Support and Rank Varieties of Totally Acyclic Complexes

TL;DR

This paper extends the equivalence of support and rank varieties from modules over group algebras and complete intersections to the broader setting of totally acyclic complexes over complete intersection rings, revealing a fundamental geometric relationship.

Contribution

It establishes that support and rank varieties are equivalent in the triangulated category of totally acyclic complexes over complete intersection rings, generalizing previous results.

Findings

Support and rank varieties are equivalent for totally acyclic complexes.

The result generalizes known equivalences from modules to complexes.

Provides a geometric perspective on acyclic complexes over complete intersections.

Abstract

Support and rank varieties of modules over a group algebra of an elementary abelian p-group have been well studied. In particular, Avrunin and Scott showed that in this setting, the rank and support varieties are equivalent. Avramov and Buchweitz proved an analogous result for pairs of modules over arbitrary commutative local complete intersection rings. In this paper we study support and rank varieties in the triangulated category of totally acyclic chain complexes over a complete intersection ring and show that these varieties are also equivalent.