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December 2021 On the equivalence of the integral symbolic and adic topologies
Reza Naghipour, Monireh Sedghi
Rocky Mountain J. Math. 51(6): 2171-2178 (December 2021). DOI: 10.1216/rmj.2021.51.2171

Abstract

Let R be a commutative Noetherian ring and I an ideal of R. The purpose of this paper is to show that the topologies defined by the integral filtration {Im¯}m1 and the symbolic integral filtration {Im}m1 are equivalent whenever Q¯(I) consists all of the minimal prime ideals of I. As an application of this result, by using the Jacobian theorem of Lipman and Sathaye we deduce that the symbolic integral topology {Im}m1 is equivalent to the I-adic topology whenever R is a regular ring. Also, applying these results we provide extensions of classical results of Hartshorne and Zariski on the equivalence of symbolic and adic topologies.

Citation

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Reza Naghipour. Monireh Sedghi. "On the equivalence of the integral symbolic and adic topologies." Rocky Mountain J. Math. 51 (6) 2171 - 2178, December 2021. https://doi.org/10.1216/rmj.2021.51.2171

Information

Received: 10 May 2020; Revised: 24 March 2021; Accepted: 30 March 2021; Published: December 2021
First available in Project Euclid: 22 March 2022

Digital Object Identifier: 10.1216/rmj.2021.51.2171

Subjects:
Primary: 13B22

Keywords: ideal topology , integral closure , quasiunmixed ring , quintasymptotic prime , regular ring , symbolic power

Rights: Copyright © 2021 Rocky Mountain Mathematics Consortium

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Vol.51 • No. 6 • December 2021
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