Groups '93 Galway / St Andrews: From stable equivalences to Rickard equivalences for blocks with cyclic defect

Let G and H be two finite groups, p a prime number. Let 0 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 0. 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, 0). We say that M induces a stable equivalence between A and B if