Cohomological jump loci and duality in local algebra

TL;DR

This paper introduces cohomological jump loci as a higher order support theory for dg modules over local dg algebras, generalizing support varieties and revealing duality invariance of homological invariants.

Contribution

It develops a new support theory for dg modules over various local algebra contexts, extending the concept of support varieties and establishing duality properties.

Findings

Cohomological jump loci are closed under Grothendieck duality.

Homological invariants like Betti degree are preserved under duality.

The theory applies to complete intersection rings, exterior algebras, and certain group algebras.

Abstract

In this article a higher order support theory, called the cohomological jump loci, is introduced and studied for dg modules over a Koszul extension of a local dg algebra. The generality of this setting applies to dg modules over local complete intersection rings, exterior algebras and certain group algebras in prime characteristic. This family of varieties generalizes the well-studied support varieties in each of these contexts. We show that cohomological jump loci satisfy several interesting properties, including being closed under (Grothendieck) duality. The main application of this support theory is that over a local ring the homological invariants of Betti degree and complexity are preserved under duality for finitely generated modules having finite complete intersection dimension.