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.
- D. J. Benson, Representations and cohomology. II, Cambridge Studies in Advanced Mathematics, vol. 31, Cambridge University Press, Cambridge, 1991. Cohomology of groups and modules. MR 1156302
- Charles W. Curtis and Irving Reiner, Methods of representation theory. Vol. I, Pure and Applied Mathematics, John Wiley & Sons, Inc., New York, 1981. With applications to finite groups and orders. MR 632548
- Hans-Werner Henn, Cohomology of groups and unstable modules over the Steenrod algebra, Advanced course on classifying spaces and cohomology of groups, Centre de Recerca Matemàtica Bellaterra (Spain). http://www-irma.u-strasbg.fr/~henn/notes.ps
- Hans-Werner Henn, Jean Lannes, and Lionel Schwartz, Localizations of unstable $A$-modules and equivariant mod $p$ cohomology, Math. Ann. 301 (1995), no. 1, 23–68. MR 1312569, DOI 10.1007/BF01446619
- Steffen König and Alexander Zimmermann, Derived equivalences for group rings, Lecture Notes in Mathematics, vol. 1685, Springer-Verlag, Berlin, 1998. With contributions by Bernhard Keller, Markus Linckelmann, Jeremy Rickard and Raphaël Rouquier. MR 1649837, DOI 10.1007/BFb0096366
- Andrei Marcus, On equivalences between blocks of group algebras: reduction to the simple components, J. Algebra 184 (1996), no. 2, 372–396. MR 1409219, DOI 10.1006/jabr.1996.0265
- Jeremy Rickard, Morita theory for derived categories, J. London Math. Soc. (2) 39 (1989), no. 3, 436–456. MR 1002456, DOI 10.1112/jlms/s2-39.3.436
- Jeremy Rickard, Splendid equivalences: derived categories and permutation modules, Proc. London Math. Soc. (3) 72 (1996), no. 2, 331–358. MR 1367082, DOI 10.1112/plms/s3-72.2.331
- Raphaël Rouquier and Alexander Zimmermann, A Picard group for derived module categories, accepted by Proc. London Math. Soc.
- Lionel Schwartz, Unstable modules over the Steenrod algebra and Sullivan’s fixed point set conjecture, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1994. MR 1282727
- Alexander Zimmermann, Auto-equivalences of derived categories acting on cohomology, to appear in Archiv der Mathematik.
- Alexander Zimmermann, Cohomology of groups and splendid equivalences of derived categories, Math. Proc. Cambridge Phil. Soc. 131 (2001) 459-472.
- 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