Gorenstein injective dimension and Tor-depth of modules

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 com plete 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 greater than or equal to 1 Ext(i)(E, M) = 0 for all indeco mposable injective R-modules E not equal E(k), then the depth of the ring i s equal to the sum of the Gorenstein injective dimension and Tor-depth of R I. As a consequence, we get that this formula holds over a d-dimensional Go rnstein local ring for every nonzero cosyzygy of a finitely generated R-mod ule and thus in particular each such n(th) cosyzygy has its Tor-depth equal to the depth of the ring whenever n greater than or equal to d.