Cohomological supports over derived complete intersections and local rings

TL;DR

This paper develops a cohomological support theory for DG modules over Koszul complexes, generalizing known support varieties and providing new proofs and applications in the derived category of local rings.

Contribution

It introduces a unified support framework for DG modules over Koszul complexes, extending support varieties to broader contexts and offering new insights into derived categories.

Findings

Supports encode (co)homological information about DG modules

Provides new proofs of classical results for complete intersections

Answers a question of Jorgensen negatively

Abstract

A theory of cohomological support for pairs of DG modules over a Koszul complex is investigated. These specialize to the support varieties of Avramov and Buchweitz defined over a complete intersection ring, as well as support varieties over an exterior algebra. The main objects of study are certain DG modules over a polynomial ring; these determine the aforementioned cohomological supports and are shown to encode (co)homological information about pairs of DG modules over a Koszul complex. The perspective in this article leads to new proofs of well-known results for pairs of complexes over a complete intersection. Furthermore, these cohomological supports are used to define a support theory for pairs of objects in the derived category of an arbitrary commutative noetherian local ring. Finally, we calculate several examples and provide an application by answering a question of D. Jorgensen in the negative.