Associated Primes and Syzygies 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 $As{s}_R(M)\cap As{s}_R(N)=\varnothing$, then $M{\otimes }_{R}N$ 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 $k$.

To appear in Journal of Commutative Algebra

References in corpus (3)

• Homological dimensions of rigid modules
• Bounds on depth of tensor products of modules
• Perfect linkage of Cohen--Macaulay modules over Cohen--Macaulay rings