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

2012/11/07 by Cuong, Nguyen Tu, Van Hoang, Nguyen
#13C15 #13D07 #13D45 #Commutative Algebra (math.AC) #FOS: Mathematics

paper · doi:10.48550/arxiv.1211.1477

Abstract

Let (R,\frakm) be a Noetherian local ring, I an ideal of R and N a finitely generated R-module. Let k≥-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⊆ \Spec R we set S_≥k=\p∈ S|dim(R/\p)≥k. We first prove in this paper that \Ass_R(H^j_I(N))_{≥ k} is a finite set for all j≤r. Let \fN=⊕_{n≥ 0}N_n be a finitely generated graded \fR-module, where \fR is a finitely generated standard graded algebra over R₀=R. Let r be the eventual value of \depth_k(I,N_n). Then our second result says that for all l≤r the sets \bigcup_j≤l\Ass_R(H^j_I(N_n))_≥k are stable for large n.

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