Polytopes, Rings, and K-Theory
This book treats the interaction between discrete convex geometry, commutative ring theory, algebraic K-theory, and algebraic geometry. The basic mathematical objects are lattice polytopes, rational cones, affine monoids, the algebras derived from them, a
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Springer Monographs in Mathematics
WINFRIED BRUNS JOSEPH GUBELADZE
Springer Monographs in Mathematics
For further volumes: http://www.springer.com/series/3733
Winfried Bruns · Joseph Gubeladze
Polytopes, Rings, and K-Theory
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Winfried Bruns FB Mathematik/Informatik Universit¨at Osnabr¨uck Osnabr¨uck Germany [email protected]
Joseph Gubeladze Department of Mathematics San Francisco State University San Francisco, CA 94132 USA [email protected]
ISBN 978-0-387-76355-2 e-ISBN 978-0-387-76356-9 DOI 10.1007/b105283 Springer Dordrecht Heidelberg London New York Library of Congress Control Number: 2009926997 Mathematics Subject Classification (2000): 05E99, 11D04 11H06 13B21 13B26 c Springer Science+Business Media, LLC 2009 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)
To our children and grandchildren
Annika, Nils, Julia, Linus, Charlotte, and Sophie Nino, Ana, Jimsher, Tamar, and Lazare
Preface
For every mathematician, ring theory and K-theory are intimately connected: algebraic K-theory is largely the K-theory of rings. At first sight, polytopes, by their very nature, must appear alien to surveyors of this heartland of algebra. But in the presence of a discrete structure, polytopes define affine monoids, and, in their turn, affine monoids give rise to monoid algebras. Their spectra are the building blocks of toric varieties, an area that has developed rapidly in the last four decades. From a purely systematic viewpoint, “monoids” should therefore replace “polytopes” in the title of the book. However, such a change would conceal the geometric flavor that we have tried to preserve through all chapters. Before delving into a description of the contents we would like to mention three general features of the book: () the exhibiting of interactions of convex geometry, ring theory, and K-theory is not the only goal; we present some of the central results in each of these fields; () the exposition is of constructive (i. e., algorithmic) nature at many places throughout the text—there is no doubt that one of the driving forces behind the current popularity of combinatorial geometry is the quest for visualization and computation; () despite the large amount of information from various fields, we have strived to keep the polytopal perspective as the ma
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