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arXiv · 1602.08625

Associated Primes and Syzygies of Linked Modules

Olgur Celikbas·Mohammad T. Dibaei·Mohsen Gheibi·Arash Sadeghi·Ryo Takahashi
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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 $k$.

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Olgur Celikbas, Mohammad T. Dibaei, Mohsen Gheibi, Arash Sadeghi, Ryo Takahashi. 2016-02-27. Associated Primes and Syzygies of Linked Modules. https://arxiv.org/abs/1602.08625

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