Gorenstein injective dimension and Tor-depth of modules

Abstract.

The main aim of this paper is to obtain a dual result to the now well known Auslander-Bridger formula for G-dimension. We will show that if R is a complete Cohen-Macaulay ring with residue field k, and M is a non-injective h-divisible Ext-finite R-module of finite Gorenstein injective dimension such that for each $i \geq 1$ Extⁱ (E,M) = 0 for all indecomposable injective R-modules $E \neq E(k)$, then the depth of the ring is equal to the sum of the Gorenstein injective dimension and Tor-depth of M. As a consequence, we get that this formula holds over a d-dimensional Gorenstein local ring for every nonzero cosyzygy of a finitely generated R-module and thus in particular each such n ^th cosyzygy has its Tor-depth equal to the depth of the ring whenever $n \geq d$.