Necessary conditions for the depth formula over Cohen-Macaulay local rings

TL;DR

This paper explores the necessary conditions for the depth formula to hold over Cohen-Macaulay local rings, linking it to the vanishing of higher Tor modules and examining the role of Ext modules.

Contribution

It establishes that the depth formula holds if and only if higher Tor modules vanish under certain conditions, and analyzes the connection with Ext modules.

Findings

Depth formula holds iff higher Tor modules vanish.

Vanishing of Ext modules relates to the depth of tensor products.

Provides conditions linking depth, Tor, and Ext modules.

Abstract

Let $R$ be a Cohen-Macaulay local ring and let $M$ and $N$ be non-zero finitely generated $R$-modules. We investigate necessary conditions for the depth formula $\depth(M)+\depth(N)=\depth(R)+\depth(M\otimes_{R}N)$ to hold. We show that, under certain conditions, $M$ and $N$ satisfy the depth formula if and only if $\Tor_{i}^{R}(M,N)$ vanishes for all $i\geq 1$. We also examine the relationship between good depth of $M\otimes_RN$ and the vanishing of $\Ext$ modules, with various applications.