Introduction

Let G and H be two finite groups, p a prime number. Let (O) be a complete discrete valuation ring with residue field k of characteristic p and with field of fractions K of characteristic 0, “big enough” for G and H. Let A and B be two blocks of G and H over O.

Let M be a (A ⊗ B°)-module, projective as A-module and as B-module, where B° denotes the opposite algebra of B. We denote by M* the (B ⊗ A°)- module Homo(M, O).

We say that M induces a stable equivalence between A and B if

M ⊗BM* ≅ A ⊗ projectives as (A ⊗ A°) – modules and

M* ⊗AM ≅ B ⊗ projectives as (B ⊗ B°) – modules.

Let C be a complex of (A ⊗ B°)-modules, all of which are projective as A-modules and as B°-modules.

Denoting by C* the O-dual of C, we say that C induces a Rickard equivalence between A and B if C ⊗BC* is homotopy equivalent to A as complexes of (A ⊗ A°)-modules and C* ⊗AC is homotopy equivalent to B as complexes of (B ⊗ B°)-modules.

By [Ri4, 5.5] j from a complex C inducing a Rickard equivalence between A and B, one can construct a module M inducing a stable equivalence between A and B as follows : In the derived bounded category of A ⊗ B°, the complex C is isomorphic to a complex with only one term which is not projective as (A ⊗ B°)-module, V in degree –n and then the n-th Heller translate (syzygy) M = Ωn(V) induces a stable equivalence between A and B.