\title{Representations of finite groups:\\ Local cohomology and support}
\author{Dave Benson, Srikanth B. Iyengar, Henning Krause}
These are the notes from an Oberwolfach Seminar which we ran from 23--29
May 2010. There were 24 graduate student and postdoctoral participants.
Each morning consisted of three lectures, one from each of the organisers.
The afternoons consisted of problem sessions, apart from Wednesday which
was reserved for the traditional hike to St.~Roman. We have tried to be
reasonably faithful to the lectures and problem sessions in these notes,
and have added only a small amount of new material for clarification.
The seminar focused on recent developments in classification methods in
commutative algebra, group representation theory and algebraic topology.
These methods were initiated by Hopkins back in 1987 \cite{Hopkins:1987a},
with the classification of the thick subcategories of the derived category
of bounded complexes of finitely generated projective modules over a
commutative noetherian ring $R$, in terms of specialisation closed subsets
of $\Spec R$. Neeman \cite{Neeman:1992a} (1992) clarified Hopkins' theorem
and used analogous methods to classify the localising subcategories of the
derived category of unbounded complexes of modules in terms of arbitrary
subsets of $\Spec R$. In 1997, Benson, Carlson and Rickard
\cite{Benson/Carlson/Rickard:1997a} proved the thick subcategory theorem
for modular representation theory of a finite $p$-group $G$ over an
algebraically closed field $k$ of characteristic $p$. Namely, the thick
subcategories of the stable category of finitely generated $kG$-modules
are classified by the specialisation closed subsets of the homogeneous
non-maximal prime ideals in $H^*(G,k)$, the cohomology ring. The
corresponding theorem for the localising subcategories of the stable
category of all $kG$-modules has only recently been achieved, in the paper
\cite{Benson/Iyengar/Krause:bik3} by the three organisers of the seminar.