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

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 $i\ge 1$. We also examine the relationship between good depth of $M{\otimes }_{R}N$ and the vanishing of $\Ext$ modules, with various applications.

Some results on semi-dualizing modules are added at the end