Tag - Support varieties

Petter Bergh: Support varieties for finite tensor categories

28th December, 2022

This is a report on recent and ongoing joint work with Julia Plavnik and Sarah Witherspoon, where we have developed a theory of cohomological support varieties for finite tensor categories. Under suitable finite generation conditions - conjectured to hold for all finite tensor categories - the varieties encode homological properties of the objects, as in the classical case for group algebras. For example, the dimension of a variety equals the complexity of the corresponding object, so that the objects having trivial support varieties are precisely the projective ones. Moreover, every potential variety is actually the support variety of some object, and the support variety of an indecomposable object is connected. I will also discuss the so-called tensor product property for varieties.

Jonathan Kujawa: Support varieties for Lie superalgebras  

26th July, 2022

Support varieties are a method which uses cohomology to bring commutative algebra and algebraic geometry to places where it doesn't obviously belong. Lie superalgebras are a graded analogue of Lie algebras and they seem well adapted this technology. In this talk, I will give an overview of recent efforts to use support varieties to study the representation theory of Lie superalgebras.

Jon Carlson: Idempotent modules and endomorphisms

14th February, 2022

We work in the group algebra kG of a finite group scheme defined over a field of characteristic p>0. The stable category stmod(kG) of finitely generated kG-modules is a tensor triangulated category of compact objects. Associated to any thick tensor ideal subcategory C is a distinguished triangle

→ E → k → F →

where E and F are idempotent modules in stable category StMod(kG) of all kG-modules. Tensoring with F is the localization functor associated to C. Indeed, the endomorphism ring of F is the endomorphism ring of the trivial module in the localized category. In this lecture we will discuss some results on the nature of the endomorphism ring of the module F and some strange variant support varieties for modules.

Jon Carlson: Idempotent modules and endomorphisms

11th January, 2022

We work in the group algebra kG of a finite group scheme defined over a field of characteristic p > 0. The stable category stmod(kG) of finitely generated kG-modules is tensor triangulated category of compact objects. Associated to any thick tensor ideal subcategory 𝒞 is a distinguished triangle  → E → k → F → where E and F are idempotent modules in the stable category StMod(kG) of all kG-modules. Tensoring with F is the localization functor associated to 𝒞. Indeed, the endomorphism ring of F is the endomorphism ring of the trivial module in the localized category. In this lecture, we will discuss some results on the nature of the endomorphism ring of the module F and some strange variant support varieties for modules.

Milen Yakimov: Non-commutative Tensor Triangular Geometry and Support Varieties for Hopf Algebras

16th October, 2021

We will describe a theory of noncommutative tensor triangular geometry for monoidal triangulated categories. It is aimed at investigating support varieties for finite dimensional Hopf algebras via non-commutative Balmer spectra. We will state effective reconstruction theorems for these spectra and an intrinsic characterization of those categories whose support variety maps satisfy the tensor product property. As an application, we obtain a treatment of the Benson-Witherspoon Hopf algebras, which previously eluded approaches of this kind, and a proof of a recent conjecture of Negron and Pevtsova that the cohomological support maps of the Borel subalgebras of all Lusztig small quantum groups possess the tensor product property. This is joint work with Daniel Nakano (University of Georgia) and Kent Vashaw (MIT).

Daniel Nakano: Non-commutative Tensor Triangular Geometry and Applications

27th September, 2021

In this talk, I will show how to develop a general non-commutative version of Balmer's tensor triangular geometry that is applicable to arbitrary monoidal triangulated categories (MΔC). Insights from non-commutative ring theory are used to obtain a framework for prime, semiprime, and completely prime (thick) ideals of an MΔC, K, and then to associate to K a topological space: the Balmer spectrum Spc(K). We develop a general framework for (noncommutative) support data, coming in three different flavors, and show that Spc(K) is a universal terminal object for the first two notions (support and weak support). The first two types of support data are then used in a theorem that gives a method for the explicit classification of the thick (two-sided) ideals and the Balmer spectrum of an MΔC. The third type (quasi support) is used in another theorem that provides a method for the explicit classification of the thick right ideals of K, which in turn can be applied to classify the thick two-sided ideals and Spc(K). Applications will be given for quantum groups and non-cocommutative finite-dimensional Hopf algebras studied by Benson and Witherspoon.

Julia Plavnik: Tensor product and support varieties for finite tensor categories

3rd May, 2021

In this talk, we will consider the support variety theory for Hopf algebras and finite tensor categories. We will start by presenting the basic definitions and properties of these algebraic structures (Hopf algebras and tensor categories) and then we will introduce the theory of support varieties for them. We will analyze questions about projectivity and tensor products by using support theory and we will illustrate this via some examples. If time allows, we will discuss some deeper results involving complexity, realization, and connectedness of the varieties.

Brian Boe: Complexity and Support Varieties for Type P Lie Superalgebras

5th April, 2021

We compute the complexity, z-complexity, and support varieties of the (thick) Kac modules for the Lie superalgebras of type P. We also show the complexity and the z-complexity have geometric interpretations in terms of support and associated varieties; these results are in agreement with formulas previously discovered for other classes of Lie superalgebras. Our main technical tool is a recursive algorithm for constructing projective resolutions for the Kac modules. The indecomposable projective summands which appear in a given degree of the resolution are explicitly described using the combinatorics of weight diagrams. Surprisingly, the number of indecomposable summands in each degree can be computed exactly: we give an explicit formula for the corresponding generating function. I wrote an iOS app to implement the combinatorics quickly and graphically, and I’ll be demoing live some of the interesting features of these resolutions.

Julia Pevtsova: Support theories for the stable module category of a finite group scheme

10th August, 2016

We'll study the global structure of the stable module category StMod G or, equivalently, the category of singularities of representations of a finite group scheme G over a field of positive characteristic p. The goal of the lectures will be to classify the tensor ideal localizing subcategories in StMod G. The techniques involved in the classification include the theories of support and cosupport in modular representation theory, detection of projectivty for modules, Benson-Iyengar-Krause theory of local cohomology functors, and new methods inspired by commutative algebra which allow to relate local cohomology at closed and arbitrary points. This is based on joint work with Eric Friedlander and Dave Benson, Srikanth Iyengar and Henning Krause.