Gorenstein Injective Dimension and Dual Auslander–Bridger Formula#

Abstract

An R-module M satisfies the dual Auslander–Bridger formula if Gid R M + Tor − depth M = depth R. Let R be a complete Cohen–Macaulay local ring with residue field k and M be a non-injective Ext-finite R-module such that for all i ≥ 1 and all indecomposable injective R-modules E ≠ E(k). If M has finite Gorenstein injective dimension, then we will prove that M satisfies the dual Auslander–Bridger formula if either Ext-depth M > 0 or Ext-depth M = 0 and Gid R M ≠ 1. We denote Gorenstein injective envelope of M by G(M). If R is a Gorenstein local ring and M is a non-Gorenstein injective finitely generated R-module and G(M) is reduced, then this formula holds for and its cosyzygy. As the last result, if R is a regular local ring, then dual Auslander–Bridger holds for any non-injective Ext-finite R − module.