On the depth and reflexivity of tensor products

TL;DR

This paper investigates the depth properties of tensor products of complexes over local rings and explores conditions for reflexivity of tensor products of modules, especially over hypersurface rings.

Contribution

It introduces new depth criteria for tensor products of complexes and characterizes when tensor products of modules are reflexive over hypersurface rings.

Findings

Identifies conditions for reflexivity of tensor products of modules.

Establishes depth bounds for tensor products of complexes.

Analyzes reflexivity of tensor products of prime ideals.

Abstract

In this paper we study the depth of tensor products of homologically finite complexes over commutative Noetherian local rings. As an application of our main result, we determine new conditions under which nonzero tensor products of finitely generated modules over hypersurface rings can be reflexive only if both of their factors are reflexive. A result of Asgharzadeh shows that nonzero symbolic powers of prime ideals in a local ring cannot have finite projective dimension, unless the ring in question is a domain. We make use of this fact in the appendix and consider the reflexivity of tensor products of prime ideals over hypersurface rings.