When Are the Local Cohomology Modules Finitely Generated?

Abstract

Let R be a commutative Noetherian ring, 𝔞 an ideal of R, and M an R-module. We prove that for a fixed non-negative integer n, the nth local cohomology module is finitely generated if and only if is 𝔞-cofinite and . This enables us to establish the Noetherian property of local cohomology modules in several cases. Finally, we obtain a new characterization of the cohomological dimensions.

Key Words:

• Cofinite modules
• Cohomological annihilators
• Cohomological dimension
• Local cohomology modules

2000 Mathematics Subject Classification:

• 13D45
• 13C15
• 13E05

ACKNOWLEDGMENT

The authors are deeply grateful to the referee for carefully reading of the manuscript and helpful suggestions. Also, we would like to thank the Centre of Excellence in Analysis on Algebraic Structures, Ferdowsi University of Mashhad (CEAAS) for the financial support.

Notes

Communicated by A. K. Singh.