{Q} ≃ C_{Q} ⊗_{O} k, and this is a Rickard

complex of kC_{G}(Q)e_{Q}-C_{H}(Q)f_{Q}-bimodules. Moreover, bOGc is a direct summand of OGi ⊗_{OP} jOH,

and V_{Q} is a direct sum of summands of OGi ⊗_{OQ} jOH because it is obtained from lifting, inflating

and inducing a projective bimodule. Thus Theorem A.1 applies, showing that X induces a stable equivalence. □

Acknowledgement. The author would like to thank the referee for his valuable comments.

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Markus Linckelmann Department of Mathematical Sciences Meston Building Aberdeen, AB24 3UE

United Kingdom