The Use of Ultraproducts in Commutative Algebra
In spite of some recent applications of ultraproducts in algebra, they remain largely unknown to commutative algebraists, in part because they do not preserve basic properties such as Noetherianity. This work wants to make a strong case against these prej
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1999
Hans Schoutens
The Use of Ultraproducts in Commutative Algebra
123
Hans Schoutens City University of New York CUNY Graduate Center Department of Mathematics 365 Fifth Avenue New York, NY, 10016 USA [email protected]
ISBN: 978-3-642-13367-1 e-ISBN: 978-3-642-13368-8 DOI: 10.1007/978-3-642-13368-8 Springer Heidelberg Dordrecht London New York Lecture Notes in Mathematics ISSN print edition: 0075-8434 ISSN electronic edition: 1617-9692 Library of Congress Control Number: 2010930621 Mathematics Subject Classification (2010): 60G51, 60E07, 60J80, 45K05, 65N30, 28A78, 60H05, 60G57, 60J75, 26A33 c Springer-Verlag Berlin Heidelberg 2010 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, reuse of illustrations, recitation, broadcasting, reproduction on microfilm 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. Violations are liable to prosecution under the German Copyright Law. 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. Cover design: SPi Publisher Services Printed on acid-free paper springer.com
To my mother, Jose Van Passel, for giving me wisdom; to my father, Louis Schoutens, for giving me knowledge; to my teacher, Pierre Gevers, for giving me the love for mathematics; to my mentor, Jan Denef, for giving me inspiration; and to my one and true love, Parvaneh Pourshariati, for giving me purpose.
Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
2
Ultraproducts and Ło´s’ Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Ultraproducts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Ultrafilters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Ultraproducts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.3 Properties of Ultraproducts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Model-theory in Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Formulae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Satisfaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3 Constructible Sets . . . . . . . . . . . . . . . . . . .
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