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.

Key Words:

• Gorenstein injective
• Gorenstein
• Cohen–Macaulay
• Depth

MSC:

• 13D01
• 13D05
• 13C11
• 13C15
• 13H05
• 13H10
• 18G15

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.