I’ll define what it means for a tensor triangulated category to be locally regular, and discuss this condition and its implications for the stable category of a finite group.
Tag - Category theory
Carlson’s Connectedness Theorem for cohomological support varieties is a fundamental result which states that the support variety for an indecomposable module of a finite group is connected. For monoidal triangulated categories, the Balmer spectrum is an intrinsic geometric space associated to the category which generalizes the notion of cohomological support for finite groups. In this talk, we will discuss a generalization of the Carlson Connectedness Theorem: that the Balmer support of any indecomposable object in a monoidal triangulated category with a thick generator is a connected subset of the Balmer spectrum. This is shown by proving a version of the Chinese remainder theorem in this context, that is, giving a decomposition for a Verdier quotient of a monoidal triangulated category by an intersection of coprime thick tensor ideals.
Describing the thick ideals of a monoidal triangulated category is a key component of the analysis of the category. We will show how this can be done by non-commutative tensor triangular geometry (NTTG), thus extending the celebrated Balmer’s theorem from the symmetric case. We will then use NTTG to analyse the stable categories of finite tensor categories, which play an important role in representation theory, mathematical physics and quantum computing. We will present general results linking this approach to the traditional one through cohomological support, based on a notion of categorical centers of cohomology rings of monoidal triangulated categories.
In the 90s, Nakajima and Grojnowski identified the total cohomology of the Hilbert schemes of points on a smooth projective surface with the Fock space representation of the Heisenberg algebra associated to its cohomology lattice. Later, Krug lifted this to derived categories and generalized it to the symmetric quotient stacks of any smooth projective variety.
On the other hand, Khovanov introduced a categorification of the free boson Heisenberg algebra, i.e., the one associated to the rank 1 lattice. It is a monoidal category whose morphisms are described by a certain planar diagram calculus which categorifies the Heisenberg relations. A similar categorification was constructed by Cautis and Licata for the Heisenberg algebras of ADE type root lattices.
We show how to associate the Heisenberg 2-category to any smooth and proper DG category and then define its Fock space 2-representation. This construction unifies all the results above and extends them to what can be viewed as the generality of arbitrary non-commutative smooth and proper schemes.
In this talk, we show that the homotopy category of (small) dg categories and the homotopy category of A∞-categories are equivalent (even from a higher categorical viewpoint). We will discuss several issues related to the various notions of unity and provide several applications. The main ones are about the uniqueness of enhancements for triangulated categories and a full proof of a claim by Kontsevich and Keller concerning a description of the category of internal Homs for dg categories.
Pollitz gave a characterization of complete intersection rings in terms of the triangulated structure of their derived category, akin to the Auslander-Buchsbaum-Serre characterization of regular rings. In this talk, we will explore how to bring this characterization back to the world of modules, and discuss the role of cohomological support varieties in solving this problem.
The goal of this talk will be to present the results from my recent joint work with Vova Sosnilo and Christoph Winges, where we prove that every spectrum is the (non-connective) K-theory spectrum of a stable category. Our main application of this is the disproof of a conjecture by Antieau-Gepner-Heller about a non-connective version of the theorem of the heart in the non-noetherian setting; but I will also try to mention other perspectives on this result.
We discuss reflectivity of colocalizing subcategories of triangulated categories under suitable set-theoretical assumptions. In earlier joint work with Gutierrez and Rosicky, we proved that if K is any locally presentable category with a stable model category structure, then Vopenka's principle implies that every full subcategory L of the homotopy category of K closed under products and fibres is reflective. Moreover, if L is colocalizing, then the reflection is exact. Using recent progress in large-cardinal theory, we show that the statement that every full subcategory closed under products and fibres is reflective is, in fact, equivalent to the so-called weak Vopenka principle. Hence this statement cannot be proved using only the ZFC axioms.
We consider the (infinity) category of excisive (aka polynomial) functors from Spectra to Spectra. Understanding this category is a basic problem in functor calculus. We will approach it from the perspective of tensor triangular geometry. Day convolution equips the category of excisive functors with the structure of a rigid monoidal triangulated category. We describe completely the Balmer spectrum of this category, i.e., its spectrum of prime tensor ideals. This leads to a Thick Subcategory Theorem for excisive functors. A key ingredient in the proof is a blueshift theorem for the generalized Tate construction associated with the family of non-transitive subgroups of products of symmetric groups. If there is time, I will say something about work in progress to extend these results to more general functor categories.
Conilpotent Koszul duality, as formulated by Positselski and Lefevre-Hasegawa, gives an equivalence (of model categories, or of ∞-categories) between augmented dg algebras and conilpotent dg-coalgebras. One should think of this as a non-commutative version of the Lurie-Pridham correspondence: indeed in characteristic zero, cocommutative conilpotent dg coalgebras are Koszul dual to dg Lie algebras, and this is precisely the correspondence between formal moduli problems and their tangent complexes. I'll talk about a global analogue where the conilpotency assumption is removed; geometrically this corresponds to non-commutative formal moduli problems modelled on profinite completions, rather than pro-Artinian completions. Global Koszul duality is best expressed as a Quillen equivalence between curved dg algebras and curved dg coalgebras, and in both categories the weak equivalences are defined using an auxiliary object, the Maurer-Cartan dg category of a curved dg algebra.
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