SPLENDID DERIVED EQUIVALENCES FOR BLOCKS OF FINITE P-SOLVABLE GROUPS

M. E. Harris, M. Linckelmann

June 1997

Typeset by AMS-TEX

1 Introduction

Since the remarkable discovery of the relevance of derived equivalences in the theory of p−blocks of finite groups - where p is a prime - by J. Rickard in [15],[16], various attempts have been made to understand this phenomenon. In particular, J. Rickard defines in [18] a certain class of derived equivalences between the derived module categories of p−blocks of finite groups that he calls splendid equivalences (and that we are going to call splendid derived equivalences in this paper) which take into account the local structure; that is, which under suitable hypotheses induce a family of derived equivalences at all “local levels” of the considered p−blocks (see [18] for a more detailed motivation). The main condition for a derived equivalence to be splendid is that it is given by a two-sided tilting complex consisting of p−permutation bimodules (see 1.3 and 1.4 below for the precise terminology).

Now in [18, 7.4] J. Rickard gives an example of a p−block of a finite p−nilpotent group, that is Morita equivalent to the group algebra of its defect groups through a Morita equivalence which is not induced by a p−permutation bimodule, but which induces a derived equivalence isomorphic to a splendid derived equivalence

We extend here Rickard’s technique to show a similar result for blocks of finite p−solvable groups with abelian defect groups. Throughout the paper, p is a prime, O a complete discrete valuation ring having an algebraically closed residue field k of characteristic p, and by a block of a finite group G we always mean a primitive idempotent b of the center Z(OG) of the group algebra of G over O.

Theorem 1.1. Let G be a finite p−solvable group, b a block of G having an abelian defect group P and c the Brauer correspondent of N_{G}(P). Then OGb and ON_{G}(P)c are Morita equivalent through a Morita equivalence inducing a derived equivalence which is isomorphic to a splendid derived equivalence.

Since isotypy is an equivalence relation and “induction” as in Proposition 3.1 induces an isotypy, the above theorem and [18, 6.1, 6.2] imply:

Corollary 1.2. The blocks OGb and ON_{G}(P)c are isotypic.

We should point out that the fact that OGb and ON_{G}(P)c in 1.1 are Morita equivalent, is well-known: this follows in particular from Dade’s work in [5] or from Puig’s structure theorem in [14] on blocks of finite p−solvable groups that describes the source algebras of such blocks, independent of whether P is abelian or not, and also implies Corollary 1.2. Moreover, Broué’s conjecture [1, 3.7] - asserting that for any finite group G having an abelian Sylow-p−subgroup P, the principal blocks of G and N_{G}(P) are splendidly derived equivalent - holds if G is p−solvable, since in that case the corresponding block algebras are actually naturally isomorphic (cf. [1, 1.4 and 3.7]). Our theorem asserts that we have splendid derived equivalences for corresponding non principal blocks.

In fact, theorem 1.1 will be a formal consequence of some of Puig’s methods and of Rickard’s idea of describing endo-permutation modules as the degree zero homology of certain bounded complexes of permutation modules, called endo-split p−permutation resolutions. In this paper, we restrict ourselves to blocks with abelian defect groups, since at this stage, it is not known in general whether all endopermutation kP −modules for possibly non abelian P arising from a block of a finite p−solvable group have endo-split p−permutation resolutions.

Also, Rickard’s technique is not limited to finite p−solvable groups as can be easily seen from the proof we give below, since the main idea is to show: given a finite p−group P, an endo-permutation OP −module V having an endo-split p−permutation resolution and A an interior P −algebra satisfying suitable conditions, then, setting S = End_{O}(V), the natural Morita equivalence between A and S ⊗_{O} A is, as a derived

equivalence, isomorphic to a splendid derived equivalence.

However, for this paper, we develop the technique only as far as needed for theorem 1.1. We use this occasion to give a short and almost self-contained ad hoc description of blocks of finite p−solvable groups with abelian defect groups, since the abelian case is easier than the general case (treated by Puig in [14]) and since Puig’s results are unpublished at the time of this writing.

By an O−algebra we always mean an associative unitary algebra A which is finitely generated and free as O−module; we denote by K^{b}(A) the homotopy category of bounded complexes of finitely generated A−modules. We freely use some standard results and concepts such as the Brauer correspondence for blocks of finite groups, relative traces (introduced by J. A. Green), the Brauer homomorphism and induction of interior algbras (due to L. Puig), which can, for instance, be found in J. Thévenaz’ book [20]. Recall that an interior P −algebra, where P is a finite group, is an O−algebra A endowed with a group homomorphism σ : P ⟶ A^{×}. The opposite algebra A^{0} of A is then considered as interior P −algebra via the group homomorphism sending u ∈ P to σ(u^{−1}).

We often write ⊗ for the tensorproduct ⊗_{O} over O.

In the sequel, an A−module is always a finitely generated O−free left module, unless stated otherwise.

We recall from Rickard [18] the following definitions (slightly extended, for our purpose, to interior P −algebras, cf. [13, 2.11]):

Definition 1.3 Let P be a finite group, A, B be symmetric interior P −algebras over O and T a bounded complex of A−B−bimodules all of whose terms are projective as left A−modules and as right B−modules. If we have isomorphisms

T ⊗_{B} T^{∗} ≅ A and T^{∗} ⊗_{A} T ≅ B

in the homotopy categories of complexes of A − A−bimodules and B − B−bimodules, respectively, then T is called a Rickard tilting complex. If moreover all terms of T are isomorphic to direct sums of direct summands of the A−B−bimodules A ⊗_{OQ} B, where Q runs over the set of subgroups of P, we call T a splendid tilting

complex.

Observe that, with the notation of the above definition, we have an isomorphism of A⊗_{O}B^{0}−modules A ⊗_{OQ} B ≅ (A ⊗ B^{0}) ⊗_{OΔQ} O, where ΔQ = {(u, u)}_{u∈Q}, and therefore,

if G, H are finite groups and P is a common p−subgroup of G, H, the relatively ΔP −projective trivial source O(G × H)−modules are precisely those which are isomorphic to direct sums of direct summands of the O(G × H)−modules OG ⊗_{OQ} OH,

where Q runs over the set of subgroups of P.

Definition 1.4 Let P be a finite group, U an OP −module. An endosplit (p−permutation) resolution of U is a bounded complex of (p−permutation) OP −modules X having homology concentrated in degree zero, together with an isomorphism from H_{0}(X) to U, such that the complex X^{∗} ⊗_{O} X is split as a complex of

OP −modules with respect to the diagonal action of P on X^{∗} ⊗_{O} X.

We need the follwing lemma (a consequence of the proof of [18, 7.5]):

Lemma 1.5. Let L be a finite group and P a normal p−subgroup of L. Let U be an OP −module having an endo-split resolution X.

(i) There is a natural O−algebra isomorphism End_{K^{b}(OP)}(X) ≅ End_{OP} (U); in particular, for any direct summand U′ of U there is a direct summand X′ of X, unique up to homotopy equivalence, such that X′ is an endo-split resolution of U′.

(ii) If X is L−stable then Ind_{P}^{L}(X) is an endo-split resolution of Ind_{P}^{L}(U).

Proof. It is well-known that there is a natural isomorphism End_{K^{b}(OP)}(X) ≅ H_{0}(Hom_{OP} (X, X)) and clearly End_{O}(U) ≅ H_{0}(Hom_{O}(X, X)). Since the complex Hom_{O}(X, X) is split for the diagonal action of P, taking fixpoints commutes with taking homology, which implies the isomorphism End_{K^{b}(OP)}(X) ≅ End_{OP} (U). Any isomorphism class of indecomposable direct summands of U corresponds to a conjugacy class of primitive idempotents in End_{OP} (U), thus to a conjugacy class of primitive idempotents in End_{K^{b}(OP)}(X), which lifts to a conjugacy class of primitive idempotents in End_{C^{b}(OP)}(X) (cf.[20,(3.2)]) and hence determines a unique isomorphism class of indecomposable direct summands of X not homotopic to zero. For (ii) we observe that Res_{P}^{L}(Ind_{P}^{L}(X)) is, by the stability hypothesis, a direct sum of [L : P] complexes isomorphic to X, and therefore, Ind_{P}^{L}(X^{∗}) ⊗ Ind_{P}^{L}(X) ≅ Ind_{P}^{L}(X^{∗} ⊗ Res_{P}^{L}(Ind_{P}^{L}(X)) is again a split complex. The lemma follows. □

The crucial point now is the following observation, due to Rickard [18, 7.1, 7.2], which is based on Dade’s classification [3], [4] of endo-permutation kP −modules for finite abelian p−groups P (the stability condition is not stated explicitely in [18] but follows easily from the proof given there, and the statement on the determinant is a consequence of the well-known fact that for odd p any OP −permutation module has determinant 1):

Lemma 1.6. Let L be a finite group and P a normal abelian p−subgroup of L. Let \overline{V} be an L−stable endo-permutation kP −module having an indecomposable direct summand with vertex P. Then there is an endo-permutation OP −module V such that \overline{V} ≅ k⊗_{O}V and such that V has an L−stable endo-split p−permutation resolution X_{V}.

Moreover, if p is odd, the module V has determinant 1 while if p = 2, the determinant of V takes values in {+1, −1}.

We finish this section with a slight generalisation of [4, 12.1]:

Proposition 1.7. Let P be a finite p−group V and V′ endo-permutation OP −modules, each having an indecomposable direct summand with vertex P. We have k ⊗ V′ ≅ k ⊗ V if and only if there is an OP −module U with rank one such that V′ ≅ U ⊗ V.

Proof. Suppose that k ⊗ V′ ≅ k ⊗ V. Let W and W′ be indecomposable direct summands of V and V′, respectively, having vertex P. By [3,6.7] we have k ⊗ W′ ≅ k⊗W and therefore, by [4, 12.1], we have W′ ≅ U⊗W for some OP −module U of rank one. Moreover, any indecomposable direct summand T of V with vertex Q for some subgroup Q of P is isomorphic to a direct summand of Ind_{Q}^{P}Res_{Q}^{P}(W) and unique up to isomorphism by [3, 6.7]. The same statement applied to any indecomposable direct summand T′ of V′ with vertex Q shows that T′ ≅ U ⊗ T. Finally, it follows from [3, 6.3] that reduction modulo J(O) induces a vertex preserving bijection between indecomposable direct summands of V and of k ⊗ V. Putting the above statements together shows that V′ ≅ U ⊗ V. The converse is trivial. □

2 Quoted results on O^{×}−groups

Following the terminology of [13, section 5], an O^{×}−group is a group Ĝ endowed with a group homomorphism O^{×} ⟶ Ĝ whose image lies in Z(Ĝ); we usually denote by λ̂ the image of λ ∈ O^{×} in Ĝ, if no confusion arises, and by Ǧ the opposite O^{×}−group, which as an abstract group is equal to Ĝ, but endowed with the homomorphism O^{×} ⟶ Ǧ sending λ ∈ O^{×} to λ̌ = (λ̂)^{−1}.

In our context, O^{×}−groups arise from the action of a finite group G on a matrix algebra S over O by algebra automorphisms (we write ^{x}s for the action of x ∈ G on s ∈ S): by the Skolem-Noether theorem, any automorphism of S is inner, and hence, for any x ∈ G there is s ∈ S such that ^{x}t = sts^{−1} for all t ∈ S. The group Ĝ of all such pairs (x, s) ∈ G × S^{×} satisfying ^{x}t = sts^{−1} for all t ∈ S becomes an O^{×}−group with the group homomorphism sending λ ∈ O^{×} to λ̂ = (1_{G}, λ1_{S}). We call Ĝ the O^{×}−group defined by the action of G on S; clearly Ĝ is a central O^{×}−extension of G, since the obvious short exact sequence of groups

1 ⟶ O^{×} ⟶ Ĝ ⟶ G ⟶ 1

is exact.

The following well-known facts imply that in our context, it will mainly be sufficient to deal with k^{×}−groups; first, by [19, Ch. II, section 4, Prop. 8],

2.1 the short exact sequence 1 ⟶ 1+J(O) ⟶^{ι} O^{×} ⟶^{π} k^{×} ⟶ 1 splits naturally, where ι is the inclusion and π the canonical surjection;

second,

2.2 if S has O−rank prime to p and P is a finite p−group acting on S, the action of P lifts to a unique group homomorphism σ : P ⟶ S^{×} whose image is contained in ker(det : S^{×} ⟶ O^{×}); in particular, the central O^{×}−extension defined by the action of P on S splits naturally as the map sending u ∈ P to (u, σ(u)) ∈ P̂ is clearly a section of the canonical surjection P̂ ⟶ P; and third (see e.g. [13, 5.22] for more general results),

2.3 if E is a p′-group acting on S, the O^{×}− group defined by the action of E on S is naturally isomorphic to Ê × (1 + J(O)), where Ê is the k^{×}−group defined by the action of E on k ⊗_{O} S.

Furthermore, we recall the definition of the “twisted” group algebra of an O^{×}−group (cf. [13, 5.12] or [20, 10.4]):

2.4 if Ĝ is an O^{×}−group, we denote by O_{∗}Ĝ the quotient of the group algebra OĜ by the ideal generated by the set of elements λ.x̂ − 1_{O}.(λ̂x̂), where λ runs over O^{×} and x̂ runs over Ĝ; clearly O_{∗}Ĝ is O−free of rank |G|, since the image in O_{∗}Ĝ of any subset of order |G| of Ĝ mapping onto G under the canonical surjection Ĝ ⟶ G is an O−basis of O_{∗}Ĝ.

It is easy to check that

2.5 if Ĝ is a central O^{×}−extension of a finite group G, there is a unique algebra homomorphism OG ⟶ O_{∗}Ĝ ⊗ O_{∗}Ǧ mapping x ∈ G to x̂ ⊗ x̂, where x̂ is any element of Ĝ (hence of Ǧ, too) which lifts x.

We denote by Ind_{ΔG}^{Ĝ×Ǧ} the obvious induction functor from the category of OG−modules to that of O_{∗}Ĝ⊗O_{∗}Ǧ−modules defined by the algebra homomorphism in 2.5; note that we can consider any O_{∗}Ĝ⊗O_{∗}Ǧ−module as an O_{∗}Ĝ−O_{∗}Ĝ−bimodule since (O_{∗}Ǧ)^{0} ≅ O_{∗}Ĝ, and then it is easy to check that

2.6 for any two OG−modules U, V there is a unique isomorphism of O_{∗}Ĝ − O_{∗}Ĝ−bimodules

Ind_{ΔG}^{Ĝ×Ǧ}(U) ⊗_{O_{∗}Ĝ} Ind_{ΔG}^{Ĝ×Ǧ}(V) ≅ Ind_{ΔG}^{Ĝ×Ǧ}(U ⊗ V)

mapping ((x̂, 1)⊗u)⊗((1, ŷ)⊗v) to (x̂, ŷ)⊗(u⊗v) , where x̂, ŷ ∈ Ĝ, u ∈ U, v ∈ V.

The last statement in this section is an immediate consequence of Puig’s version [12, (e)] of Dade’s splitting theorem of fusion in endo-permutation modules:

Lemma 2.7. Let S be a matrix algebra over k, P a finite p−group and E a subgroup of Aut(P) acting on S. Suppose there is a group homomorphism P ⟶ S^{×} compatible with the actions of E on P and S, that S has a P −stable k−basis and that S(P) ≠ 0. Then the k^{×}−groups defined by the actions of E on S and on S(P) are isomorphic.

3 The structure of blocks of finite p−solvable groups with abelian defect groups

Blocks of finite p−solvable groups were systematically first studied by P. Fong [6], who determined their character theory. A complete description of their source algebras has been given by Puig in [14] (which is a development of his results announced in [11]); their algebra structure over O follows also from work of Dade [5] and over k from Külshammer [8]. We recall here the structure of blocks of finite p−solvable groups with abelian defect groups as far as we need it (we make no uniqueness discussion); for the convenience of the reader we sketch proofs of all statements and the methods we use are an amalgam of those used by the above authors.

We refer to [20,section 47] for the definition of the Brauer category of a block of a finite group and to [20,section 16] for the induction of interior algebras.

The first proposition is a “reduction step” to the case where G = O_{p′,p,p′}(G).

Proposition 3.1. Let G be a finite p−solvable group and b a block of G.

There is a subroup H of G and an H−stable block e of O_{p′}(H) such that the following hold:

(i) We have O_{p′}(G) ⊂ H and OGb ≅ Ind_{H}^{G}(OHe) as interior G−algebras; in particular, e is a block of H and we have b = T r_{H}^{G}(e).

(ii) Any Sylow-p−subgroup P of H is a defect group of e as a block of H and of the block b of G; moreover, the Brauer categories of b and e are equivalent.

(iii) The block e of H is of principal type; that is, Br_{Q}(e) is a block of kC_{H}(Q) for any subgroup Q of P.

(iv) If c is the block of N_{G}(P) corresponding to b and f the block of N_{H}(P) corresponding to e, then f is a block of O_{p′}(N_{H}(P)) and ON_{G}(P)c ≅ Ind_{N_{H}(P)}^{N_{G}(P)}(ON_{H}(P)f)

as interior N_{G}(P)−algebras; in particular, we have c = T r_{N_{H}(P)}^{N_{G}(P)}(f).

(v) If P is abelian, then H = O_{p′,p,p′}(H).

For the proof of 3.1 we need two lemmas (the first one, given here without proof , can be found in [7, Ch. 6, Thm. 3.3]):

Lemma 3.2. Let G be a finite p−solvable group and Q a Sylow-p−subgroup of O_{p′,p}(G). We have C_{G}(Q) ⊂ O_{p′,p}(G); in particular, if G has abelian Sylowp−subgroups then G = O_{p′,p,p′}(G).

Lemma 3.3. Let G be a finite group, e a G-stable block of O_{p′}(G) and Q a Sylowp−subgroup of O_{p′,p}(G). If C_{G}(Q) ⊂ O_{p′,p}(G) then e is a block of G having any Sylow-p−subgroup P of G as a defect group.

Proof. Set N = O_{p′,p}(G). Since any block of N is a linear combination of p′−elements of N it lies in OO_{p′}(G), and therefore e is a block of N having Q as a defect group. If b is any block of G such that be = b, then Q is contained in a defect group of b (see Dade [2]); in particular, Br_{Q}(b) ≠ 0. By the same argument, if e − b ≠ 0 then Br_{Q}(e − b) ≠ 0. However, Br_{Q}(e) is an N_{G}(Q)−stable block of kN_{N}(Q), thus still a block of kN_{G}(Q), since any block of kN_{G}(Q) lies in kC_{G}(Q) = kC_{N}(Q) (cf. 3.2). Therefore Br_{Q}(b) = Br_{Q}(e), which forces b = e by the previous argument, and so e is indeed a block of G. Now P acts by conjugation on OO_{p′}(G)e, which is a matrix algebra of rank prime to p having a P −stable O−basis (as it is a direct summand of OO_{p′}(G)) and therefore P stabilises at least one element of any such P −stable O−basis of OO_{p′}(G)e. This means Br_{P} (OO_{p′}(G)e) ≠ 0, and hence Br_{P} (e) ≠ 0. It follows that P is a defect group of e as a block of G. □

Proof of 3.1. We prove (i) and (ii) by induction on |G|. Let d be a block of O_{p′}(G) such that bd ≠ 0 and H the stabiliser of d in G. Observe that the different G−conjugates of d are pairwise orthogonal, and hence T r_{H}^{G}(d) is an idempotent in Z(OG) satisfying bT r_{H}^{G}(d) = b. If G = H then b = d by 3.3, hence (i) holds trivially and (ii) follows again from 3.3. If H is strictly smaller than G, we have OGT r_{H}^{G}(d) ≅ Ind_{H}^{G}(OHd), and therefore there is a block e of H such that OGb ≅ Ind_{H}^{G}(OHe). Since this induction of interior algebras preserves the equivalence class of the Brauer categories, applying induction to H and e instead of G and b yields (i) and (ii). For (iii) recall that since C_{G}(Q) ⊂ O_{p,p′}(G), it is a well-known consequence of Thompson’s A×B−lemma [7, Ch. 5, Theorem 3.4], that we have C_{O_{p′} (G)}(R) = O_{p′}(C_{G}(R)), where R is any subgroup of P. Moreover, e is again a block of O_{p′}(G)R with R as a defect group. Thus Br_{R}(e) is a block of N_{O_{p′} (G)R}(R) = C_{O_{p′} (G)}(R)R = O_{p′}(C_{G}(R))R. Again, as Br_{R}(e) involves only p′−elements, it is in fact a block of O_{p′}(C_{G}(R)) which is C_{G}(R)−stable as e is G−stable. Therefore, by the first statement applied to Br_{R}(e) and C_{G}(R) instead of e and G, respectively, Br_{R}(e) is a block of C_{G}(R). For (iv), observe that e has P as a defect group as block of H, and therefore, since b = T r_{H}^{G}(e), an easy calculation (cf. Thévenaz [20, 11.10]) shows that Br_{P} (b) = T r_{N_{H}(P)}^{N_{G}(P)}(Br_{P} (e));

hence c = T r_{N_{H}(P)}^{N_{G}(P)}(f). It follows easily that f is a block of O_{p′}(N_{H}(P)) and that

ON_{G}(P)c ≅ Ind_{N_{H}(P)}^{N_{G}(P)}(ON_{H}(P)f). Statement (v) is an immediate consequence of (ii) and 3.2. □

Remark 3.4 A finite group G satisfying C_{G}(Q) ⊂ O_{p′,p}(G), where Q is a Sylowp−subgroup of O_{p′,p}(G) is called p-constrained in [7,Ch. 8, section 1]. It can be shown that then for any p−subgroup R of G, the subgroup C_{G}(R) is again p−constrained. Therefore the proof of 3.1 (iii) works in the more general situation of lemma 3.3, and shows that the block e of G is of principal type.

Proposition 3.5. Let G be a finite group having an abelian Sylow-p−subgroup P such that G = O_{p′,p,p′}(G). Let b be a G−stable block of O_{p′}(G). Set H = N_{G}(P) and S = OO_{p′}(G)b. Denote by Ĝ the O^{×}−group opposite to that defined by the action of G on S, and set L̂ = Ĝ/O_{p′}(G), where we identify O_{p′}(G) with its natural image {(x, xb)}_{x∈O_{p′} (G)} in Ĝ.

(i) Then b is a block of G having P as a defect group, and there is a unique algebra isomorphism

OGb ≅ S ⊗ O_{∗}L̂

mapping xb to s ⊗ \overline{(x, s)}, where x ∈ G, s ∈ S^{×} such that (x, s) ∈ Ĝ and \overline{(x, s)} is its image in L̂.

(ii) The action of P on S lifts to a unique group homomorphism σ : P ⟶ S^{×} such that det(σ(u)) = 1 for all u ∈ P, there is a p′−subgroup E of Aut(P) and a central O^{×}−extension Ê of E such that we have an isomorphism of O^{×}−groups

L̂ ≅ P ⋊ Ê

mapping \overline{(u, σ(u))} to u for all u ∈ P; in particular, the isomorphism in (i) maps ub to σ(u) ⊗ \overline{(u, σ(u))} for any u ∈ P.

(iii) Let c be the block of H corresponding to b. Then c is a block of O_{p′}(H), we have O_{p′}(H) = O_{p′}(C_{G}(P)) ⊂ O_{p′}(G) and, setting T = OO_{p′}(H)c and denoting by Ĥ the O^{×}− group opposite to that defined by the action of H on T, the inclusion H ⊂ G induces an isomorphism H/O_{p′}(H) ≅ G/O_{p′}(G) which lifts to an isomorphism of O^{×}−groups Ĥ/O_{p′}(H) ≅ Ĝ/O_{p′}(G); in particular, we have again an algebra isomorphism

OHc ≅ T ⊗ O_{∗}L̂,

which, for any u ∈ P, maps uc to 1_{T} ⊗ \overline{(u, σ(u))}.

Proof. (i) By 3.3, b is a block of G and has P as a defect group. Since the O^{×}−group Ĝ is opposite to that defined by the action of G on S, the image s⊗\overline{(x, s)} of xb, where x ∈ G and s ∈ S^{×} such that (x, s) ∈ Ĝ, does not depend on the choice of s. It is easy to check that this is an algebra homomorphism having as an inverse the unique algebra homomorphism mapping s′ ⊗ \overline{(x, s)} to xs^{−1}s′, where x ∈ G, s ∈ S^{×} such that (x, s) ∈ Ĝ and s′ ∈ S.

(ii) Since the rank of S is prime to p, the existence and uniqueness of the lifting σ : P ⟶ S^{×} of the action of P on S with determinant 1 is just statement 2.2. As moreover the image of P in L = G/O_{p′}(G) is a normal Sylow-p−subgroup of L, it has a complement E in L, and hence L̂ ≅ P ⋊ Ê for a suitable O^{×}−extension Ê of E.

(iii) It is an elementary verification to see that O_{p′}(H) = O_{p′}(C_{P} (G)) ⊂ O_{p′}(G). Clearly c is a block of O_{p′}(H) (cf. 3.1 (iii)) and we have k ⊗_{O} T ≅ S(P). By 2.7,

the k^{×}−groups defined by the actions of H on k ⊗ S and k ⊗ T are isomorphic. Thus the isomorphism H/O_{p′}(H) ≅ G/O_{p′}(G) lifts to an isomorphism between their k^{×}−extensions given by the actions of H on k⊗T and G on k⊗S. Thus Ĥ/O_{p′}(H) ≅ P ⋊Ẽ, where Ẽ is a central O^{×}−extension of E such that the k^{×}−groups Ẽ/(1+J(O)) and Ê/(1 + J(O)) are isomorphic. Since E is a p′−group, it follows from 2.3 that the O^{×}−groups Ẽ and Ê are isomorphic, hence that Ĥ/O_{p′}(H) ≅ Ĝ/O_{p′}(G), and then the last statement is an immediate consequence of (i) applied to H, c, T instead of G, b, S, respectively. □

Remark 3.6 From 3.1 and 3.5 it is easy to obtain the structure of the source algebras of any block b of a finite p−solvable group G having an abelian defect group P as determined by Puig in [14]; indeed, since the algebra induction in 3.1 (i) leaves the source algebras invariant, we may assume that G = O_{p′,p,p′}(G), and then, using the notation and hypotheses of 3.5, the source algebras of OGb and OHc are isomorphic to iSi ⊗ O_{∗}(P ⋊ Ê) and O_{∗}(P ⋊ Ê), respectively, where i is a primitive idempotent in S^{P} belonging to the unique local point of P on S (since S(P) is a matrix algebra over k).

Remark 3.7 With the notation of 3.5, the block b considered as block of O_{p′,p}(G) = O_{p′}(G)P is in fact a nilpotent block with P as a defect group, and hence the situation in 3.5 is a particular case of a p′−extension of a nilpotent block treated in [10].

More generally, the situation of a p−constrained group as in 3.3 can be viewed as a particular case of an extension (not necessarily p′) of a nilpotent block; indeed, with the notation of 3.3, the block e considered as block of O_{p′,p}(G) = O_{p′}(G)Q is again nilpotent with Q as a defect group. Thus the reduction proposition 3.1 together with the work of Külshammer-Puig in [9] can be used to give a description of blocks of finite p−solvable groups with arbitrary defect groups (see Puig [14]).

4 Proof of theorem 1.1

Let G be a finite p−solvable group and b a block of G having an abelian defect group.

In order to prove 1.1, we first observe that the algebra induction occurring in 3.1 (i) is well-known to induce a vertex and source preserving Morita equivalence which can be regarded as being given by the OGb − OHe − p−permutation bimodule OGe and its dual eOG (actually, it induces an isomorphism of source algebras, cf. [13, 2.14]), and hence, by 3.1,

4.1 we may assume that G = O_{p′,p,p′}(G) and that b is a G−stable block of O_{p′}(G).

Then 3.5 applies, and , with the notation of 3.5, since P acts trivially on T,

4.2 it suffices to show that there is a Morita equivalence between the interior P −algebras S ⊗ O_{∗}L̂ and O_{∗}L̂ which induces a derived equivalence isomorphic to a splendid derived equivalence.

By 3.5 (ii), we have an isomorphism of interior P −algebras S ≅ End_{O}(V) for some OP −module V having determinant 1, and then V is an endo-permutation OP −module, since S has a P −stable O−basis as it is a direct summand of OO_{p′}(G) (with respect to the action of P by conjugation). The uniqueness of the lifting σ : P ⟶ S^{×} of the action of P on S with determinant 1 implies that the isomorphism class of V is E−stable; that is, _{α}V ≅ V for any α ∈ E. Thus, by 1.6, k ⊗_{O} V lifts to an E−stable endo-permutation OP −module W which has an E−stable

endo-split p−permutation resolution X_{W} (at this stage one could avoid lifting k ⊗ V by working over k and applying Rickard’s lifting theorem in [17] for splendid tilting complexes at the end of this proof, as was also pointed out by the referee). By 1.7 we have W ≅ U ⊗_{O} V for some OP −module of O−rank 1 which is then necessarily

E−stable, as both V and W are so. In fact, since the O−rank of V is prime to p, if p is odd we have V ≅ W as both have determinant one by 1.6, and if p = 2, the character λ : P ⟶ O of U takes values in {1, −1}.

In any case there is an obvious algebra isomorphism S ≅ End_{O}(W) which makes W into an S−module, and we denote by τ : P ⟶ S^{×} the group homomorphism mapping u ∈ P to the element of S^{×} corresponding to the action of u on W through that isomorphism; thus τ(u) = λ(u)σ(u) for any u ∈ P.

We should point out that the two interior P −algebra structures on S⊗OP mapping u ∈ P to σ(u) ⊗ u and τ(u) ⊗ u are isomorphic through the algebra automorphism of S ⊗ OP sending s ⊗ u to sλ(u) ⊗ u, where s ∈ S, u ∈ P. Moreover, we have W ⊗ W^{∗} ≅ V ⊗ V^{∗} ≅ S as S − S−bimodules. Then W^{∗} ⊗ X_{W} is an endo-split resolution of W^{∗} ⊗ W; thus, by 1.5 (ii), Ind_{P}^{L}(W^{∗} ⊗ X_{W}) is an endo-split resolution of Ind_{P}^{L}(W^{∗} ⊗ W). Since the rank of W is prime to p, W^{∗} ⊗ W has a trivial direct summand, thus Ind_{P}^{L}(W^{∗} ⊗ W) has a trivial direct summand, too, since E is a p′−group. Therefore, again by 1.5 (i), Ind_{P}^{L}(W^{∗} ⊗ X_{W}) has a direct summand Y which is an endo-split resolution of the trivial OL−module. Set X = Ind_{ΔL}^{L̂×Ľ}(Y). By 2.6, we have

X^{∗} ⊗_{O_{∗}L̂} X ≅ Ind_{ΔL}^{L̂×Ľ}(Y^{∗} ⊗ Y)

and this is a split complex with homology concentrated in degree zero isomorphic to Ind_{ΔL}^{L̂×Ľ}(O) ≅ O_{∗}L̂, since Y^{∗} ⊗ Y is split with homology concentrated in degree zero isomorphic to the trivial OL−module. Thus X is a Rickard tilting complex of O_{∗}L̂ − O_{∗}L̂−bimodules, and hence,

4.3 the complex W ⊗X is a Rickard tilting complex of (S⊗O_{∗}L̂)−O_{∗}L̂−bimodules with homology concentrated in degree zero isomorphic to W ⊗ O_{∗}L̂.

In order to show that W ⊗ X is in fact a splendid tilting complex (in the sense of 1.3) it remains to show that the terms of W ⊗ X are sums of summands of bimodules of the form (S ⊗ O_{∗}L̂) ⊗_{OQ} O_{∗}L̂, with Q running over the set of subgroups of P, where

here S ⊗ O_{∗}L̂ is considered as interior P −algebra with structural homomorphism mapping u ∈ P to τ(u) ⊗ u (we identify P to its obvious image in O_{∗}L̂). Observe first that

4.4 for any OP −module M the natural S−S−bimodule isomorphism S ≅ W ⊗W^{∗} extends to a natural isomorphism of S ⊗ OP −modules

(S ⊗ OP) ⊗_{OP} M ≅ W ⊗ (W^{∗} ⊗ M),

where the tensorproduct over OP has to be understood via the diagonal action of u ∈ P as multiplication with τ(u) ⊗ u on S ⊗ OP.

By our construction the complex W ⊗ X, considered as complex of S ⊗ O_{∗}L̂ ⊗ O_{∗}Ľ−modules, is isomorphic to a direct summand of the complex

W ⊗ Ind_{ΔP}^{L̂×Ľ}(W^{∗} ⊗ X_{W}) ≅ W ⊗ (O_{∗}L̂ ⊗ O_{∗}Ľ) ⊗_{OΔP} (W^{∗} ⊗ X_{W}) ≅

(S ⊗ O_{∗}L̂ ⊗ O_{∗}Ľ) ⊗_{S⊗OΔP} (W ⊗ (W^{∗} ⊗ X_{W})),

where the last isomorphism comes from the standard isomorphism S ⊗_{S} W ≅ W.

This last complex is now by 4.4 applied to S ⊗ OΔP and X_{W} instead of S ⊗ OP and M, respectively, isomorphic to the complex

4.5 (S ⊗ O_{∗}L̂ ⊗ O_{∗}Ľ) ⊗_{OP} X_{W},

with P acting diagonally on each of the terms of this tensorproduct. Since X_{W} consists of OP −permutation modules, all terms of the complex in 4.5 are sums of modules of the form

(S ⊗ O_{∗}L̂ ⊗ O_{∗}Ľ) ⊗_{OQ} O ≅ (S ⊗ O_{∗}L̂) ⊗_{OQ} O_{∗}L̂

with Q running over the set of subgroups of P. This completes the proof of 1.1. □

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Morton E. Harris University of Minnesota School of Mathematics 105 Vincent Hall 206 Church St SE

Minneapolis, MN 55455-0487

and Markus Linckelmann

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

FRANCE