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http://dx.doi.org/10.1090/gsm/087 Twenty-Four Hours of Local Cohomology This page intentionally left blank Twenty-Four Hours of Local Cohomology Srikanth B. Iyengar Graham J. Leuschke Anton Leykln Claudia Miller Ezra Miller Anurag K. Singh Uli Walther Graduate Studies in Mathematics Volume 87 |p^S|\| American Mathematical Society %\yyyw^/? Providence, Rhode Island Editorial Board David Cox (Chair) Walter Craig N. V.Ivanov Steven G. Krantz The book is an outgrowth of the 2005 AMS-IMS-SIAM Joint Summer Research Con• ference on "Local Cohomology and Applications" held at Snowbird, Utah, June 20-30, 2005, with support from the National Science Foundation, grant DMS-9973450. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation. 2000 Mathematics Subject Classification. Primary 13A35, 13D45, 13H10, 13N10, 14B15; Secondary 13H05, 13P10, 13F55, 14F40, 55N30. For additional information and updates on this book, visit www.ams.org/bookpages/gsm-87 Library of Congress Cataloging-in-Publication Data Twenty-four hours of local cohomology / Srikanth Iyengar.. [et al.]. p. cm. — (Graduate studies in mathematics, ISSN 1065-7339 ; v. 87) Includes bibliographical references and index. ISBN 978-0-8218-4126-6 (alk. paper) 1. Sheaf theory. 2. Algebra, Homological. 3. Group theory. 4. Cohomology operations. I. Iyengar, Srikanth, 1970- II. Title: 24 hours of local cohomology. QA612.36.T94 2007 514/.23—dc22 2007060786 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294, USA. Requests can also be made by e-mail to [email protected]. © 2007 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. @ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10 9 8 7 6 5 4 3 2 1 12 11 10 09 08 07 To our teachers This page intentionally left blank Contents Preface Introduction Lecture 1. Basic Notions §1. Algebraic sets §2. Krull dimension of a ring §3. Dimension of an algebraic set §4. An extended example §5. Tangent spaces and regular rings §6. Dimension of a module Lecture 2. Cohomology §1. Sheaves §2. Cech cohomology §3. Calculus versus topology §4. Cech cohomology and derived functors Lecture 3. Resolutions and Derived Functors §1. Free, projective, and flat modules §2. Complexes §3. Resolutions §4. Derived functors Lecture 4. Limits §1. An example from topology Vlll Contents §2. Direct limits §3. The category of diagrams §4. Exactness §5. Diagrams over diagrams §6. Filtered posets §7. Diagrams over the pushout poset §8. Inverse limits Lecture 5. Gradings, Filtrations, and Grobner Bases §1. Filtrations and associated graded rings §2. Hilbert polynomials §3. Monomial orders and initial forms §4. Weight vectors and flat families §5. Buchberger's algorithm §6. Grobner bases and syzygies Lecture 6. Complexes from a Sequence of Ring Elements §1. The Koszul complex §2. Regular sequences and depth: a first look §3. Back to the Koszul complex §4. The Cech complex Lecture 7. Local Cohomology §1. The torsion functor §2. Direct limit of Ext modules §3. Direct limit of Koszul cohomology §4. Return of the Cech complex Lecture 8. Auslander-Buchsbaum Formula and Global Dimension §1. Regular sequences and depth redux §2. Global dimension §3. Auslander-Buchsbaum formula §4. Regular local rings §5. Complete local rings Lecture 9. Depth and Cohomological Dimension §1. Depth §2. Cohomological dimension Contents IX §3. Arithmetic rank Lecture 10. Cohen-Macaulay Rings §1. Noether normalization §2. Intersection multiplicities §3. Invariant theory §4. Local cohomology Lecture 11. Gorenstein Rings §1. Bass numbers §2. Recognizing Gorenstein rings §3. Injective resolutions of Gorenstein rings §4. Local duality §5. Canonical modules Lecture 12. Connections with Sheaf Cohomology §1. Sheaf theory §2. Flasque sheaves §3. Local cohomology and sheaf cohomology Lecture 13. Projective Varieties §1. Graded local cohomology §2. Sheaves on projective varieties §3. Global sections and cohomology Lecture 14. The Hartshorne-Lichtenbaum Vanishing Theorem Lecture 15. Connectedness §1. Mayer-Vietoris sequence §2. Punctured spectra Lecture 16. Polyhedral Applications §1. Polytopes and faces §2. Upper bound theorem §3. The h-vector of a simplicial complex §4. Stanley-Reisner rings §5. Local cohomology of Stanley-Reisner rings §6. Proof of the upper bound theorem Lecture 17. D-modules X Contents §1. Rings of differential operators §2. The Weyl algebra §3. Holonomic modules §4. Grobner bases Lecture 18. Local Duality Revisited §1. Poincare duality §2. Grothendieck duality §3. Local duality §4. Global canonical modules Lecture 19. De Rham Cohomology §1. The real case: de Rham's theorem §2. Complex manifolds §3. The algebraic case §4. Local and de Rham cohomology Lecture 20. Local Cohomology over Semigroup Rings §1. Semigroup rings §2. Cones from semigroups §3. Maximal support: the Ishida complex §4. Monomial support: Zd-graded injectives §5. Hartshorne's example Lecture 21. The Frobenius Endomorphism §1. Homological properties §2. Frobenius action on local cohomology modules §3. A vanishing theorem Lecture 22. Curious Examples §1. Dependence on characteristic §2. Associated primes of local cohomology modules Lecture 23. Algorithmic Aspects of Local Cohomology §1. Holonomicity of localization §2. Local cohomology as a D-module §3. Bernstein-Sato polynomials §4. Computing with the Frobenius morphism Lecture 24. Holonomic Rank and Hyper geometric Systems Contents xi §1. GKZ A-hypergeometric systems 247 §2. Rank vs. volume 250 §3. Euler-Koszul homology 251 §4. Holonomic families 254 Appendix. Injective Modules and Matlis Duality 257 §1. Essential extensions 257 §2. Noetherian rings 260 §3. Artinian rings 263 §4. Matlis duality 265 Bibliography 269 Index 277 This page intentionally left blank Preface This book is an outgrowth of the summer school Local cohomology and its interactions with algebra, geometry, and analysis that we organized in June 2005 in Snowbird, Utah. This was a joint program under the AMS-IMS- SIAM Summer Research Conference series and the MSRI Summer Graduate Workshop series. The school centered around local cohomology, and was in• tended for graduate students interested in various branches of mathematics. It consisted of twenty-four lectures by the authors of this book, followed by a three-day conference. We thank our co-authors for their support at all stages of the workshop. In addition to preparing and delivering the lectures, their enthusiastic par• ticipation, and interaction with the students, was critical to the success of the event. We also extend our hearty thanks to Wayne Drady, the AMS conference coordinator, for cheerful and superb handling of various details. We profited greatly from the support and guidance of David Eisenbud and Hugo Rossi at MSRI, and Jim Maxwell at AMS. We express our thanks to them, and to our Advisory Committee: Mel Hochster, Craig Huneke, Joe Lipman, and Paul Roberts. We are also indebted to the conference speakers: Markus Brodmann, Ragnar-Olaf Buchweitz, Phillippe Gimenez, Gennady Lyubeznik, Paul Roberts, Peter Schenzel, Rodney Sharp, Ngo Viet Trung, Kei-ichi Watanabe, and Santiago Zarzuela. Finally, we thank the AMS and the MSRI for their generous support in hosting this summer school, and the AMS for publishing this revised version of the "Snowbird notes". Anurag K. Singh and Uli Walt her xiii This page intentionally left blank Introduction Local cohomology was invented by Grothendieck to prove Lefschetz-type theorems in algebraic geometry. This book seeks to provide an introduction to the subject which takes cognizance of the breadth of its interactions with other areas of mathematics. Connections are drawn to topological, geo• metric, combinatorial, and computational themes. The lectures start with basic notions in commutative algebra, leading up to local cohomology and its applications. They cover topics such as the number of defining equations of algebraic sets, connectedness properties of algebraic sets, connections to sheaf cohomology and to de Rham cohomology, Grobner bases in the com• mutative setting as well as for D-modules, the Probenius morphism and characteristic p methods, finiteness properties of local cohomology modules, semigroup rings and polyhedral geometry, and hypergeometric systems aris• ing from semigroups. The subject can be introduced from various perspectives. We start from an algebraic one, where the definition is elementary: given an ideal a in a Noetherian commutative ring, for each module consider the submodule of elements annihilated by some power of a. This operation is not exact, in
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