Combinatorics and Commutative Algebra
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Springer Science+Business Media, LLC
Richard P. Stanley
Combinatorics and Commutative Algebra
Springer Science+Business Media, LLC 1983
Author: Richard P. Stanley Mathematics Department, 2-375 Massachusetts Institute of Technology Cambridge, MA 02139
Library of Congress Cataloging in PubUcation Data Stanley, Richard P., 1944Combinatorics and commutative algebra. (Progress in mathematics ; v. 41) Bibliography: p. 1. Commutative algebra. 2. Combinatorial analysis. I. Title. 11. Series: Progress in mathematics ; 41. QA251.3.S72 1983 512'.24 83-17915
CIP-Kurztitelaufnahme der Deutschen Bibliothek StanIey, Ricbard P.: Combinatorics and commutative algebra/ Richard P. Stanley. - Boston; Basel; Stuttgart : Birkhäuser, 1983. (Progress in mathematics ; 41) NE:GT ISBN 978-0-8176-3112-3 ISBN 978-1-4899-6752-7 (eBook) DOI 10.1007/978-1-4899-6752-7 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without prior permission of the copyright owner.
© Springer Science+Business Media New York 1983 Originally published by Birkhäuser Boston, Inc in1983. Softcover reprint of the hardcover 1st edition 1983 ABCDEFGHIJ
TABlE OF CONTENTS Preface Notation Chapter 0
vii viii BACKGROUND § §
§
Chapter I
10 2. 3 0
7 22
NONNEGATIVE INTEGRAL SOLUTIONS TO LINEAR EQUATIONS l. 2. 3.
4. 5. 6.
7. 8. §
9.
§1O. § ll. § 12. § 13.
Chapter II
Combinatorics Commutative algebra and homological algebra Topo 1ogy
Integer stochastic matrices (magie squares) Graded algebras and modules Elementary aspects of JIJ -sol utions to linear equations Integer stochastic matrices again Dimension, depth, and Cohen-Macaulay modules local cOhomology Local cohomology of the modules Mffi'>',0: Reci proc ity Reciprocity for integer stochastic matrices Rational points in integral polytopes Free resolutions Duality and canonical modules A final look at linear equations
30 31
33 37 39 43 45
50
51
52 54
56 60
THE FACE RING OF f\ SH1PLICIAL COMPLEX 1. 2. 3. 4. 5. 6. 7. 8.
Elementary properties of the face ring f-vectors and h-vectors of complexes and multicomplexes Cohen-Macaulay complexes and the Upper Bound Conjecture Homological properties of face rings Gorenstein face rings Gorenstein Hilbert functions Canonical modules of face rings Buchsbaum complexes
62 64 68 70 74 77 80 84
86
References
v
PREFACE
These notes are based on aseries of eight lectures given at the University of Stockholm during April and May, 1981.
They were intended
to give an overview of two topics from "combinatorial commutative algebra", viz., (1) solutions to 1 inear equations in nonnegative . integers (wh ich is equivalent to the theory of invariants of a torus acting 1 inearly on a polynomial ring), and (2) the face ring of a simpl i ci al compl ex.
In order to give a broad perspect ive many detai ls
and special ized topics have been regretfully omitted.
In general,
proofs have been provided only for those res
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