Tag - Support varieties

Kent Vashaw: A Chinese remainder theorem and Carlson theorem for monoidal triangulated categories

6th June, 2024

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. In this talk, we will discuss a generalization, where it is proved that the Balmer support for an arbitrary monoidal triangulated category satisfies the analogous property. 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.

Dave Benson: The nucleus and the singularity category of cochains on the classifying space

30th May, 2024

The definition of the nucleus was originally formulated in joint work with Carlson and Robinson, to capture the supports of modules with no cohomology. This definition works in various contexts such as finite groups, restricted Lie algebras, and more generally, suitable triangulated categories of modules. In the finite group context it has a characterization in terms of subgroups whose centralizer is not p-nilpotent. In the restricted Lie algebra context, it is described in terms of the Richardson orbit, at least for large primes. Recent work with Greenlees has highlighted a connection with the singularity category of the cochains on the classifying space, in the group theoretic context. My plan is to give an introduction to these ideas.

Vera Serganova: Sylow theorems and support varieties for supergroups

30th May, 2024

We discuss support variety theory for quasireductive algebraic supergroups, i.e. supergroups with reductive even part over complex numbers. The corresponding categories of representations are Frobenius and share many properties of representations of finite groups in positive characteristic. It is desirable to describe Balmer spectrum of related triangulated symmetric monoidal categories. Our approach involves so called homological odd elements and certain tensor functors associated to them. On the way we encounter analogues of p-groups and Sylow subgroups for supergroups. We prove projectivity detection for our support theory and present other related results. We also explore connections with homological support theory developed by B. Boe, J. Kujawa and D. Nakano.

Kent Vashaw: A Chinese remainder theorem and Carlson’s theorem for monoidal triangulated categories

27th May, 2024

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.

Milen Yakimov: Non-commutative tensor triangular geometry and finite tensor categories

27th May, 2024

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.

Eloísa Grifo: Searching for modules that are not virtually small

2nd May, 2024

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.

Julia Plavnik: Remarks on the tensor product property for support varieties for finite tensor categories

22nd April, 2024

For non-semisimple tensor categories satisfying some finiteness conditions, support varieties are meaningful geometric invariants of objects. Their theory began in the work of Quillen and Carlson on finite group representations. In more recent years, the theory of support varieties was generalized in many directions, including representations of finite-dimensional Hopf algebras and self-injective algebras, and objects in finite tensor categories and triangulated categories, among others.

In this talk, we will start by introducing the definition of support varieties for finite tensor categories and some of their basic properties. We will also present some conditions under which the tensor product property holds for support varieties, and we will present some applications to certain Hopf algebras. We will also discuss a construction of non-semisimple finite tensor categories with finitely generated cohomology for which the tensor product property does not hold for support varieties.

Vera Serganova: Invariant integration on homogeneous superspaces

1st September, 2023

We use Berezin integral in the category of CS-manifolds to construct an invariant integral for the ring of regular functions on a homogeneous affine supervariety G/K. This construction has several applications in representation theory of G. We will explain how it is used in the proof of projectivity detection for support varieties and for description of stable categories for defect 1 supergroups. We also see how this integral can be used to generalize some classical statements from modular representation theory of finite groups to supergroups in characteristic zero.

Milen Yakimov: On the spectrum and support theory of a finite tensor category

4th April, 2023

Finite tensor categories are important generalizations of the categories of finite-dimensional modules of finite-dimensional Hopf algebras. There are two support theories for them, the cohomological one and one based on the noncommutative Balmer spectrum of the corresponding stable module category. We will describe general results linking the two types of support via a new notion of categorical center of the cohomology ring of a finite tensor category and will state a conjecture giving the exact relation. The construction and results will be illustrated with various examples.

Henning Krause: Central support for triangulated categories

7th February, 2023

Various notions of support have been studied in representation theory (by Carlson, Snashall-Solberg, Balmer, Benson-Iyengar-Krause, Friedlander-Pevtsova, Nakano-Vashaw-Yakimov, to name only few). My talk offers some new and unifying perspective: For any essentially small triangulated category the centre of its lattice of thick subcategories is introduced; it is a spatial frame and yields a notion of central support. A relative version of this centre recovers the support theory for tensor triangulated categories and provides a universal notion of cohomological support. Along the way we establish Mayer-Vietoris sequences for pairs of central subcategories.