Progress in Commutative Algebra 1 Progress in Commutative Algebra 1 Combinatorics and Homology edited by Christopher Francisco Lee Klingler Sean Sather-Wagstaff Janet C. Vassilev De Gruyter Mathematics Subject Classification 2010 13D02, 13D40, 05E40, 13D45, 13D22, 13H10, 13A35, 13A15, 13A05, 13B22, 13F15 An electronic version of this book is freely available, thanks to the support of libra- ries working with Knowledge Unlatched. KU is a collaborative initiative designed to make high quality books Open Access. More information about the initiative can be found at www.knowledgeunlatched.org This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivs 4.0 License. For details go to http://creativecommons.org/licenses/by-nc-nd/4.0/. ISBN 978-3-11-025034-3 e-ISBN 978-3-11-025040-4 Library of Congress Cataloging-in-Publication Data A CIP catalog record for this book has been applied for at the Library of Congress. Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available in the Internet at http://dnb.dnb.de. ” 2012 Walter de Gruyter GmbH & Co. KG, Berlin/Boston Typesetting: Da-TeX Gerd Blumenstein, Leipzig, www.da-tex.de Printing: Hubert & Co. GmbH & Co. KG, Göttingen ϱ Printed on acid-free paper Printed in Germany www.degruyter.com Preface This collection of papers in commutative algebra stemmed out of the 2009 Fall South- eastern American Mathematical Society Meeting which contained three special ses- sions in the field: Special Session on Commutative Ring Theory, a Tribute to the Memory of James Brewer, organized by Alan Loper and Lee Klingler; Special Session on Homological Aspects of Module Theory, organized by Andy Kustin, Sean Sather-Wagstaff, and Janet Vassilev; and Special Session on Graded Resolutions, organized by Chris Francisco and Irena Peeva. Much of the commutative algebra community has split into two camps, for lack of a better word: the Noetherian camp and the non-Noetherian camp. Most researchers in commutative algebra identify with one camp or the other, though there are some notable exceptions to this. We had originally intended this to be a Proceedings Volume for the conference as the sessions had a nice combination of both Noetherian and non- Noetherian talks. However, the project grew into two Volumes with invited papers that are blends of survey material and new research. We hope that members from the two camps will read each others’ papers and that this will lead to increased mathematical interaction between the camps. As the title suggests, this volume, Progress in Commutative Algebra I, contains combinatorial and homological surveys. Contributions to this volume are written by speakers in the second and third sessions. To make the volume more complete, we have complemented these papers by articles on three topics which should be of broad interest: Boij–Söderburg theory, the current status of the homological conjectures and crepant resolutions. The collection represents the current trends in two of the most active areas of commutative algebra. Of course, the divisions we have outlined here are slightly artificial, given the interdependencies between these areas. For instance, much of combinatorial commutative algebra focuses on the topic of resolutions, a homological topic. So, we have not officially divided the volume into two parts. The combinatorial papers document some of the increasing focus in commutative algebra recently on the interaction between algebra and combinatorics. Specifically, one can use combinatorial techniques to investigate resolutions and other algebraic structures as with the papers of Fløystad on Boij–Söderburg theory, of Geramita, Har- bourne and Migliore and of Cooper on Hilbert Functions, of Clark on Minimal Poset Resolutions and of Mermin on Simplicial Resolutions. One can also utilize algebraic vi Preface invariants to understand combinatorial structures like graphs, hypergraphs, and sim- plicial complexes such as in the paper of Morey and Villarreal on Edge Ideals. Homological techniques have become indispensable tools for the study of Noethe- rian rings. These ideas have yielded amazing levels of interaction with other fields like algebraic topology (via differential graded techniques as well as the foundations of homological algebra), analysis (via the study of D-modules), and combinatorics (as described in the previous paragraph). The homological articles we have included in this volume relate mostly to how homological techniques help us better understand rings and singularities both Noetherian and non-Noetherian such as in the papers by Roberts, Yao, Hummel and Leuschke. Enjoy! March 2012 Sean Sather-Wagstaff Chris Francisco Lee Klingler Janet C. Vassilev Table of Contents Preface.............................................. v Gunnar Fløystad Boij–Söderberg Theory: Introduction and Survey 1 The Boij–Söderberg Conjectures . ........................ 5 1.1 Resolutions and Betti Diagrams . ........................ 5 1.2 The Positive Cone of Betti Diagrams . .................... 6 1.3 Herzog–Kühl Equations ............................. 7 1.4 Pureresolutions.................................. 8 1.5 LinearCombinationsofPureDiagrams.................... 9 1.6 The Boij–Söderberg Conjectures ........................ 11 1.7 Algorithmic Interpretation ............................ 12 1.8 Geometric Interpretation ............................. 12 2 The Exterior Facets of the Boij–Söderberg Fan and Their Supporting Hyper- planes............................................ 13 2.1 The Exterior Facets . ............................. 13 2.2 The Supporting Hyperplanes . ........................ 15 2.3 Pairings of Vector Bundles and Resolutions . ................ 21 3 The Existence of Pure Free Resolutions and of Vector Bundles with Supernat- ural Cohomology ..................................... 24 3.1 TheEquivariantPureFreeResolution..................... 24 3.2 EquivariantSupernaturalBundles....................... 29 3.3 Characteristic Free Supernatural Bundles . ................ 29 3.4 The Characteristic Free Pure Resolutions . ................ 30 3.5 Pure Resolutions Constructed from Generic Matrices ........... 33 4 Cohomology of Vector Bundles on Projective Spaces ............... 35 4.1 Cohomology Tables . ............................. 35 4.2 The Fan of Cohomology Tables of Vector Bundles . ........... 37 4.3 Facet Equations . ................................. 38 5 Extensions to Non-Cohen–Macaulay Modules and to Coherent Sheaves . 41 5.1 Betti Diagrams of Graded Modules in General ................ 42 5.2 Cohomology of Coherent Sheaves . .................... 43 6 FurtherTopics....................................... 45 6.1 The Semigroup of Betti Diagrams of Modules ................ 45 6.2 Variations on the Grading ............................ 49 viii Table of Contents 6.3 PosetStructures.................................. 50 6.4 ComputerPackages................................ 50 6.5 ThreeBasicProblems............................... 51 Anthony V. Geramita, Brian Harbourne and Juan C. Migliore Hilbert Functions of Fat Point Subschemes of the Plane: the Two-fold Way 1 Introduction . ....................................... 55 2 Approach I: Nine Double Points ............................ 58 3 ApproachI:PointsonCubics ............................. 67 4 ApproachII:PointsonCubics............................. 74 Susan Morey and Rafael H. Villarreal Edge Ideals: Algebraic and Combinatorial Properties 1 Introduction . ....................................... 85 2 Algebraic and Combinatorial Properties of Edge Ideals . ........... 86 3 Invariants of Edge Ideals: Regularity, Projective Dimension, Depth ...... 90 4 Stability of Associated Primes .............................104 Jeff Mermin Three Simplicial Resolutions 1 Introduction . .......................................127 2 Background and Notation ................................128 2.1 Algebra........................................128 2.2 Combinatorics...................................129 3 TheTaylorResolution..................................130 4 Simplicial Resolutions . .................................132 5 TheScarfComplex....................................135 6 TheLyubeznikResolutions...............................136 7 Intersections . .......................................138 8 Questions .........................................139 Timothy B. P. Clark A Minimal Poset Resolution of Stable Ideals 1 Introduction . .......................................143 2 Poset Resolutions and Stable Ideals . ........................147 3 The Shellability of PN .................................150 4 The topology of PN and properties of D.PN / ...................155 5 ProofofTheorem2.4 ..................................160 6 A Minimal Cellular Resolution of R=N .......................163 Susan M. Cooper Subsets of Complete Intersections and the EGH Conjecture 1 Introduction . .......................................167 Table of Contents ix 2 Preliminary Definitions and Results . ........................168 2.1 The Eisenbud–Green–Harris Conjecture and Complete Intersections . 168 2.2 Some Enumeration .................................171 3 Rectangular Complete Intersections . ........................173 4 SomeKeyTools .....................................177 4.1 PairsofHilbertFunctionsandMaximalGrowth...............177 4.2 Ideals Containing Regular Sequences . ....................178 5 Subsets