Vanishing of Tor, and why we care about it

TL;DR

This paper surveys the recent progress in understanding the vanishing of Tor modules over local rings, highlighting its implications for module structure, torsion, and depth properties.

Contribution

It provides a comprehensive overview of developments in the study of Tor vanishing over the past twenty years, emphasizing new results and open questions.

Findings

Connections between Tor vanishing and module torsion properties

Implications of Tor vanishing for module depth and structure

Progress in characterizing rings and modules with vanishing Tor

Abstract

Given finitely generated modules $M$ and $N$ over a local ring $R$, the tensor product $M\otimes_RN$ typically has nonzero torsion. Indeed, the assumption that the tensor product is torsion-free influences the structure and vanishing of the modules $\Tor^R_i(M,N)$ for all $i\geq 1$. In turn, the vanishing of $\Tor^R_i(M,N)$ imposes restrictions on the depth properties of the modules $M$ and $N$. These connections made their first appearance in Auslander's 1961 paper "Modules over unramified regular local rings". We will survey the literature on these topics, with emphasis on progress during the past twenty years.