Torsion in tensor products, and tensor powers, of modules

TL;DR

This paper explores the relationship between torsion properties of tensor products of modules over complete intersections and the vanishing of Tor, providing new insights into module depth and tensor power behaviors.

Contribution

It establishes conditions under which torsion-freeness of tensor products implies vanishing of Tor, especially for tensor powers of a single module.

Findings

Vanishing of Tor relates to torsion-free tensor products.

Tensor powers of modules exhibit specific vanishing properties.

Conditions linking Serre properties to Tor vanishing are identified.

Abstract

For finitely generated modules M and N over a complete intersection R, the vanishing of Tor_i^R(M,N) for all i> 0 gives a tight relationship among depth properties of M, N and their tensor product. Here we concentrate on the converse and show, under mild conditions, that the tensor product of M and N being torsion-free (or satisfying higher Serre conditions) forces vanishing of Tor. Special attention is paid to the case of tensor powers of a single module.