Monomial Ideals
This book demonstrates current trends in research on combinatorial and computational commutative algebra with a primary emphasis on topics related to monomial ideals. Providing a useful and quick introduction to areas of research spanning these fields, Mo
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Editorial Board S. Axler K.A. Ribet
For other titles published in this series, go to www.springer.com/series/136
Jürgen Herzog r Takayuki Hibi
Monomial Ideals
Jürgen Herzog Universität Duisburg-Essen Fachbereich Mathematik Campus Essen Universitätsstraße 2 D-45141 Essen Germany [email protected]
Editorial Board S. Axler Mathematics Department San Francisco State University San Francisco, CA 94132 USA [email protected]
ISSN 0072-5285 ISBN 978-0-85729-105-9 DOI 10.1007/978-0-85729-106-6
Takayuki Hibi Department of Pure and Applied Mathematics Graduate School of Information Science and Technology Osaka University Toyonaka, Osaka 560-0043 Japan [email protected]
K.A. Ribet Mathematics Department University of California, Berkeley Berkeley, CA 94720-3840 USA [email protected]
e-ISBN 978-0-85729-106-6
Springer London Dordrecht Heidelberg New York British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library Library of Congress Control Number: 2010937479 Mathematics Subject Classification (2010): 13D02, 13D40, 13F55, 13H10, 13P10 © Springer-Verlag London Limited 2011 Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms of licenses issued by the Copyright Licensing Agency. Enquiries concerning reproduction outside those terms should be sent to the publishers. The use of 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 laws and regulations and therefore free for general use. The publisher makes no representation, express or implied, with regard to the accuracy of the information contained in this book and cannot accept any legal responsibility or liability for any errors or omissions that may be made. Cover design: VTEX, Vilnius Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)
To our wives Maja and Kumiko, our children Susanne, Ulrike, Masaki and Ayako, and our grandchildren Paul, Jonathan, Vincent, Nelson, Sofia and Jesse
Preface
Commutative algebra has developed in step with algebraic geometry and has played an essential role as the foundation of algebraic geometry. On the other hand, homological aspects of modern commutative algebra became a new and important focus of research inspired by the work of Melvin Hochster. In 1975, Richard Stanley [Sta75] proved affirmatively the upper bound conjecture for spheres by using the theory of Cohen–Macaulay rings. This created another new trend of commutative algebra, as it turned out that commutative algebra supplies basic methods in the algebraic study of combinatorics on convex polytopes and simplici
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