Annihilators of Local Cohomology Modules

Abstract

Let (R, 𝔪) be a commutative Noetherian complete local ring, M a nonzero finitely generated R-module of dimension n, and I be an ideal of R. In this paper we calculate the annihilator of the top local cohomology module . Also, if (R, 𝔪) is a Noetherian local Cohen–Macaulay ring of dimension d and I is a nonzero proper ideal of R, then we calculate the annihilator of the first nonzero local cohomology module . Finally, we show that if R is an arbitrary Noetherian ring, I an ideal of R, and M is a nonzero finitely generated R-module with cd(I, M) = t ≥ 0, then there exists a submodule N of M such that . This is a generalization of the main result of Bahmanpour, A'zami, and Ghasemi [1] for all ideals of an arbitrary Noetherian ring R.

Key Words:

• Cohen–Macaulay ring
• Cohomological dimension
• Local cohomology
• Minimal primary decomposition

2010 Mathematics Subject Classification:

• 13D45

ACKNOWLEDGMENTS

The author is deeply grateful to the referee for a very careful reading of the manuscript and many valuable suggestions.

Notes

Communicated by G. Leuschke.