Dual of the auslander-bridger formula and GF-perfectness

Abstract

Ext-finite modules were introduced and studied by Enochs and Jenda. We prove under some conditions that the depth of a local ring is equal to the sum of the Gorenstein injective dimension and Tor-depth of an Ext-finite module of finite Gorenstein injective dimension. Let (R, m) be a local ring. We say that an R-module M with dimR M = n is a Grothendieck module if the n-th local cohomology module of M with respect to m, Hmn (M), is non-zero. We prove the Bass formula for this kind of modules of finite Gorenstein injective dimension and of maximal Krull dimension. These results are dual versions of the Auslander-Bridger formula for the Gorenstein dimension. We also introduce GF-perfect modules as an extension of quasi-perfect modules introduced by Foxby.