Support Varieties and cohomology of Verdier quotients of stable category of complete intersection rings

TL;DR

This paper develops a support variety theory for Verdier quotients of the stable category of maximal Cohen-Macaulay modules over complete intersection rings, extending cohomological results and conjectures to these quotients.

Contribution

It introduces a support variety framework for Verdier quotients of stable categories and applies it to prove cohomological symmetry results and conjectures in this setting.

Findings

Support variety theory extends to Verdier quotients of stable categories.

Cohomological symmetry and conjectures hold in the quotient categories.

Applications include generalizations of Auslander-Reiten and Murthy's results.

Abstract

Let $(A,\mathfrak{m})$ be a complete intersection with $k = A/\mathfrak{m}$ algebraically closed. Let CMS(A) be the stable category of maximal CM $A$-modules. For a large class of thick subcategories $\mathcal{S}$ of CMS(A) we show that there is a theory of support varieties for the Verdier quotient $\mathcal{T} = $ CMS(A)$/\mathcal{S}$. As an application we show that the analogous version of Auslander-Reiten conjecture, Murthys result, Avramov-Buchweitz result on symmetry of vanishing of cohomology holds for $\mathcal{T}$.