On the finiteness and stability of certain sets of associated primes ideals of local cohomology modules

TL;DR

This paper proves the finiteness and stability of certain associated prime sets of local cohomology modules over Noetherian local rings, extending understanding of their structure and behavior in graded modules.

Contribution

It establishes the finiteness of associated primes in specified subsets and demonstrates their stability in graded modules for large indices.

Findings

Ass_R(H^j_I(N))_{≥k} is finite for all j ≤ r

The sets of associated primes stabilize for large n in graded modules

Provides new insights into the structure of local cohomology modules

Abstract

Let $(R,\frak{m})$ be a Noetherian local ring, $I$ an ideal of $R$ and $N$ a finitely generated $R$-module. Let $k{\ge}-1$ be an integer and $ r=\depth_k(I,N)$ the length of a maximal $N$-sequence in dimension $>k$ in $I$ defined by M. Brodmann and L. T. Nhan ({Comm. Algebra, 36 (2008), 1527-1536). For a subset $S\subseteq \Spec R$ we set $S_{{\ge}k}={\p\in S\mid\dim(R/\p){\ge}k}$. We first prove in this paper that $\Ass_R(H^j_I(N))_{\ge k}$ is a finite set for all $j{\le}r$}. Let $\fN=\oplus_{n\ge 0}N_n$ be a finitely generated graded $\fR$-module, where $\fR$ is a finitely generated standard graded algebra over $R_0=R$. Let $r$ be the eventual value of $\depth_k(I,N_n)$. Then our second result says that for all $l{\le}r$ the sets $\bigcup_{j{\le}l}\Ass_R(H^j_I(N_n))_{{\ge}k}$ are stable for large $n$.