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Transactions of the American Mathematical Society

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ISSN 1088-6850 (online) ISSN 0002-9947 (print)

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Self-tilting complexes yield unstable modules
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by Alexander Zimmermann PDF
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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Additional Information
  • Alexander Zimmermann
  • Affiliation: Faculté de Mathématiques et CNRS (LAMFA FRE 2270), Université de Picardie, 33 rue St Leu, 80039 Amiens Cedex, France
  • MR Author ID: 326742
  • Email: Alexander.Zimmermann@u-picardie.fr
  • Received by editor(s): August 28, 2001
  • Published electronically: February 25, 2002
  • © Copyright 2002 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 354 (2002), 2707-2724
  • MSC (2000): Primary 16E30, 20J06, 55S10, 18E30
  • DOI: https://doi.org/10.1090/S0002-9947-02-02996-3
  • MathSciNet review: 1895199