Journal of Commutative Algebra

Associated primes and syzygies of linked modules

Olgur Celikbas, Mohammad T. Dibaei, Mohsen Gheibi, Arash Sadeghi, and Ryo Takahashi

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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 $\mathrm {Ass}_R(M)$ and $\mathrm {Ass}_R(N)$ are disjoint, then $M\otimes _RN$ is isomorphic to direct sum of copies of $R/\mathfrak {a}$, where $\mathfrak {a}$ is a Gorenstein ideal of $R$ of grade $g+1$. We give a criterion for the depth of a local ring $(R,\mathfrak {m},k)$ 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 $(R,\mathfrak {m},k)$ in terms of the homological dimensions of the modules linked to the syzygies of $k$.

Article information

J. Commut. Algebra, Volume 11, Number 3 (2019), 301-323.

First available in Project Euclid: 3 December 2019

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 13D07: Homological functors on modules (Tor, Ext, etc.)
Secondary: 13D02: Syzygies, resolutions, complexes 13C40: Linkage, complete intersections and determinantal ideals [See also 14M06, 14M10, 14M12] 13D05: Homological dimension

Linkage of modules geometrically linked ideals Huneke-Wiegand conjecture torsion-free module vanishing of $\mathrm {Tor}$ homological dimension.


Celikbas, Olgur; Dibaei, Mohammad T.; Gheibi, Mohsen; Sadeghi, Arash; Takahashi, Ryo. Associated primes and syzygies of linked modules. J. Commut. Algebra 11 (2019), no. 3, 301--323. doi:10.1216/JCA-2019-11-3-301.

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