\title
Stable endomorphism rings of idempotent E-modules \endtitle
\author Peteris Daugulis \endauthor
\affil Department of Mathematics,
University of Georgia, Athens, Georgia
30603 \endaffil
\date February 20,1998 \enddate
\abstract
Let $G$ be a finite group, $k$ an algebraically closed field of characteristic
dividing $|G|$, $C$ a thick subcategory of the stable category of finitely
generated $kG$-modules corresponding to a collection of closed homogeneous
subvarieties of the maximal ideal spectrum of $H^*(G,k)$ which is closed under
finite unions and specialization, ${\Cal E}_C$ the idempotent module
corresponding to $C$. We show that the stable endomorphism ring of ${\Cal
E}_C$ is a local ring if $C$ corresponds to a connected variety or an infinite
collection of varieties which is connected in an appropriate sense. We show
that noninvertible stable endomorphisms of ${\Cal E}_C$ are locally nilpotent
if $C$ corresponds to a connected variety given by a sequence of homogeneous
parameters. We relate the structure of the stable endomorphism ring of ${\Cal
E}_C$ to those of the finitely generated modules of the colimit and give a few
examples for elementary abelian groups of low rank.
\endabstract
\keywords stable module categories, idempotent modules \endkeywords