Associated Primes and Syzygies of Linked Modules

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 AssR(M)\AssR(N) = ;, then M 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 eld 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