Asymptotic Stability of Certain Sets of Associated Prime Ideals of Local Cohomology Modules

Abstract

Let (R, 𝔪) be a Noetherian local ring I, J two ideals of R and M a finitely generated R-module. Let k ≥ −1 and r _k  = depth _k (I, J ⁿ M/J ⁿ⁺¹ M) be the length of a maximal (J ⁿ M/J ⁿ⁺¹ M)-sequence in dimension >k in I defined by Brodmann and Nhan [4]. It is first shown that r _k becomes independent of n for large n. Then we prove in this article that the sets with k = −1 or k = 0, and are stable for large n. We also obtain similar results for modules M/J ⁿ M.

Key Words:

• Associated prime
• Depth
• Filter depth
• Generalized depth
• Local cohomology

2000 Mathematics Subject Classification:

• 13D45
• 13E05

ACKNOWLEDGMENT

This work is supported partly by the National Foundation for Science and Technology Development (Nafosted) of Vietnam.

Notes

Communicated by A. Singh.