Introduction

Abstract

This work began by answering the implicit question that Michel Broué raises in [6, 6.C, (pu 1,2,3)], namely by proving that two blocks have the same defect group and isomorphic source algebras if and only if they are Morita equivalent via a bimodule with trivial source (actually, at that time this answer was already known to Leonard Scott: see Remark 7.5 below). In the last part of that paper, Broué attempts to formulate, in terms of derived categories, the old idea of building the structure of a block from the collection of “local data”: the origin of this idea goes back to fifteen years ago when Michel and I tried to extend to the so-called Frobenius blocks with abelian kernel - blocks with abelian defect groups on which the inertial quotients act freely - the methods employed successfully in the nilpotent blocks [9]. Our first approach had been to switch from virtual characters to “virtual modules” and [22] has to be understood as a contribution to this effort: there we show that, indeed, the “construction” of virtual characters described in [8] and [19] can be performed in full generality with “virtual modules”.