Quillen stratification for the stable module category

(Revised 9 June 1998)

Author:  Wayne W. Wheeler

Abstract:  The Quillen Stratification Theorem gives a decomposition of the
prime ideal spectrum of the cohomology ring of a finite group $G$ into a
disjoint union of locally closed subsets.  If the coefficient field $k$ has
characteristic $p$, then each of these subsets corresponds to a unique
conjugacy class of elementary abelian $p$-subgroups.  The purpose of this paper
is to prove an analogous result for the stable category of $kG$-modules modulo
projectives.  The categorical equivalent of a stratification is the concept of
a recollement of categories, and we show that one can use this concept to
decompose the stable category into certain quotient categories $\Cal C_{G,E}^+$
corresponding to conjugacy classes of elementary abelian $p$-subgroups $E$ of
$G$.  Moreover, the categories $\Cal C_{G,E}^+$ are determined by the local
structure of the group $G$.