\title[Local cohomology and support]{Local cohomology and support for
\\ triangulated categories} 

\thanks{D.B.\ was supported by a Senior Scientist Prize from the
Humboldt Foundation.  S.I.\ was supported by NSF grant, DMS 0602498}

\author{Dave Benson}
\address{Dave Benson \\
Department of Mathematical Sciences\\
University of Aberdeen\\
Meston Building\\
King's College\\
Aberdeen AB24 3UE\\
Scotland U.K.}

\author{Srikanth Iyengar}
\address{Srikanth Iyengar\\ Department of Mathematics\\
University of Nebraska\\ Lincoln NE 68588\\ U.S.A.}

\author{Henning Krause}
\address{Henning Krause\\ Institut f\"ur Mathematik\\
Universit\"at Paderborn\\ 33095 Paderborn\\ Germany.}


\begin{abstract}
We propose a new method for defining a notion of support for objects
in any compactly generated triangulated category admitting small
coproducts.  This approach is based on a construction of local
cohomology functors on triangulated categories, with respect to a
central ring of operators.  Suitably specialized one recovers, for
example, the theory for commutative noetherian rings due to Foxby and
Neeman, the theory of Avramov and Buchweitz for complete intersection
local rings, and varieties for representations of finite groups
according to Benson, Carlson, and Rickard. We give explicit examples
of objects whose triangulated support and cohomological support
differ. In the case of group representations, this allows us to
correct and establish a conjecture of Benson.

\bigskip

\noindent
{\sc R\'esum\'e.}
Nous proposons une fa\c{c}on nouvelle de d\'efinir une notion de
support pour les objets d'une cat\'egorie avec petits coproduit,
engendr\'ee par des objets compacts. Cette approche est bas\'ee sur
une construction des foncteurs de cohomologie locale sur les
cat\'egories triangul\'ees relativement \`a un anneau central
d'op\'erateurs. Comme cas particuliers, on retrouve la th\'eorie pour
les anneaux noeth\'eriens de Foxby et Neeman, la th\'eorie d'Avramov
et Buchweitz pour les anneaux locaux d'intersection compl\`ete, ou les
vari\'et\'es pour les repr\'esentations des groupes finis selon
Benson, Carlson et Rickard. Nous donnons des exemples explicites
d'objets dont le support triangul\'e et le support cohomologique sont
diff\'erents. Dans le cas des repr\'esentations des groupes, ceci nous
permet de corriger et d'\'etablir une conjecture de Benson.
\end{abstract}