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Universitext Universitext Series editors Sheldon Axler San Francisco State University Carles Casacuberta Universitat de Barcelona Angus MacIntyre Queen Mary University of London Kenneth Ribet University of California, Berkeley Claude Sabbah École polytechnique, CNRS, Université Paris-Saclay, Palaiseau Endre Süli University of Oxford Wojbor A. Woyczyński Case Western Reserve University Universitext is a series of textbooks that presents material from a wide variety of mathematical disciplines at master’s level and beyond. The books, often well class-tested by their author, may have an informal, personal even experimental approach to their subject matter. Some of the most successful and established books in the series have evolved through several editions, always following the evolution of teaching curricula, into very polished texts. Thus as research topics trickle down into graduate-level teaching, first textbooks written for new, cutting-edge courses may make their way into Universitext. More information about this series at http://www.springer.com/series/223 W. Frank Moore • Mark Rogers Sean Sather-Wagstaff Monomial Ideals and Their Decompositions 123 W. Frank Moore Sean Sather-Wagstaff Department of Mathematics School of Mathematical and Statistical Wake Forest University Sciences Winston-Salem, NC, USA Clemson University Clemson, SC, USA Mark Rogers Department of Mathematics Missouri State University Springfield, MO, USA ISSN 0172-5939 ISSN 2191-6675 (electronic) Universitext ISBN 978-3-319-96874-2 ISBN 978-3-319-96876-6 (eBook) https://doi.org/10.1007/978-3-319-96876-6 Library of Congress Control Number: 2018948828 Mathematics Subject Classification (2010): 13-01, 05E40, 13-04, 13F20, 13F55 © Springer Nature Switzerland AG 2018 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, 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. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland Dedicated to the memory of Diana Taylor, with gratitude and respect. Contents Introduction................................................. xi Overview................................................. xii Audience ................................................. xiii Summary of Contents ....................................... xiv Notes for the Instructor/Independent Reader ...................... xvii Possible Course Outlines..................................... xviii Acknowledgments .......................................... xxiii Part I Monomial Ideals 1 Fundamental Properties of Monomial Ideals .................. 5 1.1 Monomial Ideals ................................... 5 1.2 Integral Domains (optional) ........................... 13 1.3 Generators of Monomial Ideals ......................... 15 1.4 Noetherian Rings (optional) ........................... 24 1.5 Exploration: Counting Monomials ....................... 28 1.6 Exploration: Numbers of Generators ..................... 31 2 Operations on Monomial Ideals ............................ 33 2.1 Intersections of Monomial Ideals ....................... 34 2.2 Unique Factorization Domains (optional).................. 40 2.3 Monomial Radicals ................................. 48 2.4 Exploration: Reduced Rings ........................... 56 2.5 Colons of Monomial Ideals ........................... 59 2.6 Bracket Powers of Monomial Ideals ..................... 64 2.7 Exploration: Saturation ............................... 69 2.8 Exploration: Generalized Bracket Powers ................. 73 2.9 Exploration: Comparing Bracket Powers and Ordinary Powers .......................................... 77 vii viii Contents 3 M-Irreducible Ideals and Decompositions ..................... 81 3.1 M-Irreducible Monomial Ideals ......................... 81 3.2 Irreducible Ideals (optional) ........................... 87 3.3 M-Irreducible Decompositions ......................... 95 3.4 Irreducible Decompositions (optional) .................... 102 3.5 Exploration: Decompositions in Two Variables, part I ........ 106 Part II Monomial Ideals and Other Areas 4 Connections with Combinatorics ............................ 115 4.1 Square-Free Monomial Ideals .......................... 115 4.2 Graphs and Edge Ideals .............................. 121 4.3 Decompositions of Edge Ideals ......................... 125 4.4 Simplicial Complexes and Stanley-Reisner Ideals ........... 131 4.5 Decompositions of Stanley-Reisner Ideals ................. 139 4.6 Facet Ideals and Their Decompositions ................... 146 4.7 Exploration: Alexander Duality ......................... 152 5 Connections with Other Areas ............................. 161 5.1 Krull Dimension ................................... 161 5.2 Vertex Covers and PMU Placement ..................... 165 5.3 Cohen-Macaulayness and the Upper Bound Theorem ......... 175 5.4 Hilbert Functions and Initial Ideals ...................... 188 5.5 Resolutions of Monomial Ideals ........................ 199 Part III Decomposing Monomial Ideals 6 Parametric Decompositions of Monomial Ideals ................ 221 6.1 Parameter Ideals and Parametric Decompositions ............ 221 6.2 Corner Elements ................................... 228 6.3 Finding Corner Elements in Two Variables ................ 241 6.4 Finding Corner Elements in General ..................... 246 6.5 Exploration: Decompositions in Two Variables, part II ....... 252 6.6 Exploration: Decompositions of Some Powers of Ideals ....... 253 6.7 Exploration: Macaulay Inverse Systems ................... 256 7 Computing M-Irreducible Decompositions .................... 261 7.1 M-Irreducible Decompositions of Monomial Radicals ........ 261 7.2 M-Irreducible Decompositions of Bracket Powers ........... 264 7.3 M-Irreducible Decompositions of Sums ................... 267 7.4 M-Irreducible Decompositions of Colon Ideals ............. 271 7.5 Computing General M-Irreducible Decompositions .......... 276 7.6 Exploration: Edge, Stanley-Reisner, and Facet Ideals Revisited ......................................... 283 7.7 Exploration: Decompositions of Saturations ................ 285 Contents ix 7.8 Exploration: Decompositions of Generalized Bracket Powers .......................................... 287 7.9 Exploration: Decompositions of Products of Monomial Ideals ........................................... 289 Part IV Commutative Algebra and Macaulay2 Appendix A: Foundational Concepts ........................... 297 A.1 Rings ........................................... 297 A.2 Polynomial Rings .................................. 302 A.3 Ideals and Generators ................................ 306 A.4 Sums of Ideals ..................................... 310 A.5 Products and Powers of Ideals ......................... 312 A.6 Colon Ideals ...................................... 315 A.7 Radicals of Ideals .................................. 316 A.8 Quotient Rings .................................... 319 A.9 Partial Orders and Monomial Orders ..................... 323 A.10 Exploration: Algebraic Geometry ....................... 327 Appendix B: Introduction to Macaulay2 ........................ 331 B.1 Rings ........................................... 331 B.2 Polynomial Rings .................................. 333 B.3 Ideals and Generators ................................ 335 B.4 Sums of Ideals ..................................... 337 B.5 Products and Powers of Ideals ......................... 338 B.6 Colon Ideals ...................................... 339 B.7 Radicals of Ideals .................................. 340 B.8 Quotient Rings .................................... 342 B.9 Monomial Orders ................................... 343 Further Reading ............................................. 349 References .................................................. 351 Index of Macaulay2 Commands, by Command .................... 355 Index of Macaulay2 Commands, by Description ................... 361 Index of Names .............................................. 371 Index of Symbols............................................. 373 Index of Terminology ......................................... 377 Introduction The Fundamental Theorem of Arithmetic states that every integer n>2 factors into a product of prime numbers in an essentially unique way. In algebra class, one learns a similar factorization result for polynomials in one