Criteria for vanishing of Tor over Complete Intersections

TL;DR

This paper investigates conditions under which the Tor functor vanishes over complete intersection rings, using eta-pairing and depth properties to establish new criteria and improve existing results.

Contribution

It introduces new vanishing criteria for Tor over complete intersections, connecting Tor vanishing with depth conditions of modules and their tensor products.

Findings

Vanishing of Tor_i(M, N) for all i>0 under certain depth conditions

Improved results over previous work by Dao

Established new links between Tor vanishing and tensor product depth

Abstract

In this paper we exploit properties of Dao's eta-pairing as well as techniques of Huneke, Jorgensen, and Wiegand to study the vanishing of Tor_i(M,N) of finitely generated modules M, N over complete intersections. We prove vanishing of Tor_i(M, N) for all positive integers i under depth conditions on M, N, and their tensor product. Our arguments improve a result of Dao and establish a new connection between the vanishing of Tor and the depth of tensor products.