Mathematics Colloquium: Cohomology and representation theory of finite-dimensional algebras

Speaker: Nicole Snashall (University of Leicester and The Open University)

Title: Cohomology and representation theory of finite-dimensional algebras

Microsoft Teams Meeting

Abstract:

In this talk I will give an overview of some of my research on the cohomology of finite-dimensional algebras. The aim is to discuss the ideas without too many technical details!

We start by using quivers (oriented graphs) to describe finite-dimensional algebras, and then introduce Hochschild cohomology and the Ext algebra.

The main part of the talk focuses around the question of finite generation: when is the Ext algebra a finitely generated algebra? and when is the Hochschild cohomology ring a finitely generated algebra? The group algebra for a finite group is an important motivating example, and it is well-known that both of these questions have an affirmative answer in this case.

In the final section we introduce support varieties and the maximal ideal spectrum of a commutative ring. Support varieties have been introduced in several contexts including for finite groups. Motivated by this, we developed a theory of support varieties for finite-dimensional algebras, and, more recently, provided a common framework in which most of the existing occurrences of support varieties fit.

This talk is based on several research papers with my co-authors.