Abstract Riemann surfaces of integral domains and spectral spaces
type: Item
instance of: scholarly article
title: Abstract Riemann surfaces of integral domains and spectral spaces
zbMATH Open document ID: 0646.13008
DOI: 10.1007/BF01774285
author: David E. Dobbs
author: Richard Fedder
author: Marco Fontana
published in: Annali di Matematica Pura ed Applicata. Serie Quarta
publication date: 1987-01-01T00:00:00Z
review text: The authors consider a domain R and the abstract Riemann surface X(R) of the valuation overrings of R. They prove that X(R) is a spectral space in the sense of \textit{M. Hochster} [Trans. Am. Math. Soc. 142, 43-60 (1969; Zbl 0184.294)]. It is also proved that the canonical projection \(f_ R: X(R)\to Spec(R)\) is a homeomorphism if and only if R is a Prüfer domain. It is remarked that \(f_ R\) is always a closed map and there are given sufficient conditions for \(f_ R\) to be open. The map \(f_ R\) is spectral. It is deduced that X(R) is homeomorphic to an inverse limit of finite \(T_ 0\)-spaces and that \(f_ R\) is a homeomorphism if and only if it is an order-isomorphism. The authors remark that the functor Spec is not invertible on the category of abstract Riemann surfaces.
Mathematics Subject Classification ID: 13F99
Mathematics Subject Classification ID: 12J20
Mathematics Subject Classification ID: 13G05
Mathematics Subject Classification ID: 13A18
Mathematics Subject Classification ID: 14A05
zbMATH DE Number: 4053764
zbMATH Keywords: valuation rings
zbMATH Keywords: abstract Riemann surface
zbMATH Keywords: spectral space
zbMATH Keywords: Spec
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01774285
OpenAlex ID: W2052292031
description: scientific article; zbMATH DE number 4053764