## 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.## References

- 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.

## 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