Rickard equivalences between Brauer blocks

Abstract

This section is similar to Section 6 in many aspects and we make no effort to avoid redundancy in order to emphasize it. Let G and G’ be finite groups, b and b’ blocks of G and G’ respectively, and M ^.. an indecomposable D(G × G’)-module which has projective restrictions to O(G × 1) and to O(1 × G’), and is associated with b ⊗ (b’)° (recall that (b’)° is the image of b’ by the antipodal isomorphism OG’ ≅ (OG’)°). Respectively denote by Mod _DGb and Mod _DG’b’ the categories of (O-finite as usual) O-free DGb-and DG’b’-modules, and as in 10.7, by Mod _DGb to Mod _DG’b’ the homotopy categories of A = OGb and A’ = OG’ b’ it is clear that M ^.. determines a functor from Mod _DG’b’ to Mod _DGb , namely the functor

$${F_{{M^{..}}}}:Mo{d_{D{G^,}{b^,}}} \to Mo{d_{DGb}}$$

(18.1.1)

mapping any O-free DG’b’-module M’ on M ^.. ⊗_OG’ M’ (cf. 10.2 provided we consider M’ as a.D(G × G’)-module with the trivial action of G and identify G’ to the normal subgroup 1 × G’ of G × G’), which is O-free too since M ^.. is a projective 0(1 × G’)-module, and any DG’b’-module homomorphism f’ : M’ → N’ where N’ is a second O-free DG’b’-module, on the DGb-module homomorphism

$$i{d_{{M^{..}}}}_{\vartheta {G^,}}^ \otimes {f^,}:{M^{..}}_{\vartheta {G^,}}^ \otimes {M^,} \to {M^{..}}_{\vartheta {G^,}}^ \otimes {N^,}$$

(18.1.2)