The prime spectra of subalgebras of affine algebras and their localizations
type: Item
instance of: scholarly article
title: The prime spectra of subalgebras of affine algebras and their localizations
zbMATH Open document ID: 0669.14001
DOI: 10.1016/0022-4049(89)90027-3
author: Robert Gilmer
author: Budh Singh Nashier
author: Warren D. Nichols
published in: Journal of Pure and Applied Algebra
publication date: 1989-01-01T00:00:00Z
review text: The paper under review studies the subaffine k-algebras (with k a field), i.e. the k-subalgebras of affine k-algebras. Specifically, one wants to determine to what extent the properties of Spec(A) are shared by Spec(R), where R is a k-subalgebra of the affine k-algebra A. The first result considers the situation where R is a k-subalgebra of \(A_ M\) (where M is a maximal ideal of the affine k-algebra A), and states that \(tr.\deg_ k(R)=\dim (R)\). Next one considers only subaffine domains R and one proves that certain Zariski open sets of Spec(R) do share properties of the prime spectra of affine domains. On the other hand, one presents examples to show that Spec(R) does not share such properties globally. In particular, one exhibits a Noetherian subpolynomial domain D satisfying the first chain condition and having the property that D/P is affine for every \(P\in Spec(D)\), but which is not itself affine over k.
Mathematics Subject Classification ID: 14A05
Mathematics Subject Classification ID: 13E99
Mathematics Subject Classification ID: 13A15
zbMATH DE Number: 4095639
zbMATH Keywords: affinity of subaffine algebras
zbMATH Keywords: prime spectra
zbMATH Keywords: Noetherian subpolynomial domain
zbMATH Keywords: first chain condition
reviewed by: Lucian Bădescu
MaRDI profile type: Publication
full work available at URL: 0022-4049(89)90027-3
OpenAlex ID: W2063143778
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description: scientific article; zbMATH DE number 4095639