Semi-algebraic Function Rings and Reflectors of Partially Ordered Rings
The book lays algebraic foundations for real geometry through a systematic investigation of partially ordered rings of semi-algebraic functions. Real spectra serve as primary geometric objects, the maps between them are determined by rings of functions as
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1712
Springer Berlin Heidelberg New York Barcelona Hong Kong London Milan Paris Singapore Tokyo
Niels Schwartz James J. Madden
Semi-algebraic Function Rings and Reflectors of Partially Ordered Rings
Springer
Authors Niels Schwartz Fakultat fur Mathematik und Informatik Universitat Passau Postfach 25 40 D-94030 Passau, Germany
James 1. Madden Department of Mathematics Louisiana State University Baton Rouge, LA 70803, USA E-mail: [email protected]
E-mail: [email protected]
Cataloging-in-Publication Data applied for
Die Deutsche Bibliothek - CIP-Einheitsaufnahme Schwartz, Niels: Semi-algebraic function rings and reflectors of partially ordered rings I Niels Schwartz; James J. Madden. - Berlin; Heidelberg; New York; Barcelona; Hong Kong; London; Milan; Paris; Singapore; Tokyo: Springer, 1999 (Lecture notes in mathematics; 1712) ISBN 3-540-66460-2
Mathematics Subject Classification (1991): 14P10, 06F25, l3J25, l8A40, 54C30 ISSN 0075-8434 ISBN 3-540-66460-2 Springer-Verlag Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1999 Printed in Germany
The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: Camera-ready TEX output by the authors SPIN: 10700220 41/3143-543210 - Printed on acid-free paper
Dedicated to
Sybilla PriefJ-Crampe
Preface Our work lays algebraic foundations for real algebraic geometry via a systematic study of reduced partially ordered rings (or, as we say, reduced porings ). Thus, we address real algebraists and real geometers as well as algebraists who are concerned with partially ordered algebraic structures. Since the class of rings that we study comprehends all rings of continuous functions with values in the real numbers, we hope that our work will also be useful for topologists using and studying such rings. The real spectrum is a functorial construction that links algebra to geometry by associating with each poring a topological space. This space should perhaps be viewed as a substrate on which geometric structures can be built, since in algebraic geometry many questions are concerned with more than mere topology. How does one impose a geometric structure? It is a well-established method to think of geometric properties by means of the mappings or tra
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