Vanishing of Tate Cohomology and Gorenstein Injective Dimension

Abstract

In this note, some sufficient conditions are given for the equality Gid _R RHom _R (M, N) = G-dim _R M + Gid _R N to hold via vanishing of Tate cohomology. For instance, let R be a commutative Gorenstein local ring, and let M and N be finite R-modules. If one has for all i ∈ ℤ, then Gid _R RHom _R (M, N) = G-dim _R M + Gid _R N. In addition, some Auslander–Buchsbaum type width (depth) formulas are given.

Key Words:

• Gorenstein injective dimension
• Gorenstein ring
• Tate cohomology
• Width of complex

2010 Mathematics Subject Classification:

• 13D02
• 13D05
• 13D07

ACKNOWLEDGMENT

The first author was partly supported by National Natural Science Foundation of China (No. 11301242), and the second author was partly supported by National Natural Science Foundation of China (No. 11261050). It is a pleasure to thank the referee for thorough and useful comments.

Notes

Communicated by T. Albu.