Stratifying modular representations of finite groups

By D. Benson, S. Iyengar, Henning Krause. 2008

180 citations recorded by the source corpus.

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.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/7a1f959441ea97a574c20ac1694ce6dff29c840b

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In Algebra and Number Theory

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Citations in the corpus

Stratification for module categories of finite group schemes (2015, methodology): “An analogous statement for group schemes arising from finite groups is proved in [8].”
Colocalizing subcategories and cosupport (2010, methodology): “We write VG for the set of all homogeneous prime ideals except the maximal ideal in the cohomology algebra H ðG; kÞ of G, and VGðMÞ for the support of any M A StModðkGÞ, defined by Benson, Carlson and Rickard [3] when k is algebraically closed, and extended in [6] to all fields.”
Classifying thick subcategories of the stable category of Cohen-Macaulay modules (2009, background): “A recent work of Benson, Iyengar and Krause [12] gives a new proof of the theorem of Benson, Carlson and Rickard.”
A tour about existence and uniqueness of dg enhancements and lifts (2016, background): “To motivate our interest in K(Inj(G)), observe that such a category played an import role in [5], which deals with modular representations of finite groups.”
Classification of resolving subcategories and grade consistent functions (2012, methodology): “The Hopkins-Neeman Theorem was extended to schemes by Thomason [25] and very recently to group algebras by Benson, Iyengar and Krause [6].”
Stratifying triangulated categories (2009, methodology): “In [8], we established an analogous classification theorem for the localizing subcategories of the stable module category of all representations of G.”
Deriving Auslander's formula (2014, background): “The homotopy category K(InjA) played an import role in work with Benson and Iyengar on modular representations of finite groups [5].”