Asymptotic stability of certain sets of associated prime ideals of local cohomology modules

TL;DR

This paper investigates the asymptotic behavior of associated prime ideals of certain local cohomology modules in Noetherian local rings, establishing their stability for large powers of an ideal.

Contribution

It proves the stability of specific sets of associated primes of local cohomology modules related to powers of an ideal, extending understanding of their asymptotic properties.

Findings

Sets of associated primes stabilize for large n.

Depth-related invariants become independent of n.

Results apply to modules M/J^nM as well.

Abstract

Let $(R,\m)$ be a Noetherian local ring $I, J$ two ideals of $R$ and $M$ a finitely generated $R-$module. It is first shown that for $k\geq -1$ the integer $r_k = \depth_k(I,J^nM/J^{n+1}M)$, it is the length of a maximal $(J^nM/J^{n+1}M)-$sequence in dimension $>k$ in $I$ defined by M. Brodmann and L. T. Nhan \cite{BN}, becomes for large $n$ independent of $n$. Then we prove in this paper that the sets $\bigcup_{j\le r_k}\Ass_R(H^j_I(J^nM/J^{n+1}M))$ with $k=-1$ or $k=0$, and $\bigcup_{j\le r_1}\Ass_R(H^j_I(J^nM/J^{n+1}M))\cup\{\m\}$ are stable for large $n$. We also obtain similar results for modules $M/J^nM$.