Local cohomology and support for triangulated categories

By D. Benson, S. Iyengar, Henning Krause. 2007

248 citations recorded by the source corpus.

We propose a new method for defining a notion of support for objects in any compactly generated triangulated category admitting small co- products. This approach is based on a construction of local cohomology func- tors on triangulated categories, with respect to a central ring of operators. Suitably specialized one recovers, for example, the theory for commutative noetherian rings due to Foxby and Neeman, the theory of Avramov and Buch- weitz for complete intersection local rings, and varieties for representations of finite groups according to Benson, Carlson, and Rickard. We give explicit examples of objects whose triangulated support and cohomological support dif- fer. In the case of group representations, this allows us to correct and establish a conjecture of Benson.

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

View Paper: https://www.semanticscholar.org/paper/154b9eabb9e3508ec31c32fa3f2ee646abe72176

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Citations in the corpus

Representations of Algebras and Related Topics (2011, background): “(StMod(kG)) = VG r{0}, suppH∗(G,k)(KInj(kG)) = VG. The following theorem summarises the properties of support for R-linear triangulated categories; confer Theorems 2.2 and 4.3. Proofs can be found in [8]. Theorem 8.6. Let X, Y and Z be objects in T. Then (i) X = 0 if and only if suppR(X) = ∅. (ii) s”
Colocalizing subcategories and cosupport (2010, methodology): “It involves a sequence of changes of category, for which reason it has been necessary to develop a theory of cosupport for objects in triangulated categories, along the lines for the one for support in our earlier work [4, 5, 6].”
Subcategories of singularity categories via tensor actions (2011, background): “In [Kra05] Krause identifies Qρ with taking K-injective resolutions, where the K-injectives are the objects of the colocalizing subcategory of K(InjR) generated by the bounded below complexes of injectives (such resolutions exist; see, for example, [BN93]).”
Constructing modules with prescribed cohomological support (2007, methodology): “ theories with desirable properties under a small set of conditions on a ring, its module(s), and a ring of central cohomological operators. The properties that have to be established are clariﬁed in [9] by Benson, Iyengar, and Krause, who investigate a notion of support for triangulated categories ”
Classifying thick subcategories of the stable category of Cohen-Macaulay modules (2009, unknown): “For example, see [7, 8, 9, 43, 15, 11, 38, 13, 20, 53].”
Triangulated Categories: Localization theory for triangulated categories (2008, unknown): “ﬁeld, Ravenel, Thomason-Trobaugh, Yao, and others. The cohomological localization functors commuting with localization functors of graded modules have been used to set up local cohomology functors in [7]. 6. Well generated triangulated categories 6.1. Regular cardinals. A cardinal αis called regular”
Stratifying triangulated categories (2009, methodology): “Local cohomology and support The foundation for this article is the work in [7] where we constructed analogs of local cohomology functors and support from commutative algebra for triangulated categories.”
Stratification for module categories of finite group schemes (2015, methodology): “This section provides a quick summary of the techniques developed in [6,9], with a focus on modules over finite group schemes.”
Singular support of coherent sheaves and the geometric Langlands conjecture (2012, methodology): “This construction is a variation of [BIK].”
Generalized tensor idempotents and the telescope conjecture (2011, background): “D. J. Benson, S. B. Iyengar and H. Krause, ‘Local cohomology and support for triangulated categories’, Ann.”
Spectra, spectra, spectra – Tensor triangular spectra versus Zariski spectra of endomorphism rings (2010, background): “The following result is new although known to some experts (announced for instance in talks by Julia Pevtsova, joint work with Paul Smith; see also Benson et al. [7]) : , the quotient K/J is equivalent to the stable category kG – stab, whose spectrum is described in [1, Cor. 5.10] as ϕ : Proj(H • (G”