Syzygies and tensor product of modules

TL;DR

This paper applies the New Intersection Theorem to establish conditions under which the syzygy properties of modules are preserved through tensor products over local complete intersection rings, revealing new syzygy relationships.

Contribution

It proves a novel result linking tensor product syzygy levels and vanishing Tor conditions over complete intersection rings.

Findings

If the tensor product is an (n+c)th syzygy and Tor vanishes, then both modules are n-th syzygies.

The result extends understanding of syzygy behavior in module tensor products.

Provides conditions for modules to be syzygies based on their tensor product and Tor vanishing.

Abstract

We give an application of the New Intersection Theorem and prove the following: let $R$ be a local complete intersection ring of codimension $c$ and let $M$ and $N$ be nonzero finitely generated $R$-modules. Assume $n$ is a nonnegative integer and that the tensor product $M\tensor_{R}N$ is an $(n+c)$th syzygy of some finitely generated $R$-module. If Tor$^{R}_{>0}(M,N)=0$, then both $M$ and $N$ are $n$th syzygies of some finitely generated $R$-modules.