Associated Primes and Syzygies of Linked Modules

TL;DR

This paper explores the properties of linked modules over Gorenstein rings, establishing conditions for their tensor products and linking homological dimensions to ring characteristics.

Contribution

It introduces new criteria relating linked modules, associated primes, and homological dimensions to characterize local rings.

Findings

Tensor product of linked modules decomposes into sums of Gorenstein ideals.

A criterion for the depth of a local ring based on linked syzygy modules.

Characterization of local rings via homological dimensions of linked modules.

Abstract

Motivated by the notion of geometrically linked ideals, we show that over a Gorenstein local ring $R$, if a Cohen-Macaulay $R$-module $M$ of grade $g$ is linked to an $R$-module $N$ by a Gorenstein ideal $c$, such that $Ass_R(M)\cap Ass_R(N)=\emptyset$, then $M\otimes_RN$ is isomorphic to direct sum of copies of $R/a$, where $a$ is a Gorenstein ideal of $R$ of grade $g+1$. We give a criterion for the depth of a local ring $(R,m,k)$ in terms of the homological dimensions of the modules linked to the syzygies of the residue field $k$. As a result we characterize a local ring $(R,m,k)$ in terms of the homological dimensions of the modules linked to the syzygies of $k$.