Vanishing of relative homology and depth of tensor products

TL;DR

This paper extends the conditions under which the depth formula holds for modules over Gorenstein local rings, using vanishing of Tate homology and weaker Tor-independence assumptions.

Contribution

It generalizes the depth formula to cases with $ ext{G}$-relative Tor-independence and partial Tate homology vanishing, broadening previous results.

Findings

Depth formula holds under $ ext{G}$-relative Tor-independence with vanishing Tate homology for $i o -\infty$.

Provides a necessary condition for the depth formula involving $ ext{G}$-relative homology.

Analyzes the depth of tensor products of modules over Gorenstein rings.

Abstract

For finitely generated modules $M$ and $N$ over a Gorenstein local ring $R$, one has $depth M + depth N= depth(M\otimes N) +depth R$, i.e., the depth formula holds, if $M$ and $N$ are Tor-independent and Tate homology $\hat{Tor}_{i}^{R}(M,N)$ vanishes for all $i\in\mathbb{Z}$. We establish the same conclusion under weaker hypotheses: if $M$ and $N$ are $\mathcal{G}$-relative Tor-independent, then the vanishing of $\hat{Tor}_{i}^{R}(M,N)$ for all $i\le 0$ is enough for the depth formula to hold. We also analyze the depth of tensor products of modules and obtain a necessary condition for the depth formula in terms of $\mathcal{G}$-relative homology.