\title{Self-tilting complexes yield unstable modules}
\author{Alexander Zimmermann}
\subjclass%[2000]
{16E30, 20J06, 55S10, 18E30}
\address{Facult\'e de Math\'ematiques et CNRS (LAMFA FRE 2270)\\
Universit\'e de Picardie\\
33 rue St Leu\\ 80039 Amiens Cedex\\ FRANCE}
\urladdr{http://www.mathinfo.u-picardie.fr/alex/azim.html}
\date{August 2001, revised version December 2001}

\sloppy

\begin{document}

\begin{abstract}
Let $G$ be a group and let $R$ be 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 natural 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.
\end{abstract}