Ein semialgebraischer Beweis der topologischen Form des Hauptsatzes von Zariski. (A semi-algebraic proof of the topological form of Zariski's main theorem)
type: Item
instance of: scholarly article
title: Ein semialgebraischer Beweis der topologischen Form des Hauptsatzes von Zariski. (A semi-algebraic proof of the topological form of Zariski's main theorem)
zbMATH Open document ID: 0689.14005
DOI: 10.1007/BF01153582
author: Roland Huber
published in: Manuscripta Mathematica
publication date: 1988-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/155302
review text: Let R be a real closed field, \(C=R(\sqrt{-1})\) its algebraic closure. If X is a C-variety then X(C), the set of C-rational points of X, is a semi- algebraic space (using the identification \(C=R^{2n})\) in the sense of \textit{H. Delfs} and \textit{M. Knebusch} [``Locally semialgebraic spaces'', Lect. Notes Math. 1173 (1985; Zbl 0582.14006)]. Similar to the construction of the sheaf of Nash functions on the real spectrum of a ring [cf. \textit{M. F. Roy} in Géometrie algébrique réelle et formes quadratiques, Journ. S. M. F., Univ. rennes 1981, Lect. Notes Math. 959, 406-432 (1982; Zbl 0497.14009)] a sheaf of isoalgebraic functions is defined on X(C). The space X(C) together with this sheaf is an isoalgebraic space. Locally isoalgebraic spaces are obtained by gluing isoalgebraic spaces together. A locally isoalgebraic space X carries a natural structure of a locally semi-algebraic space which is denoted by \(| X|\). The main result is: If X is a connected normal isoalgebraic space and \(A\subset X\) is locally semi-algebraic, dim(A)\(\leq \dim | X| -2\), then \(X\setminus A\) is connected. From this the topological Zariski main theorem and the Riemann extension theorem can be deduced.
Mathematics Subject Classification ID: 14Pxx
Mathematics Subject Classification ID: 14A05
Mathematics Subject Classification ID: 14F45
zbMATH DE Number: 4128966
zbMATH Keywords: Nash function
zbMATH Keywords: normal variety
zbMATH Keywords: real closed field
zbMATH Keywords: isoalgebraic space
zbMATH Keywords: locally semi-algebraic space
zbMATH Keywords: topological Zariski main theorem
zbMATH Keywords: Riemann extension theorem
reviewed by: Niels Schwartz
MaRDI profile type: Publication
cites work: On periodic points
cites work: Locally semialgebraic spaces
cites work: Q5515607
cites work: Q5285618
cites work: Further Pathologies in Algebraic Geometry
cites work: Q2551490
OpenAlex ID: W2074844191
description: scientific article; zbMATH DE number 4128966