Syzygies and tensor product of modules

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${}_{>0}^R(M,N)=0$, then both $M$ and $N$ are $n$th syzygies of some finitely generated $R$-modules.

some typos are fixed; to appear in Mathematische Zeitschrift

Cited by in corpus (4)

• Vanishing of Tor, and why we care about it
• Bounds on depth of tensor products of modules
• On the depth and reflexivity of tensor products
• On the vanishing of (co)homology for modules admitting certain filtrations