Pointed groups on D G-interior algebras and Higman embeddings

Abstract

In order to develop for the Rickard equivalences between blocks the same analysis that we do for the Morita ones in Sections 6 and 7 above, we need to extend to OG-interior algebras the notion of pointed group on an OG-interior algebra (cf. 2.7). Although this extension can be expressed in a few words - the pointed groups on a DG-interior algebra A are just the usual pointed groups on the DG-interior algebra C_° (A) - in this section we develop the relationship between pointed groups and induction further than it has been done for (OG-interior algebras, in particular by giving a description of the local structure of induced OG-interior algebras; moreover, since Rickard equivalences between blocks concern the homotopy categories of the blocks (cf. 10.7), we have to introduce the contractile pointed groups which are a specific notion for DG-interior algebras. Actually this extension can be more generally carried out for DH-interior G-algebras where H is a normal subgroup of G (cf. 11.4), except that it requires harder notation and, as a matter of fact, most of the definitions concern only DG-interior algebras; the interested reader will easily fill in this lack of generality.