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.
Acknowledgments
The author would like to thank Professor E. Enochs and Professor H. Zakeri for useful discussion concerning the material of this paper. Also, he would like to thank the referee for his/her careful reading of the paper.
Notes
#Communicated by I. Swanson.