Associated Primes and Syzygies of Linked Modules

2016/02/27 by Celikbas, Olgur, Dibaei, Mohammad T., Gheibi, Mohsen +2
#13C40 #13D02 #13D05 #13D07 #Commutative Algebra (math.AC) #FOS: Mathematics

paper · doi:10.48550/arxiv.1602.08625

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)∩ Ass_R(N)=∅, then M⊗_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.

Related

• Syzygies of Cohen-Macaulay modules over one dimensional Cohen-Macaulay local rings
• Linkage of ideals over a module
• Direct summands of syzygy modules of the residue class field
• Syzygies of associated graded modules
• Linkage of modules with respect to a semidualizing module