2017/08/15 by Reza Naghipour, Naghipour, Reza, Peter Schenzel +1
Computer Science · Mathematics · #13B2 #13B22 #13E05 #Algebraic Geometry and Number Theory #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics #Polynomial and algebraic computation #math.AC #msc:13B2 #msc:13B22 #msc:13E05
paper · pdf · doi:10.48550/arxiv.1708.04635
15 pages, to appear in: Rocky Mountain Journal of Mathematics
arxiv created 2017/08/15 · openalex publication_date 2017/08/15 · arxiv updated 2017/08/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let R be a commutative Noetherian ring, N a finitely generated R-module and I an ideal of R. The set Q^*(I, N), the quintasymptotic primes of I with respect to N, was originally introduced by McAdam \citeMc2. Also, the ideal Ia(N), the integral closure of I with respect to N, was introduced by R.Y. Sharp et al. in \citeSTY. The purpose of this paper is to show that, whenever S is a multiplicatively closed subset of R then the topologies defined by \(In)a(N)\n≥1 and \S((In)a(N))\n≥1 are equivalent if and only if S is disjoint from the quintasymptotic primes of I with respect to N. In addition, using this result, we also show that, if (R, \mathfrakm) is local and N is quasi-unmixed, then the local cohomology module Hdim NI(N) vanishes if and only if there exists a multiplicatively closed subset S of R such that \mathfrakm ∩ S ≠ ∅ and the topologies induced by \(In)a(N)\n≥1 and \S((In)a(N))\n≥1 are equivalent. As a special of this characterization we obtain the main result of Marti-Farre \citeMF.