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Associated primes and syzygies of linked modules. (English) Zbl 1453.13046

As the authors mentioned in the abstract, they 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 \({\mathfrak c}\) ( i.e. \(R/{\mathfrak c}\) is a Gorenstein ring) such that \(\mathrm{Ass}_RM\) and \(\mathrm{Ass}_RN\) 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 grade \(g+1\). They also 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 \(k\). And as a result they charactrize a local ring \((R,{\mathfrak m},k)\) in terms of the homological dimensions of the modules linked to the syzygies of \(k\). For the definition of linked modules, see [A. Martsinkovsky and J. R. Strooker, J. Algebra 271, No. 2, 587–626 (2004; Zbl 1099.13026)].

MSC:

13D07 Homological functors on modules of commutative rings (Tor, Ext, etc.)
13D02 Syzygies, resolutions, complexes and commutative rings
13C40 Linkage, complete intersections and determinantal ideals
13D05 Homological dimension and commutative rings

Citations:

Zbl 1099.13026
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References:

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