Maximal Cohen-Macaulay tensor products

TL;DR

This paper investigates when the tensor product of two modules over a complete intersection domain being maximal Cohen-Macaulay implies that one of the modules is maximal Cohen-Macaulay, extending known results to higher codimension cases.

Contribution

It proves a new criterion involving Tor-rigidity for when the tensor product's maximal Cohen-Macaulay property implies the same for individual modules in higher codimension complete intersections.

Findings

If R is an isolated singularity with dim(R) > codim(R), then M⊗N maximal Cohen-Macaulay implies M or N is maximal Cohen-Macaulay.

The result extends the understanding of tensor products over complete intersections beyond codimension one.

The approach uses Tor-rigidity to establish the equivalence under specified conditions.

Abstract

In this paper we are concerned with the following question: if the tensor product of finitely generated modules $M$ and $N$ over a local complete intersection domain is maximal Cohen-Macaulay, then must $M$ or $N$ be a maximal Cohen-Macaulay? Celebrated results of Auslander, Lichtenbaum, and Huneke and Wiegand, yield affirmative answers to the question when the ring considered has codimension zero or one, but the question is very much open for complete intersection domains that have codimension at least two, even open for those that are one-dimensional, or isolated singularities. Our argument exploits Tor-rigidity and proves the following, which seems to give a new perspective to the aforementioned question: if $R$ is a complete intersection ring which is an isolated singularity such that dim($R$) > codim($R$), and the tensor product $M\otimes_R N$ is maximal Cohen-Macaulay, then $M$ is maximal Cohen-Macaulay if and only if $N$ is maximal Cohen-Macaulay.