On the depth of tensor products over Cohen-Macaulay rings

TL;DR

This paper introduces new conditions related to the depth of tensor products over Cohen-Macaulay rings, establishing their equivalences and properties, and extending existing results on the vanishing of Tor modules.

Contribution

It defines the conditions $( ext{ldep})$ and $( ext{rdep})$, proves their equivalences in Cohen-Macaulay rings, and explores their behavior under various operations, extending work on Tor vanishing.

Findings

Derived $( ext{ldep})$ is equivalent to the uniform Auslander bound in Cohen-Macaulay rings.

$( ext{ldep})$ implies $( ext{rdep})$ when $R$ is Gorenstein.

The conditions behave well under regular sequences and completion, but not necessarily under localization.

Abstract

Inspired by classical work on the depth formula for tensor products of finitely generated $R$-modules, we introduce two conditions which we call $(\mathbf{ldep})$ and $(\mathbf{rdep})$ and their derived variations. We show for Cohen-Macaulay local rings that derived $(\mathbf{ldep})$ is equivalent to $\dim(R)$ being a uniform Auslander bound for $R$, and if $\dim(R)>0$ that both are equivalent to $(\mathbf{ldep})$. We introduce an analogous condition we call the \emph{uniform Buchweitz condition} and provide a corresponding theorem for the $(\mathbf{rdep})$ condition. As a consequence of these results, we show $(\mathbf{ldep})$ implies $(\mathbf{rdep})$ when $R$ is Gorenstein and that the $(\mathbf{ldep})$ and $(\mathbf{rdep})$ conditions behave well under modding out by regular sequences and completion, but we give a concrete example showing they need not localize. Using our methods, we extend work of Jorgensen by calculating the value $q_R(M,N):=\sup\{i \mid \operatorname{Tor}^R_i(M,N) \ne 0\}$ under certain conditions.