Self-tilting complexes yield unstable modules

Zimmermann, Alexander

doi:10.1090/S0002-9947-02-02996-3

Self-tilting complexes yield unstable modules
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Trans. Amer. Math. Soc. 354 (2002), 2707-2724 Request permission

Abstract:

Let $G$ be a group and $R$ a commutative ring. Let $TrPic_R(RG)$ 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 $TrPic_R(RG)$ 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.

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