Block theory via stable and Rickard equivalences (Q2776258)

The paper under review is concerned with equivalences between blocks of finite groups. It deals mainly with equivalences induced by a chain complex of bimodules. Ideally one would like to construct a Rickard equivalence (which amounts to an equivalence between the corresponding homotopy categories), but often one has only been able to construct a stable equivalence (i.e. an equivalence between the stable module categories). When the complexes involved consist of \(p\)-permutation modules with suitable projectivity properties then one speaks of a splendid (Rickard or stable) equivalence.NEWLINENEWLINENEWLINEThe main motivation for much of the recent work in the area comes from Broué's Abelian Defect Group Conjecture which asserts that a block \(A\) of a finite group \(G\) with Abelian defect group \(D\) should be splendidly Rickard equivalent to its Brauer correspondent \(B\) in \(N_G(D)\). The author gives a slightly stronger version of this conjecture which takes central extensions by \(p\)-groups and \(p'\)-automorphisms into account. One of the main results states that, in the case of principal blocks, a splendid complex induces a stable equivalence if and only if it induces, for every \(p\)-subgroup \(Q\) of \(G\), a Rickard equivalence between the principal blocks of the centralizers of \(Q\).NEWLINENEWLINENEWLINEThe author also shows how to construct stable Rickard equivalences, e.g. for principal blocks with cyclic or Klein four defect groups. For principal blocks with Abelian defect groups of rank 2 or Abelian defect groups of order 8, one only obtains a splendid stable equivalence, and in these cases it remains an open problem to lift these to splendid Rickard equivalences. One expects that, in general, splendid stable or Rickard equivalences should be constructed by induction on the group order. This requires the ``gluing'' of equivalences existing for centralizers of \(p\)-subgroups. The author explains some of the difficulties involved in this gluing process which have lead him to consider a more specialized class of complexes which he calls geometric complexes.NEWLINENEWLINEFor the entire collection see [Zbl 0977.00027].