Cohomological supports of tensor products of modules over commutative rings

TL;DR

This paper studies cohomological support varieties of modules over commutative local rings, showing that the support of a derived tensor product is the join of individual supports, extending previous results and applying broadly.

Contribution

It generalizes existing results on support varieties by establishing a join property for derived tensor products over Koszul complexes and related structures.

Findings

Support of tensor product equals join of supports

Generalizes results for Tor-independent modules

Applicable to exterior algebras over local rings

Abstract

This works concerns cohomological support varieties of modules over commutative local rings. The main result is that the support of a derived tensor product of a pair of differential graded modules over a Koszul complex is the join of the supports of the modules. This generalizes, and gives another proof of, a result of Dao and the third author dealing with Tor-independent modules over complete intersection rings. The result for Koszul complexes has a broader applicability, including to exterior algebras over local rings.