Alex Zimmermann
Cohomology of groups, abelian Sylow subgroups and splendid equivalences
Algebra Mondpellier Announcements, 2000.

Abstract. Let $G$ be a finite group and let $R$ be a complete discrete
valuation domain of characteristic 0 with residue field $k$ of characteristic 
$p$ and let $S$ be $R$ or $k$. The cohomology rings $H^*(K,S)$ for subgroups
$K$ of $G$ together with restriction to subgroups of $G$, transfer from
subgroups of $G$ and conjugation by elements of $G$ gives $H^*(-,S)$ the
structure of a Mackey functor. Moreover, the group $\text{\it HSplen}_S(K)$ 
of splendid auto-equivalences of the bounded derived category of finitely
generated $SG$-modules fixing the trivial module acts $S$-linearly on 
$H^*(K,S)$. In this note we study the compatibility of these structures and 
get some consequences when $G$ has an abelian Sylow $p$ subgroup. In 
particular we see that in case $G$ has an abelian Sylow $p$ subgroup, then
$\text{\it HSplen}_R(G)$ acts by automorphisms of the Sylow subgroup on
the cohomology.