Self-tilting complexes yield unstable modules

Abstract

Let G be a group and let R be a commutative ring. Let T rP icR(RG) be the group of isomorphism classes of standard self-equivalences of the derived category of bounded complexes of RG-modules. The subgroup HDR(G) of T rP icR(RG) consisting of self-equivalences xing 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 dened by J. Rickard act natural with respect to transfer and restriction to centralizers of p-subgroups in case R is a eld 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. Introduction In an earlier joint paper (9) with RaphaRouquier I dened the group T rP icR(A) of standard self-equivalences of a derived module category D b (A) for an R-algebra A which is projective as an R-module. For any A-module M let HDM(A) be the subgroup of T rP icR(A) which is formed by the self-equivalences mapping M to an isomorphic copy. Then, in an earlier paper (11) I showed that, under some hypothesis on M, the group HDM(A) acts in a natural way on the Ext-algebra Ext A (M; M). In case of A being a group algebra RG, with R being a eld of characteristic p and G being a nite group, then J. Rickard denes in (8) what is a splendid equivalence by some technical conditions basically by asking that the homogeneous components of a tilting complex are p-permutation modules induced from diagonal p-subgroups, and by some invertibility condition in the homotopy category. These splendid equivalences induce self-equivalences of the derived categories of centralizers of p-subgroups by the Brauer construction. In (12) I showed that then, for M = R being the trivial module, the action of those splendid equivalences commute with transfer and restriction from and to the cohomology rings of centralizers of p-subgroups. In the present paper we enlarge these properties still further. The action of self-equivalences of the bounded derived category D b (RG) on H (G; R) commutes with the action of the Steenrod algebra on H (G; R) for any prime p. As consequence of the above statements, any cohomology ring H (G; Fp) denes a functor H (G; Fp) p HD p (B0( p G)) from the modules over the group of derived self-equivalences of the principal block of the group ring Fp HD p (FpG) xing the trivial module to the category of unstable modules Up and similarly in the opposite direction. Obviously, we may restrict to splendid self-equivalences. We shall describe the composition of Lannes' T -functor with the rst functor and the im- age of free unstable modules by the second. Moreover, by a result of Lannes TV (H (G; Fp)) decomposes into direct product of cohomology rings as unstable modules. We shall prove that this decomposition is also a decomposition as modules over the action of splendid self- equivalences. This will give evidence that splendid equivalences are the correct objects to study in this context. The paper is organized as follows. Section 1 recalls the necessary denitions and properties of Even's norm map as it is used here and the denition of the Steenrod operation. In Section 2 it is shown that the normalized part of the outer automorphism group of the