Rocky Mountain Journal of Mathematics

The prime spectrum and dimension of ideal transform algebras

Kamal Bahmanpour

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Abstract

Let $(R,\mathfrak{m} )$ be a commutative Noetherian local ring of dimension $d\geq 1$, and let $I$ be a non-nilpotent ideal of $R$ such that the ideal transform functor $D_I(-)$ is exact. In this paper, it is shown that the finitely generated flat $R$-algebra $D_I(R)$ is a Noetherian ring of dimension $n=\dim R/\Gamma _I(R)-1$. Also, it is shown that, under Zariski topologies on the sets $Spec D_I(R)$ and $Spec R/\Gamma _I(R)$, there is a homeomorphism of topological spaces: \[ \widetilde {\eta ^*}:Spec D_I(R)\longrightarrow Spec R/\Gamma _I(R)\setminus V((I+\Gamma _I(R))/\Gamma _I(R)). \]

Article information

Source
Rocky Mountain J. Math., Volume 47, Number 5 (2017), 1415-1426.

Dates
First available in Project Euclid: 22 September 2017

Permanent link to this document
https://projecteuclid.org/euclid.rmjm/1506045619

Digital Object Identifier
doi:10.1216/RMJ-2017-47-5-1415

Mathematical Reviews number (MathSciNet)
MR3705759

Zentralblatt MATH identifier
06790020

Subjects
Primary: 13D45: Local cohomology [See also 14B15] 13E05: Noetherian rings and modules 14B15: Local cohomology [See also 13D45, 32C36]

Keywords
Cohomological dimension ideal transform functor local cohomology Noetherian ring Zariski topology

Citation

Bahmanpour, Kamal. The prime spectrum and dimension of ideal transform algebras. Rocky Mountain J. Math. 47 (2017), no. 5, 1415--1426. doi:10.1216/RMJ-2017-47-5-1415. https://projecteuclid.org/euclid.rmjm/1506045619


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