On stable equivalences of Morita type

Abstract

The aim of the present text is to report on some properties of stable equivalences of Morita type and derived equivalences. In the first section we describe, very roughly, a panorama of various types of equivalences between blocks of group algebras, in order to give an idea of the context of what we are going to do in the sequel. In section 2 we recall from [110] the basic properties of Broué's notion of a stable equivalence of Morita type, apply them to p-groups in section 3 and develop some further formal properties in section 4. In the last section we come back to derived equivalences and show, how the tilting complexes for blocks with dihedral defect groups (that one gets based on Erdmann's classification up to Morita equivalence of these blocks) give rise to an elementary principle of constructing tilting complexes for any symmetric algebra.