FROM STABLE EQUIVALENCES TO RICKARD EQUIVALENCES FOR BLOCKS WITH CYCLIC DEFECT

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