Stratifying triangulated categories
By D. Benson, S. Iyengar, Henning Krause. 2009
122 citations recorded by the source corpus.
A notion of stratification is introduced for any compactly generated triangulated category ⊤ endowed with an action of a graded‐commutative noetherian ring R. The utility of this notion is demonstrated by establishing diverse consequences that follow if ⊤ is stratified by R. Among them are a classification of the localizing subcategories of ⊤ in terms of subsets of the set of prime ideals in R; a classification of the thick subcategories of the subcategory of compact objects in ⊤; and results on the support of the graded R‐module of morphisms Hom*⊤(C, D) leading to analogs of the tensor product theorem for support varieties of modular representation of groups.
Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/7e48bd9fda53e31c0d646b4e9d684ed7e8b2b766
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Citations in the corpus
- Stratification for module categories of finite group schemes (2015, methodology): “This phenomenon was first discovered by Avramov and Buchweitz [1] when R is a (commutative) complete intersection ring, and is now understood to be related to the classification of localizing subcategories [7].”
- Classification of resolving subcategories and grade consistent functions (2012, result): “owing proposition, we can reduce problems on resolving subcategories to the case where the base ring is local. (It would be interesting to compare this result with the local-global principle given in [9, 11, 27].) Proposition 3.3. Let Xbe a resolving subcategory of modR. The following are equivalent”
- Local duality in algebra and topology (2015, background): “ts complement. The proof of it depends on a result of Benson, Iyengar, and Krause proved in the context of symmetric monoidal triangulated categories stratified by the action of a Noetherian ring (see [BIK11] for a definition of this concept). A natural example of a category of this type is the derive”
- Subcategories of singularity categories via tensor actions (2011, background): “escope conjecture. 1 2 GREG STEVENSON interesting work, namely the tensor triangular geometry of Balmer [5], [6], and BalmerFavi [7], and the stratifications of Benson, Iyengar, and Krause [10], [12], [11], providing conceptual frameworks uniting these classification theorems. The purpose of this pape”
- Colocalizing subcategories and cosupport (2010, background): “phic to the identity on ImΓ. 3. Localcohomologyand support In this section we recall the construction, and basic properties, of local cohomology functors and support for triangulated categories, from [4, 6]. Compact generation. An object C in a triangulated category Tadmitting setindexed coproducts ”