Auto-equivalences of derived categories acting on cohomology
Alexander Zimmermann
To appear in Archiv der Mathematik
Abstract. Let $A$ be a $k$-algebra which is projective as a $k$-module,
let $M$ be an $A$-module whose endomorphisms are given by multiplication
by central elements of $A$, and let $\text{\it TrPic}_k(A)$ be the group
of standard self-equivalences of the derived category of bounded complexes
of $A$-modules. Then we define an action of the stabilizer of $M$ in
$\text{\it TrPic}_k(A)$ on the $\text{\it Ext}$-algebra of $M$. In case $M$
is the trivial module for the group algebra $kG=A$, this defines an action
on the cohomology ring of $G$ which extends the well-known action of the
automorphism group of $G$ on the cohomology group.