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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 ideala 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.

We study Cohen-Macaulay non-Gorenstein local rings (R, m, k) admitting certain totally reflexive modules. More precisely, we give a description of the Poincaré series of k by using the Poincaré series of a non-zero totally reflexive module with minimal multiplicity. Our results generalize a result of Yoshino to higher-dimensional Cohen-Macaulay local rings. Moreover, from a quasi-Gorenstein ideal satisfying some conditions, we construct a family of non-isomorphic indecomposable totally reflexive modules having an arbitrarily large minimal number of generators.2010 Mathematics Subject Classification. 13C13, 13D40.