Rocky Mountain Journal of Mathematics

The prime spectrum and dimension of ideal transform algebras

Kamal Bahmanpour

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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

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

First available in Project Euclid: 22 September 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

Cohomological dimension ideal transform functor local cohomology Noetherian ring Zariski topology


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.

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