Stratifying modular representations of finite groups

Abstract

We classify localising subcategories of the stable module category of a finite group that are closed under tensor product with simple (or, equivalently all) modules. One application is a proof of the telescope conjecture in this context. Others include new proofs of the tensor product theorem and of the classification of thick subcategories of the finitely generated modules which avoid the use of cyclic shifted subgroups. Along the way we establish similar classifications for differential graded modules over graded polynomial rings, and over graded exterior algebras.

Classifying thick subcategories of the stable category of Cohen–Macaulay modules

“…Of course, this is a different notion from ours. On the other hand, a thick subcategory of the category of modules (over a group algebra) in the sense of Benson, Iyengar and Krause [12] is similar to ours.…”

“…(11) The R-modules having finite complexity. (12) The R-modules of lower complete intersection dimension zero.…”

“…Friedlander and Pevtsova [23] extended this classification theorem to finite group schemes. A recent work of Benson, Iyengar and Krause [12] gives a new proof of the theorem of Benson, Carlson and Rickard. A lot of other related results concerning thick subcategories of a triangulated category have been obtained.…”

Derived Witt group formalism

“…The work of Benson, Iyengar and Krause (e.g. [6]) would be one of the stimulus, where they consider subcategories of group algebras. Not much is available in the literature regarding resolving subcategories of abelian categories or that of the category Coh(X) of coherent sheaves over noetherian non-affine schemes X.…”

Telescope conjecture, idempotent ideals, and the transfinite radical

“…The precise statements and explanation of terminology are given below. Recently, a positive solution to the telescope conjecture for stable module categories over finite group algebras was announced by the authors of [4]. Their methods are, however, closely tied to group algebras and do not allow direct generalization to other self-injective artin algebras.…”