Self-tilting complexes yield unstable modules

Abstract

Abstract. Let G be a group and R a commutative ring. Let T rP ic R (RG) be the group of isomorphism classes of standard self-equivalences of the derived category of bounded complexes of RG-modules. The subgroup HD R (G) of T rP ic R (RG) consisting of self-equivalences fixing the trivial RG-module acts on the cohomology ring H * (G, R). The action is functorial with respect to R. The self-equivalences which are 'splendid' in a sense defined by J. Rickard act naturally with respect to transfer and restriction to centralizers of p-subgroups in case R is a field of characteristic p. In the present paper we prove that this action of self-equivalences on H * (G, R) commutes with the action of the Steenrod algebra, and study the behaviour of the action of splendid self-equivalences with respect to Lannes' T -functor.

Blockwise relations between triples, and derived equivalences for wreath products

“…This in turn, induces a Morita equivalence between [11,Theorem 5.1.21]. However, this equivalence need not be basic (see also [19,Remark 3.4]), so we cannot apply the results of [12] to deduce its compatibility with the Brauer map Br Q n .…”

“…Theorem 3.7 below improves [11,Theorem 5.2.12] in several ways, by taking into account all the additional structure that we deal with. As already noted by Zimmermann [19], a certain "p ′ -condition" on the order of the grading groups, which appears in [11,Theorem 5.2.12], is actually not needed in the case of derived equivalences, but is needed in the case of Rickard equivalences. Finally, Theorem 5.8 and Corollary 5.9 are the main results of this paper, and establish the compatibility of the relation ≥ b between module triples with wreath products of derived equivalences.…”