On stable equivalences with endopermutation source

Markus Linckelmann

Abstract We show that a bimodule between block algebras which has a fusion stable endopermutation module as a source and which induces Morita equivalences between centralisers of nontrivial subgroups of a defect group induces a stable equivalence of Morita type; this is a converse to a theorem of Puig. The special case where the source is trivial has long been known by many authors. The earliest instance for a result deducing a stable equivalence of Morita type from local Morita equivalences with possibly nontrivial endopermutation source is due to Puig, in the context of blocks with abelian defect groups with a Frobenius inertial quotient. The present note is motivated by an application, due to Biland, to blocks of finite groups with structural properties known to hold for hypothetical minimal counterexamples to the Z_{p}^{∗}-Theorem.

1 Introduction

Let p be a prime and O a complete discrete valuation ring having a residue field k of characteristic p; we allow the case O = k. We will assume that k is a splitting field for all block algebras which arise in this note. Following Broué [9, §5.A], given two O-algebras A, B, an A-B-bimodule M and a B-A-bomodule N, we say that M and N induce a stable equivalence of Morita type between A and B if M, N are finitely generated projective as left and right modules, and if M ⊗_{B} N ≅ A ⊕ W for some projective A⊗_{O} A^{op}-module W and N ⊗_{A} M ≅ B⊕ W′ for some projective B ⊗_{O} B^{op}-module W′. By a result of Puig in [27, 7.7.4] a stable equivalence of Morita type between block algebras of finite groups given by a bimodule with endopermutation source and its dual implies that there is a canonical identification of the defect groups of the two blocks such that both have the same local strucure and such that corresponding blocks of centralisers of nontrivial subgroups of that common defect group are Morita equivalent via bimodules with endopermutation sources. The following theorem is a converse to this result. The terminology and required background information for this statement are collected in the next two sections, together with further references.

Theorem 1.1. Let A, B be almost source algebras of blocks of finite group algebras over O having a common defect group P and the same fusion system F on P. Let V be an F-stable indecomposable endopermutation OP -module with vertex P, viewed as an OΔP -module through the canonical isomorphism ΔP ≅ P. Let M be an indecomposable direct summand of the A-B-bimodule

A ⊗_{OP} Ind_{ΔP}^{P ×P} (V) ⊗_{OP} B .

Suppose that (M ⊗_{B} M^{∗})(ΔP) ≠ {0}. Then for any nontrivial fully F-centralised subgroup Q of P, there is a canonical A(ΔQ)-B(ΔQ)-bimodule M_{Q} satisfying End_{k}(M_{Q}) ≅ (End_{O}(M))(ΔQ). Moreover, if for all nontrivial fully F-centralised subgroups Q of P the bimodule M_{Q} induces a Morita equivalence between A(ΔQ) and B(ΔQ), then M and its dual M^{∗} induce a stable equivalence of Morita type between A and B.

For V the trivial OP -module, variations of the above result have been noted by many authors. For principal blocks this was first pointed out by Alperin. A version for finite groups with the same local structure appears in Broué [9, 6.3], and the above theorem with V trivial is equivalent to [17, Theorem 3.1]. The first class of examples for this situation with potentially nontrivial V goes back to work of Puig [25]: it is shown in [25, 6.8] that a block with an abelian defect group P and a Frobenius inertial quotient is stably equivalent to its Brauer correspondent, using the fact that the blocks of centralisers of nontrivial subgroups of P are nilpotent, hence Morita equivalent to the defect group algebra via a Morita equivalence with endopermutation source. The above theorem is used in the proof of Biland [5, Theorem 4.1] or [7, Theorem 1]. For convenience, we reformulate this at the block algebra level.

Theorem 1.2. Let G, H be finite groups, and 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. For any subgroup Q of P denote by e_{Q} and f_{Q} the unique blocks of kC_{G}(Q) and kC_{H}(Q), respectively, satisfying Br_{ΔQ}(i)e_{Q} ≠ 0 and Br_{ΔQ}(j)f_{Q} ≠ 0. Denote by ê_{Q} and f̂_{Q} the unique blocks of OC_{G}(Q) and OC_{H}(Q) lifting e_{Q} and f_{Q}, respectively. Suppose that i and j determine the same fusion system F on P. Let V be an F-stable indecomposable endopermutation OP -module with vertex P, viewed as an OΔP -module through the canonical isomorphism ΔP ≅ P. Let M be an indecomposable direct summand of the OGb-OHc-bimodule

OGi ⊗_{OP} Ind_{ΔP}^{P ×P} (V) ⊗_{OP} jOH .

Suppose that M has ΔP as a vertex as an O(G × H)-module. Then for any nontrivial subgroup Q of P, there is a canonical kC_{G}(Q)e_{Q}-kC_{H}(Q)f_{Q}-bimodule M_{Q} satisfying End_{k}(M_{Q}) ≅ (End_{O}(ê_{Q}Mf̂_{Q}))(ΔQ). Moreover, if for all nontrivial subgroups Q of P the bimodule M_{Q} induces a Morita equivalence between kC_{G}(Q)e_{Q} and kC_{H}(Q)f_{Q}, then M and its dual M^{∗} induce a stable equivalence of Morita type between OGb and OHc.

The existence of canonical bimodules M_{Q} satisfying End_{k}(M_{Q}) ≅ (End_{O}(ê_{Q}Mf̂_{Q}))(ΔQ) in this Theorem is due to Biland [5, Theorem 3.15]. In the statement of Theorem 1.2 we let Q run over all nontrivial subgroups of P rather than only the fully F-centralised ones; this makes no difference here since one can always achieve Q to be fully centralised through simultaneous conjugation in G and H. By contrast, in the statement of Theorem 1.1, restricting attention to fully centralised subgroups is necessary in order to ensure that A(ΔQ) and kC_{G}(Q)e_{Q} are Morita equivalent. Another technical difference between the statements of the two theorems is that iMj will be an endopermutation OΔQ-module, while this is not clear for ê_{Q}Mf̂_{Q} because indecomposable OΔQ-summands with vertices strictly smaller than ΔQ might not be compatible. See Biland [6, Lemma 10] for more details on this issue.

Remark 1.3. The proof of Theorem 1.1 becomes significantly shorter if one assumes that V has a fusion-stable endosplit p-permutation resolution. This concept is due to Rickard [28], who also showed the existence of such resolutions for finite abelian p-groups. As a consequence of the classification of endopermutation modules, endosplit p-permutation resolutions exist for all endopermutation modules over k and unramified O belonging to the subgroup of the Dade group generated by relative syzygies. For odd p this is the entire Dade group while for p = 2 there are some endopermutation modules which do not have endosplit permutation resolutions. See [30, Theorem 14.3] for more details. We will outline how this simplifies the proof in the Remark 5.1 below.

2 Background material on blocks and almost source algebras

Let G be a finite group. For any subgroup H of G, we denote by ΔH the ‘diagonal’ subgroup ΔH = {(y, y) | y ∈ H} of H × H. Let b be a block of OG and P a defect group of b. That is, b is a primitive idempotent in Z(OG), and P is a maximal p-subgroup with the property that OP is isomorphic to a direct summand of OGb as an OP -OP -bimodule. As is customary, for any p-subgroup Q of G we denote by Br_{ΔQ} : (OG)^{ΔQ} → kC_{G}(Q) the Brauer homomorphism induced by the linear map sending x ∈ C_{G}(Q) to its image in kC_{G}(Q) and x ∈ G \ C_{G}(Q) to zero. The map Br_{ΔQ} is a surjective algebra homomorphism. More generally, for Q a p-subgroup of G and M an OG-module, we denote by M(Q) the kN_{G}(Q)-module obtained from applying the Brauer construction Br_{Q} to M. If A is an interior P -algebra, and Q a subgroup of Q, we denote by A(ΔP) the interior C_{P} (Q)-algebra obtained from applying the Brauer construction with respect to the conjugation action of P on A; our notational conventions are as in [19, §3].

Following Green [13], for any indecomposable OGb-module U, if Q is a minimal subgroup of G for which there exists an OQ-module V such that U is isomorphic to a direct summand of Ind_{Q}^{G}(V), then Q is a p-subgroup of G, the OQ-module V can be chosen to be indecomposable, in which case V is isomorphic to a direct summand of Res_{Q}^{G}(U), and the pair (Q, V) is unique up to G-conjugacy. In that situation, Q is called a vertex of U, and V an OQ-source of U, or simply a source of U of Q is determined by the context. Moreover, if R is a p-subgroup of G such that Res_{R}^{G}(U) has an indecomposable direct summand W with vertex R, then there is a vertex-source pair (Q, V) of U such that R ⊆ Q and such that W is isomorphic to a direct summand of Res_{R}^{Q}(V). By Higman’s criterion, this happens if and only of (End_{O}(U))(ΔR) ≠ {0}. See [22, Chapter 4] for an exposition of Green’s theory of vertices and sources.

Definition 2.1 (cf. [19, 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.

By [23, 3.5] (see also [19, Proposition 4.1] for a proof) there is a canonical Morita equivalence between the block algebra OGb and an almost source algebra iOGi sending an OGb-module M to the iOGi-module iM. Regarding fusion systems, we tend to follow the conventions of [18, §2]; in particular, by a fusion system on a finite p-group we always mean a saturated fusion system (in the terminology used in [2] or [11], for instance). With the notation of the previous Definition, it follows from work of Alperin and Broué [1] that the choice of an almost source idempotent i in (OGb)^{ΔP} determines a fusion system F on P such that 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}}. See e. g. [18, §2], or [2, Part IV]; note that we use here our blanket assumption that k is large enough. 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.

Proposition 2.2 (cf. [19, Proposition 4.5]). 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 of b. with associated almost source algebra A = iOGi. If Q is a fully F-centralised subgroup of P, then Br_{ΔQ}(i) is an almost source idempotent of kC_{G}(Q)e_{Q} with associated almost source algebra A(ΔQ); in particular, kC_{G}(Q)e_{Q} and A(ΔQ) are Morita equivalent.

By [19, 4.2], an almost source algebra A of a block with P as a defect group is isomorphic to a direct summand of A ⊗_{OP} A as an A-A-bimodule. Since A ⊗_{OP} A ≅ (A ⊗_{O} A^{op}) ⊗_{OP ⊗OP^{op}} OP this means that as an A ⊗_{O} A^{op}-module, A is relatively OP ⊗_{O} OP^{op}-projective. Since A, OP, and hence A ⊗_{O} A^{op}, OP ⊗_{O} OP^{op} are symmetric O-algebras, it follows that A is also relatively OP ⊗_{O} OP^{op}-injective. Tensoring a split map A → A ⊗_{OP} A by − ⊗_{A} U implies that any Amodule U is relatively OP -projective, or equivalently, isomorphic to a direct summand of A⊗_{OP} U. Vertices and sources of indecomposable OGb-modules can be read off from almost source algebras; the following result is a slight generalisation of [15, 6.3].

Proposition 2.3. Let G be a finite group, b a block of OG, P a defect group of b, and i an almost source idempotent in (OGb)^{ΔP}. Set A = iOGi. Let U be an indecomposable OGb-module, and let Q be a minimal subgroup of P such that the A-module iU is isomorphic to a direct summand of A⊗_{OQ}V for some OQ-module V. Then Q is a vertex of U, and U is isomorphic to a direct summand of OGi ⊗_{OQ} iU, or equivalently, the OQ-module V with the property that iU is isomorphic to a direct summand of A ⊗_{OQ} V can be chosen to be an indecomposable direct summand of Res_{Q}(iU).

Proof. Note that iU is an indecomposable A-module. Let Q be a minimal subgroup of P such that iU is isomorphic to a direct summand of A ⊗_{OQ} V, for some OQ-module V. Tensoring with OGi ⊗_{A} − implies that U is isomorphic to a direct summmand of OGi ⊗_{OQ} V, hence of Ind_{Q}^{G}(V). Thus Q contains a vertex of U. By general abstract nonsense (e.g. the equivalence of the statements (i) and (ii) in [10, Theorem 6.8] applied to restriction and induction between A and OQ), iU is then isomorphic to a direct summand of A ⊗_{OQ} iU, thus of A ⊗_{OQ} V for some indecomposable direct summand V of Res_{Q}(iU). The minimality of Q implies that V has Q as a vertex. But V is isomorphic to a direct summand of Res_{Q}^{G}(U), and hence Q is contained in a vertex of U. The result follows.

By a result of Puig in [24], fusion systems of blocks can be read off their source algebras; this is slightly extended to almost source algebras in [19, 5.1, 5.2].

Proposition 2.4 (cf. [19, 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.

Proposition 2.5 (cf. [19, Proposition 5.2]). Let G be a finite group, 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 the fusion system of A on P. Let Q, R be subgroups of P.

(i) Every indecomposable direct summand of A as an OQ-OR-bimodule is isomorphic to OQ ⊗_{OS} φOR 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 an OQ-OR-bimodule. In particular, F is determined by the OP -OP -bimodule structure of A.

Proposition 2.6. Let G be a finite group, b be a block of kG with defect group P, and let i be an almost source idempotent in (kGb)^{ΔP}. Denote by F the fusion system on P determined by i. Let Q, R be subgroups of P and denote by e the unique block of kC_{G}(Q) satisfying Br_{ΔQ}(i)e ≠ 0. Let φ : Q → R be an injective group homomorphism such that _{φ}kR is isomorphic to a direct summand of ekGi as an kQ-kR-bimodule. Then φ ∈ Hom_{F}(Q, R).

Proof. Let T be a fully F-centralised subgroup of P isomorphic to Q in the fusion system F. That is, if f is the unique block of kC_{G}(T) satisfying Br_{ΔT} (i)f ≠ 0, then C_{P} (T) is a defect group of kC_{G}(T)f, there is an element x ∈ G such that (T, f) = ^{x}(Q, e), and the isomorphism ψ : Q → T defined by ψ(u) = xux^{−1} is in F. Since _{φ}kR is a summand of e_{Q}kGi, multiplication by x shows that the kT -kR-bimodule _{φ∘ψ^{−1}}kS is a direct summand of xekGi = xex^{−1}kGi = fkGi. Moreover, φ is a morphism in F if and only if φ∘ψ^{−1} is. Thus, after possibly replacing (Q, e) by (T, f) we may assume that (Q, e) is fully F-centralised. By [19, Proposition 4.6] (ii) this implies that every local point of Q on kGb associated with e has a representative in ikGi. Since _{φ}kR is indecomposable as a kQ-kR-bimodule with a vertex of order |Q|, this bimodule is isomorphic to a direct summand of jkGi for some primitive local idempotent j in (kGb)^{ΔQ} appearing in a primitive decomposition of e in (kGb)^{ΔQ}. But then Br_{ΔQ}(j) ∈ kC_{G}(Q)e, and hence, after possibly replacing j with a suitable ((kGb)^{ΔQ})^{×}-conjugate, we may assume that j ∈ ikGi. It follows from Proposition 2.5 (i) that φ is a morphism in F.

Proposition 2.7. Let G be a finite group, H a subgroup of G and A an O-algebra. Let M be an OH-A-bimodule and V an OH-module. Consider V ⊗_{O} M as an OH-A bimodule with H acting diagonally on the left, consider V as a module for kΔH via the canonical isomorphism ΔH ≅ H and consider Ind_{ΔH}^{G×H}(V) as an OG-OH-bimodule. We have a natural isomorphism of OG-A-bimodules

Ind_{H}^{G}(V ⊗_{O} M) ≅ Ind_{ΔH}^{G×H}(V) ⊗_{OH} M

sending x ⊗ (v ⊗ m) to ((x, 1) ⊗ v) ⊗ m, where v ∈ V and m ∈ M.

Proof. This is a straightforward verification.

3 On fusion-stable endopermutation modules

Let P be a finite p-group. Following Dade [12] a finitely generated O-free OP -module V is an endopermutation module if End_{O}(V) ≅ V ⊗_{O} V^{∗} is a permutation OP -module, with respect to the ‘diagonal’ action of P. See Thévenaz [30] for an overview on this subject and some historic background, leading up to the classification of endopermutation modules. We will use without further comment some of the basic properties, due to Dade, of endopermutation modules - see for instance [29, §28]. If V is an endopermutation OP -module having an indecomposable direct summand with vertex P, then for any two subgroups Q, R of P such that Q is normal in R, there is an endopermutation kR/Q-module V′ = Defres_{R/Q}^{P}(V) satisfying End_{O}(V)(ΔQ) ≅ End_{k}(V′) as R/Q-algebras, and as interior R/Q-algebras if R ⊆ QC_{P} (Q). This construction is also known as Dade’s ‘slash’ construction.

Definition 3.1. Let P be a finite p-group and F a fusion system on P. Let Q be a subgroup of P and V be an endopermutation OQ-module. We say that V is F-stable if for any subgroup R of Q and any morphism φ : R → Q in F the sets of isomorphism classes of indecomposable direct summands with vertex R of the OQ-modules Res_{R}^{Q}(V) and _{φ}V are equal (including the possibility that both sets may be empty).

With the notation of 3.1, the property of V being F-stable does not necessarily imply that Res_{R}^{Q}(V) and _{φ}V have to be isomorphic as OR-modules, where φ : R → Q is a morphism in F (so this is a slight deviation from the terminology in [21, 3.3. (1)]). What the F-stability of V means is that the indecomposable direct factors of Res_{R}^{Q}(V) and _{φ}V with vertex R, if any, are isomorphic, but they may occur with different multiplicities in direct sum decompositions (in other words, in the terminology of [21, 3.3.(2)] the class of V in the Dade group is F-stable, provided that V has an indecomposable direct summand with vertex P). By [21, 3.7], every class in D_{O}(P) having an Fstable representative has a representative W satisfying the stronger stability condition Res_{R}^{P}(W) ≅ φW for any morphism φ : R → P in F. It follows from Alperin’s fusion theorem that in order to check whether an endopermutation OP -module V with an indecomposable direct summand of vertex P is F-stable, it suffices to verify that Res_{R}^{P}(V) and _{φ}V have isomorphic summands with vertex R for any F-essential subgroup R of P and any p′-automorphism φ of R in Aut_{F}(R). In particular, if P is abelian, then an indecomposable endopermutation OP -module V with vertex P is F-stable if and only if V ≅ _{φ}V for any φ ∈ Aut_{F}(P). In the majority of cases where Definition 3.1 is used we will have Q = P. One notable exception arises in the context of bimodules, where we consider the fusion systems F × F on P × P with the diagonal subgroup ΔP playing the role of Q. The key argument exploiting the F-stability of an endopermutation OP -module V having an indecomposable direct summand with vertex P goes as follows: if Q is a subgroup of P and φ : Q → P a morphism in F, then the restriction to ΔQ of V ⊗_{O} _{φ}V^{∗} is again a permutation module, or equivalently, V ⊗_{O} V^{∗} remains a permutation module for the twisted diagonal subgroup Δ_{φ}Q = {(u, φ(u)) | u ∈ Q} of P ×P. By a result of Broué in [8], if V is a permutation OP -module, then End_{O}(V)(ΔP) ≅ End_{k}(V (P)). This is not true for more general modules, but Dade’s ‘slash’ construction from [12] for endopermutation modules yields a generalisation of this isomorphism, as follows.

Proposition 3.2. Let A be an almost source algebra of a block of a finite group algebra over O with a defect group P and fusion system F on P. Let Q be a subgroup of P and let V be an F-stable endopermutation OQ-module having an indecomposable direct summand with vertex Q. Set U = A ⊗_{OQ} V. The following hold.

(i) As an OQ-module, U is an endopermutation module, and U has a direct summand isomorphic to V. (ii) Let R be a subgroup of Q. The A-module structure on U induces an A(ΔR)-module structure on U′ = Defres_{RC_{Q}(R)/R}^{Q}(U) extending the kC_{Q}(R)-module structure on U′ such that we have an isomorphism (End_{O}(U))(ΔR) ≅ End_{k}(U′) as algebras and as A(ΔR)-A(ΔR)-bimodules.

Proof. By [19, 5.2], every indecomposable direct summand of A as an OQ-OQ-bimodule is isomorphic to OQ ⊗_{OR} _{φ}OQ for some subgroup R of Q and some morphism φ ∈ Hom_{F}(R, Q), and at least one summand of A as an OQ-OQ-bimodule is isomorphic to OQ. Thus every indecomposable direct summmand of U is isomorphic to Ind_{R}^{Q}(_{φ}V_{φ(R)}) for some subgroup R of Q and some φ ∈ Hom_{F}(R, Q), where V_{φ(R)} is an indecomposable direct summand of vertex φ(R) of Res_{φ(R)}^{Q}(V), and V is a summand of U as an OQ-module. Since V is F-stable, we have _{φ}(V_{φ(R)}) ≅ V_{R}, which implies that the restriction to OQ of U is an endopermutation OQ-module. Statement (i) follows. For statement (ii) we consider the structural algebra homomorphism A → End_{O}(U) given by the action of A on U. This is a homomorphism of interior Q-algebras. Applying the Brauer construction with respect to ΔR, where R is a subgroup of Q, yields a homomorphism of interior C_{Q}(R)-algebras A(ΔR) → (End_{O}(U))(ΔR). Since U is an endopermutation OQ-module, we have (End_{O}(U))(ΔR) ≅ End_{k}(U′) as interior C_{Q}(R)-algebras. This yields a homomorphism A(ΔR) → End_{k}(U′), hence a canonical A(ΔR)-module structure on U′ with the properties as stated.

Statement (ii) in Proposition 3.2 is particularly useful when Q is fully F-centralised, since in that case C_{P} (Q) is a defect group of the unique block e_{Q} of kC_{G}(Q) satisfying Br_{ΔQ}(i)e_{Q} ≠ 0, and the algebras A(ΔQ) and kC_{G}(Q)e_{Q} are Morita equivalent. Statement (ii) of 3.2 is essentially equivalent to a result of Biland; since we will use this for proving that the Theorems 1.1 and 1.2 are equivalent we state this and sketch a proof for the convenience of the reader.

Proposition 3.3 (Biland [5, Theorem 3.15 (i)]). 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 V be an F-stable endopermutation OQ-module having an indecomposable direct summand with vertex Q. Set X = OGi ⊗_{OQ} V. Let R be a subgroup of Q, denote by e_{R} the unique block of kC_{G}(R) satisfying Br_{ΔR}(i)e_{R} ≠ 0, and let ê_{R} be the block of OC_{G}(R) which lifts e_{R}. There is a canonical kC_{G}(R)e_{R}-module Y_{R} such that we have an isomorphism (End_{O}(ê_{R}Y))(ΔR) ≅ End_{k}(Y_{R}) as algebras and as kC_{G}(R)e_{R}-kC_{G}(R)e_{R}-bimodules.

Proof. Applying Br_{R} to the canonical algebra homomorphism OGb → End_{k}(Y) and cutting by e_{R} and ê_{R} yields an algebra homomorphism kC_{G}(R)e_{r} → (End_{O}(ê_{R}Y))(ΔR). In order to show that this is isomorphic to End_{k}(Y_{R}) for some module Y_{R} it suffices to observe that the indecomposable summands of Res_{R}(ê_{R}Y) with vertex R are all isomorphic. Note that ê_{R}Y = ê_{R}OGi ⊗_{OQ} V. Any indecomposable direct summand of ê_{R}OGi as an OR-OQ-bimodule with a vertex of order at least |R| is isomorphic to _{φ}OQ for some group homomorphism φ : R → Q induced by conjugation with an element in G. In view of the fusion stability of V, it suffices to show that φ is a morphism in F. This is an immediate consequence of 2.6, whence the result.

As mentioned earlier, there is a technical difference between the Propositions 3.2 and 3.3: statement (i) in Proposition 3.2 may not have an an analogue at the block algebra level, since it is not clear whether ê_{R}X is an endopermutation OR-module, because the indecomposable direct summands with vertex strictly contained in R might not be compatible.

4 Bimodules with fusion-stable endopermutation source

Throughout this Section we fix the following notation and hypotheses. Let G, H be finite groups, b a block of OG and c a block of OH. Suppose that b and c have a common defect group P. Let i ∈ (OGb)^{ΔP} and j ∈ (OHc)^{ΔP} be almost source idempotents. Set A = iOGi and B = jOHj. Suppose that A and B determine the same fusion system F on P. Let V be an F-stable indecomposable endopermutation OP -module with vertex P. Whenever expedient, we consider V as an OΔP -module through the canonical isomorphism ΔP ≅ P. Set

U = A ⊗_{OP} Ind_{ΔP}^{P ×P} (V) ⊗_{OP} B ,
X = OGi ⊗_{OP} Ind_{ΔP}^{P ×P} (V) ⊗_{OP} jOH .

The A ⊗_{O} B^{op}-module U corresponds to the O(G × H)-module X through the canonical Morita equivalence between A ⊗_{O} B^{op} and OGb ⊗_{O} OHc^{op}; in particular, there is a canonical bijection between the isomorphism classes of indecomposable direct summands of U and of X. This Section contains some technical statements which involve the tensor product of two bimodules. This yields a priori four module structures, and keeping track of those is essential - see Broué [10, §1] for some formal properties of quadrimodules. If the algebras under consideration are group algebras, we play this back to two actions via the usual ‘diagonal’ convention: given two finite groups G, H and two OG-OH-bimodules S, S′, we consider S ⊗_{O} S′ as an OG-OH-bimodule via the diagonal left action by G and the diagonal right action by H; explicitly, x · (s ⊗ s′) · y = xsy ⊗ xs′y, where x ∈ G, y ∈ H, s ∈ S, and s′ ∈ S′. This is equivalent to the diagonal G × H-action if we interpret the OG-OH-bimodules as O(G × H)-modules in the usual way. The following result is a bimodule version of 3.2.

Proposition 4.1. Consider U as an OΔP -module, with (u, u) ∈ ΔP acting on U by left multiplication with u and right multiplication with u^{−1}. Then, as an OΔP -module, U is an endopermutation module having V as a direct summand, and for any subgroup Q of P, the A-B-bimodule structure on U induces an A(ΔQ)-B(ΔQ)-bimodule structure on U′ = Defres_{ΔQC_{P} (Q)/Q}^{ΔP}(U) such that we have an isomorphism of A(ΔQ) ⊗_{k} B(ΔQ)^{op}-bimodules End_{O}(U)(ΔQ) ≅ End_{k}(U′).

Proof. This is the special case of 3.2 with P × P, F × F, ΔP, ΔQ, A ⊗_{O} B^{op}, instead of P, F, Q, R, A, respectively.

Theorem 4.2. Let Q be a subgroup of P, and let U′ be the A(ΔQ)-B(ΔQ)-bimodule from 4.1 such that (End_{O}(U))(ΔQ) ≅ End_{k}(U′). Then End_{B^{op}}(U) is a ΔQ-subalgebra of End_{O}(U), the algebra homomorphism

β : End_{B^{op}}(U)(ΔQ) → End_{O}(U)(ΔQ)

induced by the inclusion End_{B^{op}}(U) ⊆ End_{O}(U) is injective, and there is a commutative diagram of algebra homomorphisms

\xymatrix{End_{O}(U)(ΔQ) \ar[r]^{≅} & End_{k}(U′) \\ End_{B^{op}}(U)(ΔQ) \ar[u]^{β} \ar[r]_{γ} & End_{B(ΔQ)^{op}}(U′) \ar[u]}

where the right vertical arrow is the obvious inclusion map. In particular, the algebra homomorphism γ is injective.

Proof. For φ ∈ End_{O}(U), y ∈ Q, and u ∈ U we have

Δyφ(u) = Δy · φ(Δy^{−1} · u) = yφ(y^{−1}uy)y^{−1} ,

where Δy = (y, y). If φ ∈ End_{B^{op}}(U), then in particular φ commutes with the right action by Q, and hence we have ^{Δy}φ(u) = yφ(y^{−1}u) = ^{(y,1)}φ(u), which shows that ^{Δy}φ is again a B^{op}- homomorphism. The algebra of ΔQ-fixed points in End_{B^{op}}(U) is equal to End_{OQ⊗_{O}B^{op}}(U). The existence of a commutative diagram as in the statement is formal: if φ ∈ End_{OQ⊗_{O}B^{op}}(U), then in particular b·φ = φ·b for all b ∈ B, hence for all b ∈ B^{ΔQ}, and applying Br_{ΔQ} yields that the image of φ in End_{O}(U)(ΔQ) commutes with the elements in B(ΔQ). Since the upper horizontal map is a bimodule isomorphism, it follows that the image of φ in End_{k}(U′) commutes with the elements in B(ΔQ), hence lies in the subalgebra End_{B(ΔQ)^{op}}(U′). In order to show that β is injective, we first note that this injectivity does not make use of the left A-module structure of U but only of the left OQ-module structure. Thus we may decompose U by decomposing A as an OQ-OP -bimodule. By 2.5, every summand of A as an OQ-OP -bimodule is of the form OQ⊗_{OR}_{φ}OP for some subgroup R of Q and some homomorphism φ : R → P belonging to the fusion system F. Using the appropriate version of the isomorphism 2.7 of OP -B-modules Ind_{ΔP}^{P ×P} (V) ⊗_{OP} B ≅ V ⊗_{O} B it suffices therefore to show that applying Br_{ΔQ} to the inclusion map

Hom_{B^{op}}(OQ ⊗_{OR} _{φ}(V ⊗_{O} B), OQ ⊗_{OS} _{ψ}(V ⊗_{O} B)) ⊆
Hom_{O}(OQ ⊗_{OR} _{φ}(V ⊗_{O} B), OQ ⊗_{OS} _{ψ}(V ⊗_{O} B))

remains injective upon applying Br_{ΔQ}, where R, S are subgroups of Q and where φ ∈ Hom_{F}(R, P), ψ ∈ Hom_{F}(S, P). If one of R, S is a proper subgroup of Q, then both sides vanish upon applying Br_{ΔQ}. Thus it suffices to show that the map

Hom_{B^{op}}(_{φ}(V ⊗_{O} B),_{ψ}(V ⊗_{O} B))(ΔQ) → Hom_{O}(_{φ}(V ⊗_{O} B),_{ψ}(V ⊗_{O} B))(ΔQ)

is injective, where φ, ψ ∈ Hom_{F}(Q, P). The summands of _{φ}V, _{ψ}V with vertices smaller than Q yield summands of V ⊗_{O} B which vanish on both sides upon applying Br_{ΔQ}. The fusion stability of V implies that indecomposable summands with vertex Q of _{φ}V, _{ψ}V are all isomorphic to an indecomposable direct summand W with vertex Q of Res_{Q}^{P}(V). Thus it suffices to show that the map

End_{B^{op}}(W ⊗_{O} B)(ΔQ) → End_{O}(W ⊗_{O} B)(ΔQ)

is injective, where W is an indecomposable direct summand of Res_{Q}^{P}(V) with vertex Q. Using the natural adjunction isomorphism

End_{O}(W ⊗_{O} B) ≅ Hom_{O}(B, W^{∗} ⊗_{O} W ⊗_{O} B)

it suffices to show that the map

Hom_{B}(B, W^{∗} ⊗_{O} W ⊗_{O} B)(ΔQ) → Hom_{O}(B, W^{∗} ⊗_{O} W ⊗_{O} B)(ΔQ)

is injective. Now W^{∗} ⊗_{O} W is a direct sum of a trivial OQ-module O and indecomposable permutation OQ-modules with vertices strictly smaller than Q. Thus it suffices to show that the map

End_{B^{op}}(B)(ΔQ) → End_{O}(B)(ΔQ)

is injective. The canonical isomorphism End_{B^{op}}(B) ≅ B yields an isomorphism End_{B^{op}}(B)(ΔQ) ≅ B(ΔQ). Since B has a ΔQ-stable O-basis, it follows that End_{O}(B)(ΔQ) ≅ End_{k}(B(ΔQ)). Using these isomorphisms, the last map is identified with the structural homomorphism

B(ΔQ) → End_{k}(B(ΔQ)) ,

which is clearly injective.

Lemma 4.3. Let W be an indecomposable direct summand of U ⊗_{B} U^{∗} or of U ⊗_{OP} U^{∗}. Then W is isomorphic to a direct summand of A ⊗_{OQ} A for some fully F-centralised subgroup Q of P such that W (ΔQ) ≠ {0}. In particular, U ⊗_{B} U^{∗} and U ⊗_{OP} U^{∗} are p-permutation bimodules.

Proof. Since B is isomorphic to a direct summand of B⊗_{OP} B, it follows that U ⊗_{B} U^{∗} is isomorphic to a direct summand of U ⊗_{OP} U^{∗}. Thus it suffices to prove the statement for an indecomposable direct summnad W of U ⊗_{OP} U^{∗}. Using the isomorphisms from 2.7, we get isomorphisms as OP -OP -bimodules

Ind_{ΔP}^{P ×P} (V^{∗}) ⊗_{OP} B ⊗_{OP} B ⊗_{OP} Ind_{ΔP}^{P ×P} (V) ≅ V^{∗} ⊗_{O} B ⊗_{OP} B ⊗_{O} V ≅ (V ⊗_{O} V^{∗}) ⊗_{O} (B ⊗_{OP} B) ,

where the right side is to be understood as a tensor product of two OP -OP -bimodules with the above conventions. Every indecomposable summand of B ⊗_{OP} B as an OP -OP -bimodule is isomorphic to OP ⊗_{OQ}_{φ}OP ≅ Ind_{Δ_{φ}Q}^{P ×P}(O) for some subgroup Q of P and some morphism φ : Q → P in F, where Δ_{φ}Q = {(u, φ(u)) | u ∈ Q}. Thus any indecomposable direct summand of (V ⊗_{O} V^{∗})⊗_{O} (B ⊗_{OP} B) is isomorphic to a direct summand of an OP -OP -bimdoule of the form Ind_{Δ_{φ}Q}^{P ×P}(V ⊗_{O} V^{∗}). The restriction to Δ_{φ}Q of V ⊗_{O} V^{∗} is a permutation module thanks to the stability of V, and hence the indecomposable direct summands of Ind_{Δ_{φ}Q}^{P ×P}(V ⊗_{O} V^{∗}) are of the form OP ⊗_{OR} _{φ}OP, where R is a subgroup of Q and where we use abusively the same letter φ for the restriction of φ to any such subgroup. Thus W is isomorphic to a direct summand of A ⊗_{OR} _{φ}A, with R and φ as before. Set S = φ(R). The indecomposability of W implies that W is isomorphic to a direct summand of Ar⊗_{OR}_{φ}mA for some primitive idempotent r in A^{ΔR} and some primitive isempotent m in A^{ΔS}. By choosing R minimal, we may assume that r, m belong to local points of R and S on A, respectively. Let T be a fully F-centralised subgroup of P and let ψ : T → R be an isomorphism in F. Then, by 2.4, there are primitive idempotents n, s in A^{ΔT} such that An ≅ Am_{ψ} as A-OT -bimodules and sA ≅ _{ψ∘φ}rA as OT -A-bimodules. Thus Y is isomorphic to a direct summand of A ⊗_{OT} A. The minimality of R, hence of T, implies that ΔT is a vertex of OGi ⊗_{A} W ⊗_{B} jOH, viewed as an O(G × H)-module, and Proposition 2.3 implies that a source, which has just shown to be trivial, is a summand of W restricted to ΔT, which implies that W (ΔT) ≠ {0}.

Proposition 4.4. Let M be an indecomposable direct summand of the A-B-bimodule U. The following statements are equivalent.

(i) A is isomorphic to a direct summand of the A-A-bimodule M ⊗_{B} M^{∗}. (ii) A is isomorphic to a direct summand of the A-A-bimodule M ⊗_{OP} M^{∗}. (iii) (M ⊗_{B} M^{∗})(ΔP) ≠ {0}. (iv) (M ⊗_{OP} M^{∗})(ΔP) ≠ {0}.

Proof. Since B is isomorphic to a direct summand of the B-B-bimodule B ⊗_{OP} B, it follows that M ⊗_{B} M^{∗} is isomorphic to a direct summand of M ⊗_{OP} M^{∗}. This yields the implications (i) ⇒ (ii) and (iii) ⇒ (iv). Since A(ΔP) ≠ {0}, we trivially have the implications (i) ⇒ (iii) and (ii) ⇒ (iv). Since M is finitely generated projective as a left A-module and as a right B-module (hence also as a right OP -module), we have M ⊗_{B} M^{∗} ≅ End_{B^{op}}(M), and M ⊗_{OP} M^{∗} ≅ End_{(OP)^{op}}(M). It is well-known that if A is isomorphic to a direct summand of M ⊗_{OP} M^{∗}, then the canonical algebra homomorphism A → End_{(OP)^{op}}(M) is split injective as a bimodule homomorphism (see e.g. [14, Lemma 4] for a proof). This algebra homomorphism factors through the inclusion End_{B^{op}}(M) ⊆ End_{(OP)^{op}}(M), which implies that the canonical algebra homomorphism A → End_{B^{op}}(M) is also split injective as a bimodule homomorphism. This shows the implication (ii) ⇒ (i). Suppose that (iv) holds. Set Y = OGi ⊗_{A} M. Then Y ⊗_{OP} Y^{∗} ≅ OGi ⊗_{A} M ⊗_{OP} M^{∗} ⊗_{A} iOG. It follows from 4.3 that Y ⊗_{OP} Y^{∗} is a permutation O(P × P)-module on which OP acts freely on the left and on the right and that OP is isomorphic to a direct summand of Y ⊗_{OP} Y^{∗} ≅ End_{OP}^{op}(Y). Thus (End_{OP}^{op}(X))(ΔP) is nonzero projective as a left or right kZ(P)-module. It follows from [26, Proposition 3.8] that OGb is isomorphic to a direct summand of Y ⊗_{OP} Y^{∗} as an OGb-OGb-bimodule. Multiplying by i on the left and right implies that A is isomorphic to a direct summand of M ⊗_{OP} M^{∗}. Thus (iv) implies (ii), completing the proof.

Proposition 4.5. Let M be an indecomposable direct summand of the A-B-bimodule U. Then A is isomorphic to a direct summand of the A-A-bimodule M ⊗_{B} M^{∗} if and only if B is isomorphic to a direct summand of the B-B-bimodule M^{∗} ⊗_{A} M.

Proof. Suppose that A is isomorphic to a direct summand of M ⊗_{B}M^{∗}, but that B is not isomorphic to a direct summand of M^{∗} ⊗_{A} M. It follows from Lemma 4.3, applied to B, A, V^{∗} instead of A, B, V, respectively, that M^{∗} ⊗_{B} M is a direct sum of summands of bimodules of the form B ⊗_{OQ} B, with Q running over a family of proper subgroups of P. Thus M ⊗_{B} M^{∗} ⊗_{A} M ⊗_{B} M^{∗} is a direct sum of summands of bimodules of the form M ⊗_{OQ} M^{∗}, with Q running over a family of proper subgroups of P. In particular, we have (M ⊗_{B} M^{∗} ⊗_{A} M ⊗_{B} M^{∗})(ΔP) = {0}. But A ≅ A ⊗_{A} A is a summand of M ⊗_{B} M^{∗} ⊗_{A} M ⊗_{B} M^{∗}, hence (M ⊗_{B} M^{∗} ⊗_{A} M ⊗_{B} M^{∗})(ΔP) ≠ {0}. This contradiction shows that B is isomorphic to a direct summand of M^{∗} ⊗_{A} M. Exchanging the roles of A and B yields the converse.

In particular, if the equivalent statements in Proposition 4.5 hold, then the algebras A, B are separably equivalent (cf. [20, Definition 3.1]).

Proposition 4.6. Let M be an indecomposable direct summand of the OGb-OHc-bimodule X. The following are equivalent.

(i) OGb is isomorphic to a direct summand of the OGb-OGb-bimodule M ⊗_{OHc} M^{∗}. (ii) OHc is isomorphic to a direct summand of the OHc-OHc-bimodule M^{∗} ⊗_{OGb} M. (iii) M has vertex ΔP. If these equivalent conditions hold, then V is an OΔP -source of M.

Proof. Set A = iOGi and B = jOHj. The equivalence of (i) and (ii) is a reformulation of 4.5 at the level of block algebras, via the standard Morita equivalences between block algebras and almost source algebras. The bimodule M has a vertex ΔQ contained in ΔP, for some subgroup Q of P. If this vertex is smaller than ΔP, then (M ⊗_{OHc} M^{∗})(ΔP) = {0}, so also (iMj ⊗_{B} jM^{∗}i)(ΔP) = {0}. Thus 4.4 implies that iMj ⊗_{B} jM^{∗}i has no summand isomorphic to B, hence M ⊗_{OHc} M^{∗} has no summand isomorphic to OHc. This shows that (ii) implies (iii). Suppose that ΔP is a vertex of M. Then clearly V is a source of M. By 2.3, M has a vertex source pair (P′, V′) such that P′ ⊆ P × P and such that V′ is a direct summand of iMj as an OP′-module. It follows that as an O(P × P)-module, iMj has an indecomposable direct summand W with vertex P′ and source V′. Green’s indecomposability theorem implies that W ≅ Ind_{P′}^{P ×P} (V′) is a summand of iMj as an OP -OP -bimodule, hence of A ⊗_{OP} Ind_{ΔP}^{P ×P} (V) ⊗_{OP} B. Using the bimodule structure of A and B, it follows that P′ is a ‘twisted’ diagonal subgroup of the form {(φ(u), ψ(u)) | u ∈ P for some φ, ψ ∈ Aut_{F}(P). Since A ≅ _{φ}A as OP -A-bimodules and B ≅ B_{ψ} as B-OP -bimodules, it follows that iMj has a direct summand isomorphic to to Ind_{ΔP}^{P ×P} (V′), and then V′ ≅ V by the stability of V. But then iMj ⊗_{OP} jM^{∗}i has a summand isomorphic to Ind_{ΔP}^{P ×P} (V) ⊗_{OP} Ind_{ΔP}^{P ×P} (V^{∗}) ≅ Ind_{ΔP}^{P ×P} (V ⊗_{O} V^{∗}). Since V ⊗_{O} V^{∗} has a trivial summand, it follows that iMj ⊗_{OP} jM^{∗}i has a summand isomorphic to OP, which implies that (iMj ⊗_{OP} jM^{∗}i)(ΔP) ≠ {0}. Proposition 4.4 implies that A is isomorphic to a direct summand of iMj ⊗_{B} jM^{∗}i, and hence OGb is isomorphic to a direct summand of M ⊗_{OHc} M^{∗}, completing the proof.

5 Proof of Theorem 1.1 and of Theorem 1.2

Proof of Theorem 1.1. We use the notation and hypotheses from Theorem 1.1. Since (M ⊗_{B} M^{∗})(ΔP) ≠ {0}, it follows from 4.4 that M ⊗_{B} M^{∗} ≅ A ⊕ X for some A-A-bimodule X with the property that every indecomposable direct summand of X is isomorphic to a direct summand of A ⊗_{OQ} A for some fully F-centralised subgroup Q of P. In what follows we use the canonical isomorphism M ⊗_{B} M^{∗} ≅ End_{B}(M) and analogous versions. By 4.2, for any subgroup Q of P we have an injective algebra homomorphism

End_{B^{op}}(M)(ΔQ) → End_{B(ΔQ)^{op}}(M_{Q})

The left term is isomorphic to A(ΔQ) ⊕ X(ΔQ). If Q is nontrivial and fully F-centralised, then the right term is isomorphic to A(ΔQ) by the assumptions on M_{Q}. This forces X(ΔQ) = {0} for any nontrivial fully F-centralised subgroup Q of P. It follows from 4.3 that X is projective as an A-A-bimodule. Similarly, 4.5 implies that M^{∗} ⊗_{A} M ≅ B ⊕ Y for some B-B-bimodule Y, and the same argument with the roles of A and B exchanged shows that Y is projective.

Proof of Theorem 1.2. By multiplying the involved bimodules with almost source idempotents, it follows using the block algebra versions 3.3 and 4.6 of 4.4 and of 3.2, respectively, that Theorem 1.2 is equivalent to Theorem 1.1.

Remark 5.1. We sketch a proof of Theorem 1.1 under the additional assumption that the endopermutation OP -module V has an F-stable p-permutation resolution Y_{V} (cf. [28, §7]). That is, Y_{V} is a bounded complex of permutation OP -modules such that the complex Y_{V} ⊗_{O} Y_{V}^{∗} is split as a complex of OP -modules with respect to the diagonal action of P, and such that Y_{V} ⊗_{O} Y_{V}^{∗} has homology concentrated in degree zero and isomorphic to V ⊗_{O} V^{∗}. The F-stability means that for any subgroup Q of P and any morphism φ : Q → P in F the indecomposable summands of Res_{Q}^{P}(Y_{V}) and Res_{φ}(Y_{V}) with vertex Q (as complexes) are isomorphic (this is slightly weaker than the condition stated in [16, Theorem 1.3]). The proof of [16, Theorem 1.3] yields an indecomposable direct summand Y of the complex A ⊗_{OP} Ind_{ΔP}^{P ×P} (Y_{V}) ⊗_{OP} B such that Y ⊗_{B} Y^{∗} is split with homology concentrated in degree zero isomorphic to M ⊗_{B} M^{∗}; similarly for Y^{∗} ⊗_{A} Y. Note that Y is splendid in the sense of [16, 1.10] or [17, 1.1]. It follows from [28, §7.3] that if Q is a fully F-centralised subgroup of P, then Y (ΔQ) is a bounded complex of A(ΔQ)-B(ΔQ)-bimodules with homology concentrated in a single degree and isomorphic to a bimodule M_{Q} as in the statement of the Theorem. It follows from [16, Proposition 2.4] or [19, Theorem 9.2] that for any fully Fcentralised subgroup Q of P we have (Y ⊗_{B} Y^{∗})(ΔQ) ≅ Y (ΔQ) ⊗_{B(ΔQ)} Y (ΔQ)^{∗} and this complex is again split with homology concentrated in degree zero isomorphic to M_{Q} ⊗_{B(ΔQ)} M_{Q}^{∗}. Thus if M_{Q} induces a Morita equivalence, then A(ΔQ) M_{Q} ⊗_{B(ΔQ)} M_{Q}^{∗}. Therefore, if M_{Q} induces a Morita equivalence for all nontrivial fully F-centralised subgroups Q of P, then Y (ΔQ) induces in particular a derived equivalence for all such Q, and hence, by a result of Rouquier (see [19, Appendix] for a proof) the complex Y induces a stable equivalence. This implies that M induces a stable equivalence, providing thus an alternative proof of Theorem 1.1.

6 Appendix

In the proof of Proposition 4.4 we have made use of [26, Proposition 3.8]. The purpose of this section is to give a proof of a slightly more general result in this direction. We use without further comment the following standard properties of p-permutation modules: if U is an indecomposable OG-module with vertex P and trivial source, then the kN_{G}(P)-module U(P) is the Green correspondent of k ⊗_{O} U, and we have a canonical algebra isomorphism (End_{O}(U))(P) ≅ End_{k}(U(P)). Moreover, as a kN_{G}(P)/P -module, U(P) is the multiplicity module of U; in particular, U(P) is projective indecomposable as a kN_{G}(P)/P -module. Any p-permutation kG-module lifts uniquely, up to isomorphism, to a p-permutation OG-module. In particular, the isomorphism class of an indecomposable OG-module U with vertex P and trivial source is uniquely determined by the isomorphism class of the projective indecomposable kN_{G}(P)/P -module U(P). See e. g. [29, §27] for an expository account on p-permutation modules with further references. The following result is well-known (we include a proof for the convenience of the reader):

Proposition 6.1. Let G be a finite group, P a p-subgroup, U an indecomposable OG-module with vertex P and trivial source O, and let M be an OG-module such that Res_{P}^{G}(M) is a permutation OP -module. Set N = N_{G}(P)/P. Let α : U → M be a homomorphism of OG-modules. The following are equivalent.

(i) The OG-homomorphism α : U → M is split injective. (ii) The kN-homomorphism α(P) : U(P) → M(P) is injective.

Proof. The implication (i) ⇒ (ii) is trivial. Suppose that (ii) holds. Then α(P) : U(P) → M(P) is split injective as a kN-homomorphism because U(P) is projective, hence injective, as a kN-module. Using that soc(U(P)) is simple it follows that M has an indecomposable direct summand M′ such that the induced map β(P) : U(P) → M′(P) is still split injective, where β is the composition of α followed by the projection from M onto M′. The Brauer homomorphism applied to the algebra End_{O}(M′) maps End_{O}(M′)_{P}^{G} onto (End_{k}(M′))(P)_{1}^{N} ≅ End_{k}(M′(P))_{1}^{N} (cf. [29, (27.5)]). The summand of M′(P) isomorphic to U(P) corresponds to a primitive idempotent in End_{k}(M′(P))_{1}^{N}, hence lifts to a primitive idempotent in End_{O}(M′)_{P}^{G}. Since M′ is indecomposable, this idempotent is Id_{M′}, and hence, by Higman’s criterion, M′ has P as a vertex. But then M′ has a trivial source, and so M′(P) is indecomposable as a kN-module, hence isomorphic to U(P). By the Green correspondence this implies U ≅ M′. Composing β with the inverse of this isomorphism yields an endomorphism γ of U which induces an automorphism on U(P). Since End_{OG}(U) is local, this implies that γ is an automorphism of U, and hence that β : U → M′ is an isomorphism. It follows that α is split injective, whence the implication (ii) ⇒ (i).

Proposition 6.2. Let G be a finite group, P a p-subgroup, U an indecomposable OG-module with vertex P and trivial source, and let M be an OG-module such that Res_{P}^{G}(M) is a permutation OP - module. Set N = N_{G}(P)/P. Suppose that N has a normal p-subgroup Z such that the kN-module k ⊗_{kZ} U(P) is simple and such that M(P) is projective as a kZ-module. Let α : U → M be a homomorphism of OG-modules. The following are equivalent.

(i) The OG-homomorphism α : U → M is split injective.

(ii) There is a nonzero direct summand W of Res_{Z}^{N}(U(P)) such that the kZ-homomorphism α(P)|_{W} : W → Res_{Z}^{N}(M(P)) is injective.

Proof. The implications (i) ⇒ (ii) is trivial. Suppose that (ii) holds. Since U(P) is projective indecomposable as a kN-module, it has a simple socle and a simple top, and these are isomorphic. The restriction of U(P) to kZ remains projective, and hence the module k ⊗_{kZ} U(P) has the same dimension as the submodule U(P)^{Z} of Z-fixed points in U(P). Since Z is normal in N, it follows that U(P)^{Z} is a kN-submodule of U(P), hence that U(P)^{Z} contains the simple socle of U(P). Since k ⊗_{kZ} U(P) is assumed to be simple, hence isomorphic to the top and bottom composition factor of U(P), it follows that U(P)^{Z} is equal to the socle soc(U(P)) of U(P) as a kN-module. By the assumption (ii), the kernel of the map U(P) → M(P) does not contain U(P)^{Z}. Since the socle of U(P) as a kN-module is simple, this implies that α(P) : U(P) → M(P) is injective, and hence that α is split injective by Proposition 6.1. This shows the implication (ii) ⇒ (i).

Corollary 6.3 (Puig [26, Proposition 3.8]). Let G be a finite group, b a block of OG, P a defect group of b, and A an interior G-algebra. Suppose that the conjugation action of P on A stabilises an O-basis of A, and that Br_{ΔP} (b) · A(ΔP) · Br_{ΔP} (b) is projective as a left or right kZ(P)-module. Then the map α : OGb → A induced by the structural homomorphism G → A^{×} is split injective as a homomorphism of OGb-OGb-bimodules.

Proof. After replacing A by b·A·b we may assume that A(ΔP) is projective as a left or right kZ(P)- module. As an O(G × G)-module, OGb has vertex ΔP and trivial source. By the assumptions, A is a permutation OΔP -module. We have N_{G×G}(ΔP) = (C_{G}(P) × C_{G}(P)) · N_{ΔG}(ΔP). Set N = N_{G×G}(ΔP)/ΔP. Denote by Z the image of Z(P) × {1} in N; this is equal to the image of {1} × Z(P), normal in N, and canonically isomorphic to Z(P). Consider the induced map α(ΔP) : kC_{G}(P)Br_{ΔP} (b) → A(ΔP). If e is a block of kC_{G}(P) occurring in Br_{ΔP} (b), then kC_{G}(P)/Z(P)ē is a matrix algebra, where ē is the canonical image of e in kC_{G}(P)/Z(P). (We use here again our assumption that k is large enough.) Thus kC_{G}(P)/Z(P)ē is simple as a module over k(C_{G}(P) × C_{G}(P)). Since the blocks e arising in this way are permuted transitively by N_{G}(P), it follows that k ⊗_{kZ} kC_{G}(P)Br_{ΔP} (b) ≅ kC_{G}(P)/Z(P)c is a simple kN-module, where c is the image of Br_{ΔP} (b) in kC_{G}(P)/Z(P), or equivalently, c is the sum of the ē as above. By the assumptions, A(ΔP) is projective as a kZ-module, and hence the obvious composition of algebra homomorphisms kZ(P) → kC_{G}(P)Br_{ΔP} → A(ΔP) is injective. Thus kC_{G}(P)Br_{ΔP} (b) has a summand isomorphic to kZ, as a kZ-module, which is mapped injectively into A(ΔP) by α(ΔP). The result follows from the implication (ii) ⇒ (i) in Proposition 6.2.

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