ON SPLENDID DERIVED AND STABLE EQUIVALENCES

BETWEEN BLOCKS OF FINITE GROUPS

Markus Linckelmann

July 1998

1 Introduction

A p−block of a finite group G is an algebra coming along with some invariants which cannot, a priori, be read off the module category of the blockalgebra. These invariants are due to the presence of the group G - such as defect groups, the p−local structure, or generalized decomposition numbers. We are therefore interested in those of Morita, derived or stable equivalences between blockalgebras, which preserve the above invariants.

Any Morita equivalence between two blockalgebras of finite groups induced by an isomorphism of their source algebras has this property; this is due to L. Puig [12], [13].

For derived equivalences, J. Rickard developed in [20] a notion of derived equivalences between the module categories of p−blocks of finite groups that he calls splendid equivalences, and proves that indeed, at least for principal blocks, a splendid equivalence is compatible with the p−local structure. Starting from Rickard’s work [20] we define in [11] a notion of splendid derived equivalences which is a slight modification of Rickard’s definition of a splendid equivalence, and show then that splendid derived equivalences are compatible with the p−local structure for arbitrary blocks. Our definition suggests also what we should want to call a splendid stable equivalence.

In section 2 we prove an additional property of splendid derived equivalences - the transitivity theorem 2.1 below (answering a question asked by M. E. Harris), which claims, roughly speaking, that the tensorproduct of two splendid two-sided tilting complexes is again splendid. As a by-product, we generalize our main result Theorem 1.1 in [11].

In section 3 we give a “local” characterization of splendid stable equivalences of Morita type, motivated by an observation of J. Alperin, who proved the result in the particular case of principal blocks and restriction to a subgroup.

For the sake of completeness we include in section 4 a proof of the fact, due independently to L. Puig and L. Scott, that splendid Morita equivalences between two blocks of finite groups are precisely the Morita equivalences arising from an isomorphism of source algebras.

Section 5 is devoted to showing that R. Rouquier’s Rickard complex for blocks with cyclic defect groups constructed in [21] is in fact still splendid in our more restrictive definition of splendid tilting complexes.

As in [11], much of this work is based on an extensive use of the Brauer construction (see e.g. [11, section 2] for further reference) and on properties of fusion in block algebras (see e.g. [11, section 3]). For the convenience of the reader we give in the sections 6 and 7 below a short review of these notions and some of their basic properties we often use without further comment. We refer to [12] or [24] for Puig’s concept of pointed groups and related subjects. In fact, all background material on block theory we need in this paper and in [11] can be found in J. Thévenaz’ book [24].

Notation and terminology. We fix a prime p and a discrete complete valuation ring O having a residue field k of characteristic p. Our notation and conventions are exactly as in [11]; in particular, an O−algebra is always free of finite rank as O−module, and all modules are finitely generated unitary left modules, unless stated explicitely otherwise. Recall that an O−algebra A is symmetric if it is isomorphic to its O−dual A^{∗} = Hom_{O}(A, O) as A − A−bimodule. One of the particular features of symmetric algebras A and B is the following: if M is an A − B−bimodule and N a B − A−bimodule inducing a Morita equivalence between A and B, that is, M ⊗_{B} N ≅ A and N ⊗_{A} M ≅ B as bimodules, then necessarily N ≅ M^{∗} as B −

A−bimodule. J. Rickard proved in [18] that if A and B are derived equivalent, there is a Rickard complex of A − B−bimodules, that is, a bounded complex X of left and right projective A − B−bimodules such that the total complexes X ⊗_{B} X^{∗} and X^{∗} ⊗_{A} X

are chain homotopy equivalent to A and B, respectively, where the latter are viewed as complexes of bimodules concentrated in degree zero. Recall from Broué [3] or [4], that we say that an A − B−bimodule M induces a stable equivalence of Morita type between A and B, if M is projective as left A−module, as right B−module and if M ⊗_{B} M^{∗} ≅ A ⊕ X for some projective A − A−bimodule X and M^{∗} ⊗_{A} M ≅ B ⊕ Y

for some projective B − B−bimodule Y. See [9], [10] for some general properties of stable equivalences of Morita type.

We need some standard notions and facts from block theory: if G is a finite group, a block of OG is a primitive idempotent b of the center Z(OG) of the group algebra OG of G over O. A defect group of b is a minimal subgroup P of G such that the map OGb ⊗_{OP} OGb ⟶ OGb induced by multiplication in the block algebra OGb splits

as homomorphism of OGb − OGb−bimodules. A source idempotent in (OGb)^{ΔP} is a primitive idempotent i ∈ (OGb)^{ΔP} satisfying Br_{ΔP} (i) ≠ 0, where Br_{ΔP} : OG ⟶ kC_{G}(P) is the Brauer homomorphism (see Section 6 below for details on the Brauer construction and the Brauer homomorphism). The algebra iOGi, endowed with the group homomorphism P → (iOGi)^{×} mapping u ∈ P to ui is a source algebra of the block b (cf. [12]). By the uniqueness theorem [5, 1.8] due to Broué and Puig, for any subgroup Q of P there is a unique block e_{Q} of kC_{G}(Q) satisfying Br_{ΔQ}(i)e_{Q} = Br_{ΔQ}(i); we call e_{Q} the block of kC_{G}(Q) determined by the source idempotent i. The local structure of the block b is the information encoded in the category whose objects are the pairs (Q, e_{Q}), with Q running over the set of subgroups of P, and whose morphisms are, for any two subgroups Q, R of P, the sets E_{G}((Q, e_{Q}), (R, e_{R})) consisting of all classes φ̃ modulo inner automorphisms of R of group homomorphisms φ : Q → R for which there is an element x ∈ G such that φ(u) = xux^{−1} for all u ∈ Q and such that xe_{Q}x^{−1} = e_{xQx^{−1}}. Roughly speaking, E_{G}((Q, e_{Q}), (R, e_{R})) consists of all classes of group homomorphisms from Q to R modulo inner automorphisms of R which are induced by elements in G compatible with the local structure of the block.

Splendid versus splendid. Originally J. Rickard defined a splendid equivalence between to block algebras OGb and OHc of finite groups G, H having a common defect group P to be a derived equivalence given by a Rickard complex X of OGb − OHc−bimodules which are isomorphic to sums of direct summands of the bimodules OGb ⊗_{OQ} OHc, with Q running over the set of subgroups of P (or, equivalently, the

indecomposable direct summands of the components of X have all a vertex contained in ΔP = {(u, u)}_{u∈P} and trivial source when viewed as O(G × H)−modules).

In order to prove a compatibility theorem with the local structure of the blocks [11, 1.1] and the transitivity theorem 2.1 below for arbitrary blocks we need a slightly sharper condition on the components on X (which is fulfilled by all examples of splendid equivalences known to the author):

Definition 1.1 (cf. [11]) Let G, H be finite groups and let b, c be blocks of OG, OH, respectively, having a common defect group P. Choose source idempotents i ∈ (OGb)^{ΔP} and j ∈ (OHc)^{ΔP}. A Rickard complex X of OGb − OHc−bimodules is called a splendid tilting complex (with respect to the choice of i, j), if all components of X are isomorphic to finite direct sums of direct summands of the OGb − OHc−bimodules OGi ⊗_{OQ} jOH, where Q runs over the set of subgroups of

P. A splendid derived equivalence between OGb and OHc is a derived equivalence given by a splendid tilting complex of OGb − OHc−bimodules.

We emphasize that the notion of a splendid tilting complex is defined with respect to a choice of source idempotents; we will abusively omit to mention this choice if it is clear from the context. The main point about working relative to a choice of source idempotents is that the concept of a splendid tilting complex passes down to the source algebras of the considered blocks. Precisely, if X is a splendid tilting complex of OGb − OHc−bimodules with respect to source idempotents i, j as above, then setting A = iOGi, B = jOHj, the complex iXj is a Rickard complex of A − B−bimodules whose components are isomorphic to finite direct sums of summands of the A − B−bimodules A ⊗_{OQ} B, where Q runs over the set of subgroups of P.

Another change of terminology is that we use the adjective “splendid” as well to qualify Morita equivalences and stable equivalences of Morita type: a splendid Morita equivalence between OGb and OHc is a Morita equivalence given by an OGb− OHc−bimodule M which is isomorphic to a direct summand of OGi ⊗_{OP} jOH. Note

that putting the source idempotents is no restriction in this case: if M is an OGb − OHc−bimodule inducing a Morita equivalence between OGb and OHc then M is indecomposable, and thus, if M has a vertex contained in ΔP this vertex must then be equal to ΔP since P is a defect group of both b and c, hence M is isomorphic to a direct summand of OGb ⊗_{OP} OHc and therefore of OGi ⊗_{OP} jOH at least for

some source idempotents i, j since M is indecomposable. Similarly, a splendid stable equivalence between OGb and OHc is a stable equivalence of Morita type given by an OGb − OHc−bimodule M which is isomorphic to a direct summand of OGi ⊗_{OP} jOH.

2 Transitivity of splendid derived equivalences

The main result of this section shows that under suitable hypotheses on the local structure of the considered blocks, the tensorproduct of two splendid tilting complexes is again a splendid tiliting complex.

Theorem 2.1. Let G, H, L be finite groups and let b, c, d be blocks of G, H, L, respectively, having a common defect group P. Let i, j, l be source idempotents in (OGb)^{ΔP}, (OHc)^{ΔP}, (OLd)^{ΔP}, respectively. For any subgroup Q of P, denote by e_{Q}, f_{Q}, g_{Q} the unique block of kC_{G}(Q), kC_{H}(Q), kC_{L}(Q) determined by i, j, l, respectively. Assume that for any two subgroups Q, R of P we have

E_{G}((Q, e_{Q}), (R, e_{R})) = E_{H}((Q, f_{Q}), (R, f_{R})) = E_{L}((Q, g_{Q}), (R, g_{R})) .

Let X be a splendid tilting complex of OGb − OHc−bimodules and let Y be a splendid tilting complex of OHc − OLd−bimodules. Then the following hold:

(i) The total complex X ⊗_{OHc} Y is a splendid tilting complex of OGb −

OLd−bimodules.

(ii) For any subgroup Q of P there is an isomorphism of complexes of kC_{G}(Q)e_{Q} −

kC_{L}(Q)g_{Q}−bimodules e_{Q}(X ⊗_{OHc} Y)(ΔQ)g_{Q} ≅ e_{Q}X(ΔQ)f_{Q} ⊗_{kC_{H}(Q)f_{Q}} f_{Q}Y (ΔQ)g_{Q}.

Remark 2.2 All complexes occurring in statement 2.1 (ii) are Rickard complexes by [11,1.1]. This follows also from 2.1(ii) applied to the particular case G = L , b = d and Y = X^{∗}.

The proof of 2.1 will be an immediate consequence of the following lemma, which is based on the material developed in the Sections 2 and 3 of [11] (reviewed in the appendix, Section 7 below):

Lemma 2.3. With the notation and hypotheses of 2.1, let R, S be subgroups of P, let U be a direct summand of the OGb − OHc−bimodule OGi ⊗_{OR} jOH, V a direct

summand of the OHc−OLd−bimodule OHj ⊗_{OS} lOL and M an indecomposable direct

summand of the OGb − OLd−bimodule U ⊗_{OHc} V.

(i) There is a subgroup T of P such that C_{P} (T) is a defect group of e_{T} and such that M is isomorphic to a direct summand of OGi ⊗_{OT} lOL.

(ii) For any subgroup Q of P there is a natural isomorphism of kC_{G}(Q)e_{Q} −

kC_{L}(Q)g_{Q}−bimodules e_{Q}(U ⊗_{OHc} V)(ΔQ)g_{Q} ≅ e_{Q}U(ΔQ)f_{Q} ⊗_{kC_{H}(Q)f_{Q}} f_{Q}V (ΔQ)g_{Q}.

Proof. (i) Since M is indecomposable, it is isomorphic to a direct summand of OGi ⊗_{OR} W ⊗_{OS} lOL for some indecomposable direct summand W of jOHj as

OR − OS−bimodule. By 7.1 there is y ∈ H such that W ≅ O[RyS] ≅ OR ⊗_{OT} _{φ}(OS),

where T = R ∩ ^{y}S and φ : T ⟶ S is the group homomorphism defined by φ(u) = y^{−1}uy for all u ∈ T. It follows from 7.7 that we have φ̃ ∈ E_{H}((T, f_{T}), (S, f_{S})).

Thus M is isomorphic to a direct summand of OGi ⊗_{OT} _{φ}(lOL) and hence, again

since M is indecomposable, it is isomorphic to a direct summand of OGm ⊗_{OT} _{φ}(nOL)

for some primitive idempotents m ∈ (iOGi)^{ΔT} and n ∈ (lOLl)^{Δφ(T)}. By choosing T minimal with this property, using [11,2.6], we may assume that in fact m and n belong to local points μ and ν of T on OGb and of φ(T) on OLd, respectively, such that T_{μ} ⊂ P_{γ} and φ(T)_{ν} ⊂ P_{λ}, where γ and λ are the unique local points of P on OGb and OLd containing i and l, respectively. By 7.6 we may assume that C_{P} (T) is a defect group of e_{T}. Then C_{P} (T) is a defect group of g_{T} too by 7.8. Moreover, since we still have φ̃ ∈ E_{H}((T, f_{T}), (S, f_{S})) = E_{L}((T, g_{T}), (S, g_{S})) there is z ∈ L such that φ(u) = z^{−1}uz for all u ∈ T. Then T_{^{z}ν} ⊂ P_{λ} by the last statement of 7.6. Thus OGm ⊗_{OT} _{φ}(nOL) ≅ OGm ⊗_{OT} ^{z}nOL is isomorphic to a direct summand of OGi ⊗_{OT} lOL,

and hence M is so, too.

(ii) The proof is an amalgam of [11, 2.4] and [11, 4.1]. Let Q be a subgroup of P and T a subgroup of Q. The spaces U^{ΔQ} and U_{ΔT}^{ΔQ} are OC_{G}(Q)−OC_{H}(Q)−subbimodules of U. Thus the natural map U ⊗_{OC_{H}(Q)} V ⟶ U ⊗_{OH} V induces a map U^{ΔQ} ⊗_{OC_{H}(Q)}

V^{ΔQ} ⟶ (U ⊗_{OH} V)^{ΔQ} which in turn induces a map U(ΔQ) ⊗_{kC_{H}(Q)} V (ΔQ) ⟶

(U ⊗_{OH} V)(ΔQ). This map is a homomorphism of kC_{G}(Q) − kC_{L}(Q)−bimodules, and

whence induces a homomorphism of kC_{G}(Q)e_{Q} − kc_{L}(Q)g_{Q}−bimodules

2.3.1. e_{Q}U(ΔQ)f_{Q} ⊗_{kC_{H}(Q)f_{Q}} f_{Q}V (ΔQ)g_{Q} ⟶ e_{Q}(U ⊗_{OHc} V)(ΔQ)g_{Q}.

This homomorphism is natural in U and V. By 7.6 and 7.9 we may conjugate simultaneously (Q, e_{Q}), (Q, f_{Q}), (Q, g_{Q}) in order to obtain that C_{P} (Q) is a defect group of any of the blocks e_{Q}, f_{Q}, g_{Q}. This allows us to reduce the situation down to the source algebra level.

Let A = iOGi, B = jOHj and C = lOLl. By [12, 3.5], the algebras OGb and A are Morita equivalent via the OGb−A−bimodule OGi and its O−dual iOG. Since C_{P} (Q) is a defect group of e_{Q} and Br_{ΔC_{P} (Q)}(Br_{ΔQ}(i)) = Br_{ΔQC_{P} (Q)}(i) ≠ 0, for the same reason the algebras kC_{G}(Q)e_{Q} and Br_{ΔQ}(i)kC_{G}(Q)Br_{ΔQ}(i) = A(ΔQ) are Morita equivalent via the bimodule kC_{G}(Q)Br_{ΔQ}(i) and its k−dual. Thus, in order to show that 2.3.1 is an isomorphism, it suffices to show that the induced homomorphism of A(Q) − C(Q)−bimodules

2.3.2. (iUj)(ΔQ) ⊗_{B(ΔQ)} (jV l)(ΔQ) ⟶ (iUj ⊗_{B} jV l)(ΔQ)

is an isomorphism. Since 2.3.1 is natural in U, V, it suffices to show this for U = OGi ⊗_{OR} jOH and V = OHj ⊗_{OS} lOL by the assumptions on U and V. Then iUj =

A ⊗_{OR} B and jV l = B ⊗_{OS} C. Computing the expressions in 2.3.2 involves not the

full bimodule structure of iUj and jV l, but requires only the structure of iUj as OQ − B−bimodule and the structure of jV l as B − OQ−bimodule. Thus it suffices to show that for any indecomposable direct summand W of A as OQ−OR−bimodule the induced map

2.3.3. (W ⊗_{OR} B)(ΔQ) ⊗_{B(ΔQ)} (B ⊗_{OS} C)(ΔQ) ⟶ (W ⊗_{OR} B ⊗_{OS} C)(ΔQ)

is a linear isomorphism. The structure of W is determined in 7.1: there is a subgroup T of Q and an injective group homomorphism φ : T ⟶ R such that W ≅ OQ ⊗_{OT}_{φ}(OR). If T is a proper subgroup of Q, both sides in 2.3.3 are zero (cf. 6.1(iii)).

It remains to treat the case where Q = T. In other words, we have to show that the induced map

2.3.4. (_{φ}B)(ΔQ) ⊗_{B(ΔQ)} (B ⊗_{OR} C)(ΔQ) ⟶ (_{φ}B ⊗_{OR} C)(ΔQ)

is an isomorphism. We do this again by decomposing this homomorphism into direct summands: it suffices to show that for any primitive idempotent n ∈ B^{φ(Q)} the induced map

2.3.5. (_{φ}nB)(Q) ⊗_{B(Q)} (B ⊗_{OR} C)(Q) ⟶ (_{φ}nB ⊗_{OS} C)(Q)

is an isomorphism. If Br_{Δφ(Q)}(n) = 0 one easily sees that, again by 6.1(iii), both sides in 2.3.5 are zero. We may thus assume that n belongs to a local point ν of φ(Q) on OHc such that φ(Q)_{ν} ⊂ P_{δ}, where δ is the local point of P on OHc containing j. Now we use the fact that by 7.7 the group homomorphism φ defines a G−fusion φ̃ from (Q, e_{Q}) to (φ(Q), e_{φ(Q)}). Thus this is also an H−fusion from (Q, f_{Q}) to (φ(Q), f_{φ(Q)}). In particular, there is y ∈ H such that φ(u) = ^{y}u = yuy^{−1} for any u ∈ Q. Let now μ be the unique local point of Q on OHc such that ^{y}(Q_{μ}) = φ(Q)_{ν}. Since C_{P} (Q) is a defect group of f_{Q} we may apply 7.6; in particular, the last statement of 7.6 implies that we have Q_{μ} ⊂ P_{δ}. Thus there is m ∈ μ lying in B. Moreover, mB ≅ _{φ}(nB) as OQ − B−bimodule because of the fact that φ is induced by conjugation with y and y conjugates Q_{μ} to φ(Q)_{ν}. Thus we are reduced to showing that the induced map

2.3.6. (mB)(ΔQ) ⊗_{B(ΔQ)} (B ⊗_{OS} C)(ΔQ) ⟶ (mB ⊗_{OS} C)(ΔQ)

is an isomorphism. We have (mB)(ΔQ) = Br_{ΔQ}(m)B(ΔQ) and (mB ⊗_{OS}C)(ΔQ) =

Br_{ΔQ}(m)(B ⊗_{OS} C)(ΔQ), thus 2.3.6 is indeed an isomorphism, which concludes the

proof of 2.3(ii). □

Proof of 2.1. The fact that X ⊗_{OHc} Y is again a Rickard tilting complex is trivial, and

that it actually splendid is an immediate consequence of 2.3(i). Statement (ii) follows from 2.3(ii). □

3 A local characterization of splendid stable equivalences of Morita type

The relevance of stable equivalences of Morita type in group representation theory comes from the fact that all known stable equivalences between block algebras are in fact of Morita type. Moreover, Rickard’s work in [19] shows that if there is a derived equivalence between two symmetric algebras A and B there is also a stable equivalence of Morita type between A and B.

The next theorem is an analogue to 4.1; this should be thought of as a generalization of elementary properties of p−permutation modules and the Brauer construction (such as 6.5 below). For principal blocks and the restriction to a subgroup the equivalence statement is due to J. Alperin and the last statement saying that a splendid stable equivalence preserves the stable local category is due to Puig [16, 7.6.6], but again, we include full proofs for the convenience of the reader.

Theorem 3.1. Let G, H be finite groups, b and c be blocks of G and H, respectively, having a common defect group P. Let i, j be source idempotents in (OGb)^{ΔP}, (OHc)^{ΔP}, respectively. For any subgroup Q of P, denote by e_{Q}, f_{Q} the unique block of kC_{G}(Q), kC_{H}(Q) determined by i, j, respectively.

Let M be an indecomposable direct summand of the OGb−OHc−bimodule OGi ⊗_{OP}

jOH. The following are equivalent:

(i) The bimodule M and its O−dual M^{∗} induce a stable equivalence of Morita type between OGb et OHc.

(ii) For any non trivial subgroup Q of P the bimodule e_{Q}M(ΔQ)f_{Q} and its O−dual induce a Morita-equivalence between kC_{G}(Q)e_{Q} and kC_{H}(Q)f_{Q}, and for any two subgroups Q, R of P we have E_{G}((Q, e_{Q}), (R, e_{R})) = E_{H}((Q, f_{Q}), (R, f_{R})).

In that case, there is a bimodule isomorphism Mj ≅ OGi ⊕ U for some projective OGb − OP −bimodule U, which induces, for any non trivial subgroup Q of P, algebra isomorphisms (iOGi)^{ΔQ}/(iOGi)_{1}^{ΔQ} ≅ (jOHj)^{ΔQ}/(jOHj)_{1}^{ΔQ} and (iOGi)(ΔQ) ≅ (jOHj)(ΔQ), which in turn induce fusion compatible bijections between the sets of local points of Q on iOGi, jOHj, respectively.

For the proof we need two lemmas; the first one is due to Puig (it is in fact a corollary of [15, 14.6] and it has some variations, see e.g. [11, 3.3 (v)] or 7.7, 7.8 below; we sketch a proof for the convenience of the reader):

Lemma 3.2. (cf. Puig[15, 14.6])Let G be a finite group, P_{γ} a defect pointed group of G_{{b}} and i ∈ γ. Let φ ∈ Aut(P). As an OP −OP − bimodule, _{φ}(OP) is isomorphic to a direct summand of iOGi if and only if there is x ∈ N_{G}(P_{γ}) such that φ(u) = xux^{−1} for all u ∈ P.

Proof. If _{φ}(OP) is isomorphic to a direct summand of iOGi as OP − OP −bimodule, hence of OG, it is of the form O[P xP] for some x ∈ G, and then necessarily x ∈ N_{G}(P) because this bimodule has rank |P |. Thus _{φ}(OP) is isomorphic to a direct summand of iOGi if and only if OP is isomorphic to a direct summand of ^{x^{−1}}iOGi for some x ∈ N_{G}(P) satisfying φ(u) = xux^{−1} for all u ∈ P. Since (OP)(ΔP) = kZ(P) this is equivalent to requiring (^{x^{−1}}iOGi)(ΔP) ≠ 0; in particular, Br_{ΔP} (x^{−1}ix) and Br_{ΔP} (i) are non zero primitive idempotents belonging to the same block e_{P} of kC_{G}(P) and hence x^{−1}ix and i belong to the same local point γ (cf. [24, (40.13)]), which means that x ∈ N_{G}(P_{γ}). □

Lemma 3.3. Let G be a finite group, b a block of G, P a defect group of b and i a source idempotent in (OGb)^{ΔP}. For any subgroup Q of P denote by e_{Q} the unique block of kC_{G}(Q) determined by i. Let R be a subgroup of P and X an indecomposable direct summand of the OGb − OGb−bimodule OGi ⊗_{OR} iOG.

Then X is projective if and only if e_{Q}X(ΔQ)e_{Q} = 0 for any non trivial subgroup Q of P.

Proof. Suppose that e_{Q}X(ΔQ)e_{Q} = 0 for any non trivial subgroup Q of P. We show that then in fact X(ΔQ) = 0 for any non trivial subgroup Q of P. To do this it suffices to show that eX(ΔQ)e′ = 0 for any two blocks e, e′ of kC_{G}(Q) satisfying Br_{ΔQ}(b)e = e and Br_{ΔQ}(b)e′ = e′. Since there is x ∈ G such that ^{x}(Q, e) ⊂ (P, e_{P}) (cf. [24, (40.13)]) or, equivalently, such that ^{x}Q ⊂ P and ^{x}e = e_{^{x}Q} we may assume that e = e_{Q}. If e′ = e we are then done by the hypothesis, so assume now that e′ ≠ e.

The image of the set G ⊗ G in kG ⊗_{kR} kG is a k−basis stable under conjugation

with elements of Q. In order to compute (kG ⊗_{kR} kG)(ΔQ) we have to determine all

Q−stable elements of this basis; that is, all pairs (x, y) ∈ G×G, whose image x⊗y in kG ⊗_{kR}kG is Q−stable (cf. 6.1(i)). This is the case if for any u ∈ Q there is r_{u} ∈ R such

that ux = xr_{u} and yu^{−1} = r_{u}^{−1}y, or equivalently, 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 such that xy ∈ C_{G}(Q) and Q^{x} ⊂ R. Consequently,

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

with (x, y) as before. By the uniqueness theorem [5, 1.8], for any such pair (x, y) we have exi = e(^{x}i)x and e(^{x}i) ∈ ker(Br_{ΔQ}) unless e = ^{x}(e_{Q^{x}}); similarly, iye′ = y(i^{y})e′ and (i^{y})e′ ∈ ker(Br_{ΔQ}) unless e′ = (e_{^{y}Q})^{y}. As xy ∈ C_{G}(Q) we have ^{x}(e_{Q^{x}}) = (e_{^{y}Q})^{y}. Therefore, since e ≠ e′, at least one of e(^{x}i) or (i^{y})e′ lies in ker(Br_{ΔQ}). It follows that (ekGi ⊗_{kR} ikGe′)^{ΔQ} ⊂ ker(Br_{ΔQ}); in particular, eX(ΔQ)e′ = {0}.

This shows that X(ΔQ) = {0} for any non trivial subgroup Q of P. When viewed as k(G × G)−module, X has a vertex contained in ΔR = {(r, r)}_{r∈R} and a trivial source; thus if Q is a subgroup of R such that ΔQ is a vertex of X, necessarily X(ΔQ) ≠ {0}, hence Q = 1, which shows that X is projective.

The converse is clear by 6.1(ii). □

Proof of 3.1. If (i) holds then M ⊗_{OHc} M^{∗} ≅ OGb ⊕ X for some projective

OGb − OGb−bimodule X and M^{∗} ⊗_{OGb} M ≅ OHc ⊕ Y for some projective OHc −

OHc−bimodule Y. For any non trivial subgroup Q of P we have X(ΔQ) = 0 and thus, by 2.3(ii), we have e_{Q}M(ΔQ)f_{Q} ⊗_{kC_{H}(Q)f_{Q}} f_{Q}M(ΔQ)^{∗}e_{Q} ≅ e_{Q}(M ⊗_{OHc}

M^{∗})(ΔQ)e_{Q} ≅ kC_{G}(Q)e_{Q}, which means that e_{Q}M(ΔQ)f_{Q} and its k−dual induce a Morita equivalence between kC_{G}(Q)e_{Q} and kC_{H}(Q)f_{Q}. It remains to show the statement on fusion.

Since j is primitive in (OHc)^{ΔP} and not contained in (OHc)_{1}^{ΔP}, the OHc − OP −bimodule OHj is indecomposable non projective. Thus the OGb − OP −bimodule Mj ≅ M ⊗_{OHc}OHj is a direct sum of an indecomposable non projective

OGb − OP −bimodule and a projective OGb − OP −bimodule. By the hypotheses on M, the bimodule Mj is a direct summand of OGi ⊗_{OP} jOHj, thus the non projective

part of Mj is isomorphic to a direct summand of OGi ⊗_{OP} W for some indecompos-

able direct summand W of jOHj as OP − OP −bimodule. Consequently, on one hand M^{∗} ⊗_{OGb} Mj ≅ OHj ⊕ Y j and on the other hand the non projective part of

M^{∗} ⊗_{OGb} Mj is isomorphic to a direct summand of M^{∗}i ⊗_{OP} W. Since Br_{ΔP} (j) ≠ 0 we

have (M^{∗} ⊗_{OGb} Mj)(ΔP) ≠ 0, thus (M^{∗}i ⊗_{OP} W)(ΔP) ≠ 0. Therefore W ≅ _{φ}(OP) for

some automorphism φ of P which by 3.2 is induced by conjugation with an element y ∈ N_{H}(P_{δ}), where δ is the local point of P on OHc containing j. Thus the non projective part of Mj is isomorphic to a direct summand of OGi ⊗_{OP} _{φ}(OP) ≅ (OGi)_{φ^{−1}},

or equivalently, the non projective part of (Mj)_{φ} is isomorphic to a direct summand of OGi. But since φ is induced by conjugation with an element of N_{H}(P_{δ}) we have (OHj)_{φ} ≅ OHj (cf. 6.3) and therefore (Mj)_{φ} ≅ Mj as OGb−OP −bimodules. This shows that

3.1.1. we have Mj ≅ OGi ⊕ U for some projective OGb − OP −bimodule U.

Thus, for any non trivial subgroup Q of P, the functor M ⊗_{OHc} − induces a bijec-

tion between the sets of isomorphism classes of indecomposable non projective direct summands of OHj as OHc − OQ−bimodule and OGi as OGb − OQ−bimodule. By 7.3 this is equivalent to saying that we have a fusion compatible bijection between the sets of local points of Q on iOGi and on jOHj, for any non trivial subgroup Q of P. In particular we have E_{G}((Q, e_{Q}), (R, e_{R})) = E_{H}((Q, f_{Q}), (R, f_{R})) for any two subgroups Q, R of P, which shows indeed, that (i) implies (ii). Moreover, the isomorphism in 3.1.1 induces algebra isomorphisms

3.1.2. (jOHj)^{ΔQ}/(jOHj)_{1}^{ΔQ} ≅ \overline{End}_{OHc⊗OQ}(OHj)^{0} ≅ \overline{End}_{OGb⊗OQ}(Mj)^{0} ≅ \overline{End}_{OGb⊗OQ}(OGi)^{0} ≅ (iOGi)^{ΔQ}/(iOGi)_{1}^{ΔQ},

where the first and last isomorphism are a variation on Higman’s criterion (cf. [24, (17.3)], the second isomorphism follows from the fact that M ⊗_{OHc} − is a stable

equivalence of Morita type, and the third isomorphism follows from 3.1.1. The isomorphism in 3.1.2 induces clearly an isomorphism of k−algebras

3.1.3. (jOHj)(ΔQ) ≅ (iOGi)(ΔQ),

where Q is still running over the set of non trivial subgroups of P. This shows that (i) implies also the last statement of 3.1.

Conversely, if (ii) holds, consider the sequence of adjunction maps (cf. [3])

3.1.4. OGb ⟶_{ϵ} M ⊗_{OHc} M^{∗} ⟶_{η} OGb .

Applying for any non trivial subgroup Q of P the Brauer construction to this sequence and multiplying it with e_{Q} on the left and on the right yields a sequence

3.1.5. kC_{G}(Q)e_{Q} ⟶_{ϵ_{Q}} e_{Q}(M ⊗_{OHc} M^{∗})(ΔQ)e_{Q} ⟶_{η_{Q}} kC_{G}(Q)e_{Q}.

The middle term of this sequence is, by 2.3(ii) and the hypotheses, canonically

isomorphic to e_{Q}M(ΔQ)f_{Q} ⊗_{kC_{H}(Q)f_{Q}} f_{Q}M(ΔQ)^{∗}e_{Q} ≅ kC_{G}(Q)e_{Q}.

It follows that ϵ_{Q} and η_{Q} are mutually inverse isomorphisms. This in turn implies that the composition ηϵ is an automorphism of OGb as OGb−OGb−bimodule; indeed, otherwise we would have Im(ηϵ) ⊂ J(OGb), from which would follow Im(η_{Q}ϵ_{Q}) ⊂ J(kC_{G}(Q)e_{Q}), contradicting the fact that η_{Q}ϵ_{Q} is the identity on kC_{G}(Q)e_{Q}. In particular, η is split surjective, hence M ⊗_{OHc} M^{∗} ≅ OGb ⊕ X, where X = ker(η). We

show now that X is projective.

Observe that e_{Q}X(ΔQ)e_{Q} = 0 for any non trivial subgroup Q of P, as η_{Q} is an isomorphism. Every indecomposable direct summand X′ of X is a direct summand of M ⊗_{OHc} M^{∗}, thus of OGi ⊗_{OP} jOHj ⊗_{OP} iOG by the assumption on M, and hence

of OGi ⊗_{OR} iOG for some subgroup R of P by 2.3(i). Therefore X′ is projective by

3.3, and hence X is so, too. Exchanging the rôles of G and H shows that indeed M and M^{∗} induce a stable equivalence of Morita type between OGb and OHc, which concludes the proof of 3.1. □

Remark 3.4. The equivalence of the statements (i), (ii) in 3.1 holds in the more general situation where M is isomorphic to a direct summand of OGi ⊗_{OP}Ind_{ΔP}^{P ×P} (V) ⊗_{OP}

jOH for some endo-permutation-OΔP -module V, and then V is fusion stable. This follows from using work of Puig in [14]. In this more general situation it is then still true that if (i), (ii) hold, then M induces an equivalence between the stable local categories (cf. [16, 7.6.6]).

4 Splendid Morita equivalences

As mentioned before, it has been independently observed by L. Puig [16, 7.5.1] and L. Scott [22] (implicitly, this is also in [23]), that a Morita equivalence between two blocks of finite groups induced by a p−permutation bimodule is equivalent to an isomorphism between source algebras. For the the sake of completeness and the convenience of the reader we restate this and include a short proof.

Theorem 4.1. (L. Puig [16], L. L. Scott [22]) Let G, H be finite groups, b and c be blocks of G and H, respectively, having a common defect group P. Let i, j be source idempotents in (OGb)^{ΔP}, (OHc)^{ΔP}, respectively.

Let M be an indecomposable direct summand of the OGb−OHc−bimodule OGi ⊗_{OP}

jOH.

The following are equivalent:

(i) The bimodule M and its O−dual M^{∗} induce a Morita equivalence between OGb and OHc.

(ii) There is an algebra isomorphism φ : iOGi ⟶ jOHj mapping ui to uj for

any u ∈ P such that M ≅ OGi ⊗_{iOGi} _{φ}(jOH).

If these statements hold, there is an isomorphism of OGb − OP −bimodules OGi ≅ Mj; moreover, the correspondence sending M to the algebra isomorphism φ induces a bijection between the isomorphism classes of indecomposable direct summands of OGi ⊗_{OP} jOH inducing a Morita equivalence between OGb, OHc and the set

of ((jOHj)^{P})^{×}−conjugacy classes of algebra isomorphisms iOGi ≅ jOHj mapping ui to uj for all u ∈ P.

Proof. Since OGi and iOG induce a Morita equivalence between OGb and iOGi (cf. [12]), clearly if φ : iOGi ⟶ jOHj is an algebra isomorphism, the bimodule OGi ⊗_{iOGi} _{φ}(jOH) induces a Morita equivalence. As moreover φ maps ui to uj for

all u ∈ P this bimodule is isomorphic to a direct summand of OGi ⊗_{OP} jOH since

iOGi is isomorphic to a direct summand of iOGi ⊗_{OP} iOGi. Thus (ii) implies (i).

Next, if M is a direct summand of OGi ⊗_{OP} jOH inducing a Morita equivalence, then

Mj ≅ M ⊗_{OH} OHj is an indecomposable direct summand of the OGb−OP −bimodule

OGi ⊗_{OP} jOHj, hence of OGi ⊗_{OP} W for some indecomposable direct summand of jOHj

as OP − OP −bimodule. As M^{∗} ⊗_{OGb} Mj ≅ OHj is isomorphic to a direct summand

of M^{∗}i ⊗_{OP} W viewed as OHc − OP −bimodule we have (M^{∗}i ⊗_{OP} W)(ΔP) ≠ 0, which

forces W ≅ O[yP] for some y ∈ N_{H}(P_{δ}) by 3.2, where δ is the local point of P on OHc containing j. But then Mj ≅ M^{y}j = Mjy^{−1} and the latter is clearly isomorphic to a direct summand of OGi ⊗_{OP} O[yP y^{−1}] ≅ OGi as OGb − OP −bimodules.

Since i is primitive in (OGb)^{ΔP} the bimodule OGi is indecomposable and therefore we have an isomorphism α : Mj ≅ OGi as OGb − OP −bimodules. In particular, α is an isomorphism of left OGb−modules, hence induces an algebra isomorphism (jOHj)^{0} ≅ End_{OGb⊗1}(Mj) ≅ End_{OGb⊗1}(OGi) ≅ (iOGi)^{0} and the fact that this chain of isomorphisms maps uj to ui for all u ∈ P follows from α being also a homomorphism of right OP −modules. Therefore (i) implies (ii). The last statement follows from the fact that any automorphism of the bimodule Mj is induced by right multiplication with an invertible element in (jOHj)^{ΔP} and hence any two bimodule isomorphisms between OGi and Mj induce algebra isomorphisms between iOGi and jOHj which differ by conjugation with an invertible element in (jOHj)^{ΔP}. □

We conclude this section by some remarks on how splendid stable, derived and Morita equivalences are connected. As mentioned before, J. Rickard proved in [19] that if we have a derived equivalence between two blockalgebras OGb and OHc, there is a stable equivalence of Morita type between OGb and OHc. However, even if this derived equivalence is given by a splendid tilting complex X of OGb − OHc−bimodules, there need not be any splendid stable equivalence. This is due to the way a stable equivalence of Morita type is obtained from X: take a projective resolution π : P_{X} ⟶ X of X; that is, P_{X} is a right bounded complex of projective OGb − OHc−bimodules and π is a quasi-isomorphism, and truncate then P_{X} in some degree large enough such that the truncated complex is still quasiisomorphic to X. The bimodule occurring in the degree where P_{X} is truncated yields a stable equivalence of Morita type between OGb and OHc, but this is not a splendid stable equivalence in general. In fact, there are situations where the splendid stable equivalences are precisely the splendid Morita equivalences:

Proposition 4.2. Let G, H be finite groups, b and c be blocks of G and H, respectively, having a common defect group P. Suppose there is a non trivial subgroup Z of P such that Z ⊂ Z(G) and Z ⊂ Z(H).

Any indecomposable direct summand M of OGb ⊗_{OP} OHc inducing a stable equiv-

alence of Morita type between OGb and OHc induces actually a Morita equivalence; in particular, OGb and OHc have isomorphic source algebras in that case.

Proof. Let X be a projective OGb − OGb−bimodule such that M ⊗_{OHc} M^{∗} ≅ OGb ⊕ X. The hypotheses on M imply that any z ∈ Z acts in the same way on the left and on the right of M, M^{∗}, whence of M ⊗_{OHc} M^{∗}. But on a non zero projective

OGb − OGb−bimodule the left and right action of a non trivial element of Z never coincide. Thus X = {0}, and therefore M induces a Morita equivalence. Since M is indecomposable there are primitive idempotents i ∈ (OGb)^{ΔP}, j ∈ (OHc)^{ΔP} such that M is isomorphic to a direct summand of OGi ⊗_{OP} jOH. The idempotents i, j

are not contained in the kernel of the corresponding Brauer homomorphisms because the diagonal subgroup ΔP is a vertex of M as O(G × H)−module as P is a defect group of both b and c. Thus, by 4.1, we have an isomorphism of source algebras iOGi ≅ jOHj as interior P −algebras. □

The first part of 4.2 can obviously be generalized: if M is an indecomposable OGb − OHc−bimodule with vertex ΔP when viewed as O(G × H)−modules, with a source on which ΔZ acts trivially, and such that M induces a stable equivalence of Morita type, the argument in the proof of 4.2 shows that then M induces a Morita equivalence (however, unless the source of M is trivial, the source algebras of OGb and OHc are no longer isomorphic).

5 A splendid tilting complex for blocks with cyclic defect groups

Let G be a finite group and b a block of OG having a non trivial cyclic defect group P. Let Z be the unique subgroup of order p of P, set H = N_{G}(Z) and denote by c the unique block of OH having P as defect group such that Br_{ΔP} (b) = Br_{ΔP} (c); this makes sense, as C_{G}(P) ⊂ H. We assume that k is large enough for OHc/J(OHc) to be a direct product of matrix algebras.

Let i and j be primitive idempotents in (OGb)^{ΔP} and (OHc)^{ΔP}, respectively, such that Br_{ΔP} (i) = Br_{ΔP} (j) ≠ 0. That is, i, j are source idempotents which determine the same block of kC_{G}(P). We denote by e the unique block of kC_{G}(P) = kC_{H}(P) such that Br_{ΔP} (i)e = Br_{ΔP} (i), set E = N_{G}(P, e)/C_{G}(P) and denote by P ⋊ E the semi-direct product of P by E with E acting naturally on P. Note that E is a cyclic p′−group acting freely on P − {1}; that is, P ⋊ E is a Frobenius group with kernel P. The unit element of O(P ⋊ E) is primitive in (O(P ⋊ E))^{ΔP}; in other words, the unit element of O(P ⋊ E) is both its unique block and its unique source idempotent.

R. Rouquier constructed in [21] explicitly a two-sided Rickard tilting complex of OGb − O(P ⋊ E)−bimodules which is splendid in the sense defined by J. Rickard in [20]; that is, all of its components are direct sums of indecomposable O(G × (P ⋊ E)^{0})−modules with a vertex contained in the diagonal subgroup ΔP and trivial source.

We add here, that Rouquier’s results in [21] and the transitivity theorem 2.1 imply that there is actually a splendid tilting complex of OGb − O(P ⋊ E)−bimodules in the sense of our definition 1.1 (thus this strong variation of Broué’s abelian defect conjecture in [4] holds for blocks with cyclic defect groups).

We start by recalling the structure of OHc:

5.1. There is an indecomposable endo-permutation OP −module V with vertex P and determinant 1 such that, setting S = End_{O}(V), we have an isomorphism of interior P −algebras

jOHj ≅ S ⊗_{O} O(P ⋊ E) .

Since OHc and jOHj are Morita equivalent, this can be expressed as follows:

5.2. The OHc − O(P ⋊ E)−bimodule OHj ⊗_{OP} Ind_{ΔP}^{P ×P^{0}}(V) ⊗_{OP} O(P ⋊ E) has an

indecomposable direct summand M with vertex ΔP and source V inducing a Morita equivalence between OHc and O(P ⋊ E).

The endo-permutation module V in 5.1 is unique up to isomorphism (thanks to the condition that the determinant of V is one), and therefore V is E−stable. By Rickard’s work in [19], V has an E−stable p−permutation resolution Y_{V}; that is, Y_{V} is a bounded complex of permutation OP −modules with homology concentrated in degree zero, isomorphic to V, whose isomorphism class is E−stable, and such that Y_{V} ⊗_{O} Y_{V}^{∗} is split as complex of OP −modules with respect to the diagonal action of

P.

Thus we may apply [11, 1.3] to obtain the following:

5.3. There is a splendid tilting complex Y of OHc − O(P ⋊ E)−bimodules which is isomorphic to a direct summand of the complex OHj ⊗_{OP} Ind_{ΔP}^{P ×P^{0}}(Y_{V}) ⊗_{OP} O(P ⋊ E)

such that Y has homology concentrated in degree zero and isomorphic to M.

Recall from [10,6.1] the following well-known fact (which holds more generally in any “trivial intersection” situation):

5.4. The OGb − OHc−bimodule bOGc and its dual cOGb induce a stable equivalence of Morita type between OGb and OHc.

This stable equivalence is splendid:

5.5. Up to isomorphism, the OGb−OHc−bimodule bOGc has a unique non projective indecomposable direct summand, and this summand is isomorphic to a direct summand

of OGi ⊗_{OP} jOH.

Indeed, it is a general property of stable equivalences of Morita type [10,3.4], that bOGc has, up to isomorphism, a unique non projective direct summand. In order to show that this summand is isomorphic to a direct summand of OGi ⊗_{OP} jOH it suffices

to show that every indecomposable direct summand of iOGj is isomorphic to a direct summand of iOGi ⊗_{OP} jOHj, or equivalently, that iOGj is relatively OΔP −projective

when viewed as iOGi⊗_{O}(jOHj)^{0}−module. Any iOGi⊗_{O}(jOHj)^{0}−module is relatively

O(P × P^{0})−projective. Since iOGj is projective as left and right OP −module and since (iOGj)(ΔP) ≠ 0 (as Br_{ΔP} (i) = Br_{ΔP} (j)), it follows indeed that iOGj is relatively OΔP −projective. This shows 5.5.

The decisive step is now to apply Rouquier’s Theorem [21, 10.2], which says the following:

5.6. There is a direct summand (N, π) of a projective cover of the OGb − OHc−bimodule bOGc such that the complex

X = (.. → 0 → N ⟶_{π} bOGc → 0 → ..)

is a two-sided Rickard tilting complex of OGb − OHc−bimodules.

Since any projective indecomposable OGb − OHc−bimodule is isomorphic to a direct summand of OGi ⊗_{O} jOH (again because OGb and iOGi as well as OHc and

jOHj are Morita equivalent), 5.5 and 5.6 imply that:

5.7. the complex X is a splendid tilting complex of OGb − OHc−bimodules (with respect to the choice of the source idmpotents i, j).

It is a well-known property of blocks with abelian defect groups, that the inertial quotient E controls fusion (cf. [1]). Therefore the hypotheses of 2.1 are fulfilled for G, H, P ⋊ E (instead of G, H, L), and we obtain:

Theorem 5.8. With the notation above, the total complex X ⊗_{OHc} Y is a splendid

tilting complex of OGb − O(P ⋊ E)−bimodules.

Remark 5.9 The length of the complex X ⊗_{OHc}Y in 5.8 (i.e. the number of non zero

components) is at most twice the length of the endo-split p−permutation resolution Y_{V} of V, since X has at most length 2 (in fact, X has length 1 if and only if OGb and OHc have isomorphic source algebras iOGi ≅ jOHj).

Remark 5.10 The methods of this section apply to blocks with Klein four defect groups: for p = 2, J. Rickard showed that there is a splendid tilting complex between OA_{4} and the principal block algebra OA_{5}b_{0} of A_{5}. Using this and the classification of the source algebras of blocks with a Klein four defect group in [8], one can show, by applying exactly the same strategy as above, that for any block b of a finite group G having a defect group P ≅ ℤ_{2} ×ℤ_{2}, there is a splendid derived equivalence between OGb and ON_{G}(P)c in the sense of our definition 1.1, where c is the unique block of ON_{G}(P) such that Br_{ΔP} (c) = Br_{ΔP} (b). Again, this means that this strong version of Broué’s abelian defect conjecture [4] holds for blocks with a Klein defect group.

6 Quoted results on the Brauer construction and p−permutation modules

If P is a finite p−group, for any subgroup Q of P the Brauer construction yields a functor from the category of OP −modules to the category of kC_{P} (Q)−modules which factors through the O−stable category (cf. [9]) of OP −modules. This construction goes back to Brauer [2] in the case of group algebras and has been generalized by Broué and Puig [5] to arbitrary G−algebras, as well as to modules in Feit [6].

This functor turns out to be particularly well-behaved on the subcategory of permutation modules; we recall in this section its definition and some of its main properties.

If U is an OP −module, for subgroup Q of P we denote as usual by U^{Q} the subspace of Q−fixpoints in U. For any further subgroup R of Q, we denote by Tr_{R}^{Q} : U^{R} → U^{Q} the R−linear map sending u ∈ U^{R} to ∑_{y∈[Q/R]}yu, where [Q/R] is a

system of representatives in Q of the cosets Q/R; this definition does not depend on the choice of [Q/R]. We set U_{R}^{Q} = Im(Tr_{R}^{Q}). Clearly U_{R}^{Q} is an OC_{P} (Q)−submodule of U^{Q}, and we define U(Q) = U^{Q}/(J(O)U^{Q} + ∑_{R}U_{R}^{Q}), where in the sum R runs

over the set of proper subgroups of Q. We denote then by Br_{Q}^{U} or simply Br_{Q} if no confusion is possible, the canonical surjection U^{Q} ⟶ U(Q). Note that if U is a non zero kP −module then U^{P} is the socle of U and hence in particular non zero, too.

Given any homomorphism of OP −modules φ : U ⟶ V, for any two subgroups Q and R of P such that R ⊂ Q, clearly φ maps U_{R}^{Q} to V_{R}^{Q} and hence induces a homomorphism of kC_{P} (Q)−modules φ(Q) : U(Q) ⟶ V (Q), which makes this construction functorial.

If X is an OP − OP −bimodule and Q a subgroup of P, we mean by X(ΔQ) the above construction with respect to the diagonal subgroup ΔQ = {(u, u)}_{u∈Q} of P ×P, where we consider X as O(P × P)−module with (u, v) ∈ P × P acting on x ∈ X as uxv^{−1} (this is a slight change of notation with respect to [11], where we would write X(Q) instead of X(ΔQ)).

We recall some elementary properties of the Brauer construction (the easy and well-known proof of 6.1 is left to the reader):

Lemma 6.1. Let P be a finite p−group, U an OP −permutation module and X an indecomposable OP − OP −bimodule having a P × P −stable O−basis such that X is projective as left and right OP −module.

(i) If B is a P −stable O−basis of U then Br_{Q}^{U}(B^{Q}) is a k−basis of U(Q) for any subgroup Q of P, where B^{Q} = B ∩ U^{Q}.

(ii) The module U is projective if and only if U(Q) = 0 for any non trivial subgroup Q of P.

(iii) We have X(ΔP) ≅ kZ(P) if X ≅ OP as OP −OP −bimodule and X(ΔP) = 0 otherwise.

(iv) For any subgroup Q of P there is a natural isomorphism (X^{∗})(ΔQ) ≅ (X(ΔQ))^{∗}.

The above lemma applied to X = OG, where G is a finite group and P a p−subgroup of G yields an obvious isomorphism (OG)(ΔP) ≅ kC_{G}(P). By identification through this isomorphism, the Brauer homomorphism Br_{ΔP} : OG ⟶ kC_{G}(P) is in that case an algebra homomorphism, because its kernel is a two-sided ideal in (OG)^{P}.

Lemma 6.2. Let P be a finite p−group and U a permutation kP −module.

(i) We have ∩_{Q} ker(Br_{Q}^{U}) = U_{1}^{P}, where Q runs over the set of non trivial subgroups

of P.

(ii) If U has no non trivial projective direct summand, we have U = ∑_{Q}U^{Q}, where

Q runs over the set of non trivial subgroups of P.

Proof. (i) Let B be a P −stable k−basis of U and let u = ∑_{x∈B}λ_{x}x be an element of

U^{P}. Let x ∈ B such that λ_{x} ≠ 0 and R be the stabiliser of x in P. Clearly then Br_{R}^{U}(x) ≠ 0 and the statement follows.

(ii) If U has no non trivial projective direct summand, every element x of a P −stable k−basis B of U has a non trivial stabilisor Q_{x} in P and hence U = ∑_{x∈B}kx =

∑_{x∈B}U^{Q_{x}}. □

Lemma 6.3. Let P be a finite p−group and U a kP −module. For any non trivial subgroup Q of P we have ker(Br_{Q}^{U}) ⊂ (rad(U))^{Q}.

Proof. Let R be a proper subgroup of Q and let u ∈ U^{R}. Since k has characteristic

p we have Tr_{R}^{Q}(u) = ∑_{y∈[Q/R]}yu = ∑_{y∈[Q/R]}(y − 1)u ∈ J(kP)U. □

Lemma 6.4. Let P be a finite p−group, U, V be kP −modules and φ : U ⟶ V a kP −homomorphism.

(i) If V is a kP −permutation module having no non zero projective direct summand and φ(Q) : U(Q) ⟶ V (Q) is surjective for any non trivial subgroup Q of P then φ is surjective.

(ii) If U is a kP −permutation module having no non zero projective direct summand and φ(Q) : U(Q) ⟶ V (Q) is injective for any non trivial subgroup Q of P then φ is injective.

Proof. (i) If φ is not surjective, there is a maximal submodule M of U containing Im(φ). Since for any non trivial subgroup Q of P the map φ(Q) is surjective, we have M^{Q}+ker(Br_{Q}^{V}) = V^{Q}, thus M^{Q} = V^{Q}, since by 6.3 we have ker(Br_{Q}^{V}) ⊂ (rad(V))^{Q} ⊂ M^{Q}. Hence 6.2 (ii) yields the contradiction V = ∑_{Q}M^{Q} ⊂ M, where in the sum Q

runs over the set of non trivial subgroups of P.

(ii) The hypotheses imply that (ker(φ))^{P} is contained in ker(Br_{Q}^{U}) for any non trivial subgroup Q of P and thus in U_{1}^{P} by 6.2(i). Now U_{1}^{P} = 0 as U has no non zero projective direct summand, thus (ker(φ))^{P} = 0 and hence ker(φ) = 0. □

Proposition 6.5. Let P be a finite p−group, U, V be kP −permutation modules and φ : U ⟶ V be a kP −homomorphism. Then φ induces an isomorphism \overline{φ} : U ⟶ V in the stable category \overline{Mod}(kP) if and only if φ(Q) is an isomorphism of kC_{P} (Q)−modules for any non trivial subgroup Q of P.

Proof. We clearly may assume that U, V have no non zero projective direct summands. It follows then from 6.4 that φ is an isomorphism if and only if φ(Q) is so for any non trivial subgroup Q of P. □

Corollary 6.6. With the notation of 6.5, assume that φ(Q) is an isomorphism for any non trivial subgroup Q of P.

If φ is surjective then ker(φ) is projective and if φ is injective then coker(φ) is projective.

7 Appendix: Fusion in block algebras

Let G be a finite group, b a block of G , P_{γ} a defect pointed group of b and i ∈ γ; that is, i is a source idempotent in (OGb)^{ΔP}. For any subgroup Q of P there is a unique block e_{Q} of kC_{G}(Q) satisfying Br_{ΔQ}(i)e_{Q} ≠ 0 (cf. [5, 1.8] or [22, (40.4)]).

In the previous sections as well as in [11] we exploit frequently the interplay between fusion in OGb and its bimodule structure as OP −OP −bimodule. The basic argument - first used by Puig in [13] in order to show that the local category of b (cf.[13] ) is an invariant of its source algebra iOGi - goes as follows:

Since i commutes to P, we have a direct sum decomposition of OP −OP −bimodules OG = iOGi ⊕ iOG(1 − i) ⊕ (1 − i)OGi ⊕ (1 − i)OG(1 − i). Moreover, for any two subgroups Q, R of P we have a decomposition of OG as direct sum of indecomposable OQ−OR−bimodules OG = ⊕_{x∈[Q\G/R]}O[QxR]. Thus, by the Krull-Schmidt theorem,

7.1. for any two subgroups Q, R of P and any indecomposable direct summand W of iOGi as OQ − OR−bimodule there is x ∈ G such that W ≅ O[QxR].

With this notation, if we set S = Q ∩ ^{x}R and define the group homomorphism φ : Q ⟶ R by φ(u) = x^{−1}ux for all u ∈ S, we have an isomorphism of OQ − OR−bimodules O[QxR] ≅ OQ ⊗_{OS} _{φ}(OR) sending qxr to q ⊗ r, where q ∈ Q, r ∈ R.

We use the following notation from [13]: if Q, R are subgroups of P we denote by { \atop Hom}(Q, R) the set of equivalence classes of group homomorphisms from Q to R modulo inner automorphisms of R and by E_{G}((Q, e_{Q}), (R, e_{R})) the image in { \atop Hom}(Q, R) of all group homomorphisms φ : Q ⟶ R such that there is x ∈ G satisfying x(Q, e_{Q}) ⊂ (R, e_{R}) (where this inclusion means that ^{x}Q ⊂ R and ^{x}e_{Q} = e_{^{x}Q}; see [1], [5] or [24, section 40] for the inclusion of Brauer pairs) and φ(u) = xux^{−1} for all u ∈ Q; note that φ is in particular injective. We call the equivalence class φ̃ of φ in { \atop Hom}(Q, R) a G−fusion from (Q, e_{Q}) to (R, e_{R}). Analogously, for any two local pointed groups Q_{δ}, R_{ϵ} on OGb we denote by E_{G}(Q_{δ}, R_{ϵ}) the image in { \atop Hom}(Q, R) of all group homomorphisms φ : Q ⟶ R such that there is x ∈ G satisfying ^{x}(Q_{δ}) ⊂ R_{ϵ} and φ(u) = xux^{−1} for all u ∈ Q; again we call φ̃ a G−fusion from Q_{δ} to R_{ϵ}. Observe that if Q_{δ} ⊂ P_{γ}, R_{ϵ} ⊂ P_{γ} we have E_{G}(Q_{δ}, R_{ϵ}) ⊂ E_{G}((Q, e_{Q}), (R, e_{R})).

The maximal b−Brauer pairs are all G−conjugate (cf. [1]); thus for any b−Brauer pair (Q, e) there is x ∈ G such that ^{x}(Q, e) ⊂ (P, e_{P}). Similarly, the maximal local pointed groups on OGb are all G−conjugate (cf. [12, 1.2]); thus for any local pointed group Q_{δ} on OGb there is x ∈ G such that ^{x}(Q_{δ}) ⊂ P_{γ}. In other words, E_{G}((Q, e), (P, e_{P})) and E_{G}(Q_{δ}, P_{γ}) are non empty.

The points of any subgroup Q of G on OG correspond bijectively to the set of isomorphism classes of indecomposable direct summands of OG viewed as OQ − OG−bimodule; indeed, the Krull-Schmidt theorem implies that 7.2. for any two points δ, δ′ of Q on OG and any d ∈ δ, d′ ∈ δ′ we have dOG ≅ d′OG as OQ − OG−bimodules if and only if δ = δ′.

Since any representative of a G−fusion is injective it is the composition of an isomorphism and an inclusion. Using 7.1 and 7.2 we can characterise any G−fusion represented by an isomorphism as follows:

7.3. Let Q_{δ}, R_{ϵ} be local pointed groups on OGb such that Q ≅ R, let d ∈ δ, e ∈ ϵ and φ : Q ⟶ R a group homomorphism. We have φ̃ ∈ E_{G}(Q_{δ}, R_{ϵ}) if and only if OGd ≅ (OGe)_{φ} as OG − OQ−bimodules, and this in turn holds if and only if dOG ≅ _{φ}(eOG) as OQ − OG−bimodules.

Proof. If φ̃ ∈ E_{G}(Q_{δ}, R_{ϵ}) there is x ∈ G such that ^{x}(Q_{δ}) = R_{ϵ} and φ(u) = xux^{−1} for all u ∈ Q; in particular, ^{x}δ = ϵ and thus, possibly after replacing e by some ((OGb)^{R})^{×}−conjugate we may assume that e = xdx^{−1}. Then the map sending ad to adx^{−1} = ax^{−1}e (where a ∈ OG) is an isomorphism of OGb − OQ−bimodules OGd ≅ (OGe)_{φ}. Conversely, if OGd ≅ (OGe)_{φ} then in particular dOGd ≅ (dOGe)_{φ} as OQ− OQ−bimodules and as both terms are projective as right OQ−modules necessarily φ is injective, hence an isomorphism by the hypothesis Q ≅ R. As Br_{ΔQ}(d) ≠ 0 there is a direct summand of dOGd as OQ − OQ− bimodule isomorphic to OQ (cf. 6.1 (iii)). Thus (OQ)_{φ^{−1}} is isomorphic to a direct summand of dOGe as OQ − OR−bimodule, hence in particular (OQ)_{φ^{−1}} ≅ O[Qx^{−1}] = O[x^{−1}R] for some x ∈ G satisfying φ(u) = xux^{−1} for all u ∈ Q. But then OGd ≅ (OGe)_{φ} ≅ OGex as OG − OQ−bimodules, or, equivalently, OGdx^{−1} = OG^{x}d ≅ OGe, which by 7.2 means that ^{x}(Q_{δ}) = R_{ϵ}. □

Statement 7.3 admits a reformulation at the source algebra level in the following way. For any subgroup Q of P any primitive idempotent of (iOGi)^{Q} is still primitive in (OGb)^{Q} and from this it follows easily (cf. [13, 1.7]) that any local point δ of Q on iOGi is contained in a unique local point δ′ of Q on OGb such that Q_{δ′} ⊂ P_{γ}, and conversely, if δ′ is a local point of Q on OGb such that Q_{δ′} ⊂ P_{γ} then δ = iOGi ∩ δ′ is a local point of Q on iOGi. We will abusively denote δ and δ′ by the same letter. As moreover OGb and iOGi are Morita equivalent (cf. [12, 3.5]) , statement 7.3 is equivalent to the following:

7.4. Let Q_{δ}, R_{ϵ} be local pointed groups on iOGi such that Q ≅ R, let d ∈ δ, e ∈ ϵ and φ : Q ⟶ R a group homomorphism. We have φ̃ ∈ E_{G}(Q_{δ}, R_{ϵ}) if and only if iOGd ≅ (iOGe)_{φ} as iOGi − OQ−bimodules.

In fact, we recover Puig’s more precise theorem [13, 3.1] characterising G−fusion in iOGi as iOGi−fusion (cf. [13, 2.5]):

7.5. (Puig [13,2.12 and 3.1]) Let Q_{δ} , R_{ϵ} be local pointed groups on iOGi such that Q ≅ R, let d ∈ δ, e ∈ ϵ and φ : Q ⟶ R a group homomorphism. We have φ̃ ∈ E_{G}(Q_{δ}, R_{ϵ}) if and only if there is an invertible element a ∈ (iOGi)^{×} satisfying a(ud)a^{−1} = φ(u)e for all u ∈ Q.

.

Proof. If a exists as claimed then right multiplication by a^{−1} induces an isomorphism of iOGi − OQ−bimodules iOGd ≅ (iOGda^{−1})_{φ} = (iOGe)_{φ} and hence φ̃ ∈ E_{G}(Q_{δ}, R_{ϵ}) by 7.4. Conversely, if x ∈ G such that ^{x}(Q_{δ}) = R_{ϵ} and φ(u) = xux^{−1} for all u ∈ Q, then ^{x}δ = ϵ and hence ^{x}d = ^{c}e for some c ∈ ((OGb)^{R})^{×}, or equivalently, c^{−1}xd = ec^{−1}x. In particular, iOGd ≅ iOGe as left iOGi−modules and therefore iOG(i − d) ≅ iOG(i − e) as left iOGi−modules. Any such isomorphism and its inverse come from right multiplication with suitable elements s ∈ (i − d)OG(i − e), t ∈ (i − e)OG(i − d) satisfying st = i − d, ts = i − e, and then a = t + c^{−1}xd is invertible with inverse a^{−1} = s + dx^{−1}c and a(ud)a^{−1} = φ(u)e for all u ∈ Q. □

We recall now parts of the material of [11, section 3] that we need in this paper. We start with a (well-known) statement on defect groups of the blocks e_{Q} saying roughly speaking that via a convenient conjugation, we may assume that C_{P} (Q) is a defect group of e_{Q} without loosing any information on local points of Q on OGb (see for instance [11, 3.3(i),(ii),(iv)]):

7.6. For any subgroup Q of P the group C_{P} (Q) is contained in a defect group of e_{Q}, and there is x ∈ G such that ^{x}(Q, e_{Q}) ⊂ (P, e_{P}) and C_{P} (^{x}Q) is equal to a defect group of the block ^{x}e_{Q} = e_{^{x}Q} of kC_{G}(Q). Moreover, for any local point δ of Q on OGb such that Q_{δ} ⊂ P_{γ} we have then also ^{x}(Q_{δ}) ⊂ P_{γ}.

Proof. Since Br_{ΔP} (i) ≠ 0 we have Br_{ΔC_{P} (Q)}(Br_{ΔQ}(i)) = Br_{ΔQC_{P} (Q)}(i) ≠ 0 whence Br_{ΔC_{P} (Q)}(e_{Q}) ≠ 0 which shows that C_{P} (Q) is contained in a defect group of e_{Q}. Moreover, if (R, f) is a maximal e_{Q}−Brauer pair then (QR, f) is a b−Brauer pair. Since maximal b−Brauer pairs are all conjugate, there is x ∈ G such that ^{x}(QR, f) ⊂ (P, e_{P}). But then also^{x}(Q, e_{Q}) ⊂ (P, e_{P}), and the defect group^{x}R of^{x}e_{Q} is contained in C_{P} (^{x}Q), whence equal to C_{P} (^{x}Q) by the first part. In order to show the last statement, it suffices therefore to show that if C_{P} (Q) is a defect group of e_{Q}, then for any local point δ of Q on OGb satisfying Br_{ΔQ}(δ)e_{Q} ≠ 0 we have in fact Q_{δ} ⊂ P_{γ}. Now if C_{P} (Q) is a defect group of e_{Q} then, since Br_{ΔC_{P} (Q)}(Br_{ΔQ}(i)) ≠ 0, the algebras kC_{G}(Q)e_{Q} and Br_{ΔQ}(i)kC_{G}(Q)Br_{ΔQ}(i) are Morita equivalent, which implies that the second algebra contains some element of any point of the first algebra; in particular, it contains an element of Br_{ΔQ}(δ), which means that Q_{δ} ⊂ P_{γ}. □

As a consequence of [11, 3.3(v)] and the above arguments we have the following connection between fusion and the bimodule structure of iOGi:

7.7. Let R, S be subgroups of P, let T be a subgroup of R and φ : T ⟶ S a group homomorphism. If the OR − OS−bimodule OR ⊗_{OT} _{φ}(OS) is isomorphic to a direct

summand of iOGi then φ is injective and φ̃ ∈ E_{G}((T, e_{T}), (S, e_{S})).

Proof. Observe first that φ has to be injective because iOGi is projective as left OR−module, that _{φ}(Oφ(T)) is isomorphic to a direct summand of iOGi and whence, by 7.1, isomorphic to O[T y^{−1}] = O[y^{−1}φ(T)] for some y ∈ G satisfying ^{y}t = φ(t) for any t ∈ T. Then OT is isomorphic to a direct summand of iOGiy = iOGy^{−1}iy. In particular, we have Br_{ΔT} (iOGy^{−1}iy) = Br_{ΔT} (i)kC_{G}(T)Br_{ΔT} (y^{−1}iy) ≠ 0 by 6.1(iii). Since Br_{ΔT} (i)e_{T} = Br_{ΔT} (i) this forces Br_{ΔT} (y^{−1}iy)e_{T} ≠ 0, thus Br_{Δφ(T)}(i)(^{y}e_{T}) ≠ 0, whence ^{y}e_{T} = e_{φ(T)} = e_{^{y}T}, which concludes the proof of 7.7. □

There is a partial converse to 7.7 (cf. [11, 3.3(v)]):

7.8. Let S, T be subgroups of P and φ : T ⟶ S a group isomorphism. If φ̃ ∈ E_{G}((T, e_{T}), (S, e_{S})) and C_{P} (S) is a defect group of e_{S} then the OT − OS−bimodule φ(OS) is isomorphic to a direct summand of iOGi.

Proof. By the assumptions there is x ∈ G such that ^{x}(T, e_{T}) = (S, e_{S}) and φ(t) = ^{x}t for any t ∈ T. By 7.6, for any local point ϵ of T on OGb such that T_{ϵ} ⊂ P_{γ} we have S_{^{x}ϵ} ⊂ P_{γ}. Thus, if l ∈ ϵ, the OT − OS−bimodule lOG^{x}l is isomorphic to a direct summand of iOGi. Since Br_{ΔT} (l) ≠ 0, the OT − OT −bimodule lOGl has a direct summand isomorphic to OT by 6.1(iii). Therefore lOG^{x}l has a direct summand isomorphic to O[T x^{−1}] = O[x^{−1}S] ≅ _{φ}(OS) as claimed. □

The property of a block e_{Q} to have C_{P} (Q) as defect group can be read off the local structure of the block b. To state this properly, let H be another finite group containg P, c a block of H having P as defect group, let δ be a local point of P on OHc, let j ∈ δ and denote for any subgroup Q of P by f_{Q} the unique block of kC_{H}(Q) such that Br_{ΔQ}(j)f_{Q} = Br_{ΔQ}(j).

7.9. (cf. [11, 3.4]) Assume that E_{G}((Q, e_{Q}), (P, e_{P})) ⊃ E_{H}((Q, f_{Q}), (P, f_{P})) for any subgroup Q of P.

For any subgroup Q of P, if C_{P} (Q) is a defect group of e_{Q} then C_{P} (Q) is a defect group of f_{Q}.

Proof. Let Q be a subgroup of P. Assume that C_{P} (Q) is a defect group of e_{Q} and let R be a defect group of f_{Q}. There is a block f of kC_{C_{H}(Q)}(R) = kC_{G}(QR) such that (R, f) is a maximal f_{Q}−Brauer pair. Then (QR, f) is a c−Brauer pair such that (Q, f_{Q}) ⊂ (QR, f). Thus there is y ∈ H satisfying ^{y}(Q, f_{Q}) ⊂ ^{y}(QR, f) ⊂ (P, f_{P}) because the maximal c−Brauer pairs are H−conjugate. Note that then ^{y}f = f_{^{y}(QR)}. The assumptions imply that there is x ∈ G such that ^{x}(Q, e_{Q}) ⊂ (P, e_{P}) and xu = ^{y}u for all u ∈ Q. In particular, we have ^{x}Q = ^{y}Q, and whence ^{x}(Q, e_{Q}) ⊂ (^{y}(QR), e_{^{y}(QR)}). This means that (^{y}R, e_{^{y}(QR)}) is an ^{x}e_{Q}−Brauer pair, or, equivalently, (^{x^{−1}}(^{y}R), e_{Q(^{x^{−1}} (^{y}R))}) is an e_{Q}−Brauer pair. But then again R is conjugate to a subgroup of the defect group C_{P} (Q) of e_{Q}, whence equal to C_{P} (Q) by 7.6. □

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

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