Commutative Algebra Geometric, Homological, Combinatorial and Computational Aspects PURE AND APPLIED MATHEMATICS A Program of Monographs, Textbooks, and Lecture Notes EXECUTIVE EDITORS Earl J. Taft Zuhair Nashed Rutgers University University of Central Florida New Brunswick, New Jersey Orlando, Florida EDITORIAL BOARD M. S. Baouendi Anil Nerode University of California, Cornell University San Diego Donald Passman Jane Cronin University of Wisconsin, Rutgers University Madison Jack K. Hale Fred S. Roberts Georgia Institute of Technology Rutgers University S. Kobayashi David L. Russell University of California, Virginia Polytechnic Institute Berkeley and State University Marvin Marcus Walter Schempp University of California, Universität Siegen Santa Barbara Mark Teply W. S. Massey University of Wisconsin, Yale University Milwaukee LECTURE NOTES IN PURE AND APPLIED MATHEMATICS Recent Titles G. Lumer and L. Weis, Evolution Equations and Their Applications in Physical and Life Sciences J. Cagnol et al., Shape Optimization and Optimal Design J. Herzog and G. Restuccia, Geometric and Combinatorial Aspects of Commutative Algebra G. Chen et al., Control of Nonlinear Distributed Parameter Systems F. Ali Mehmeti et al., Partial Differential Equations on Multistructures D. D. Anderson and I. J. Papick, Ideal Theoretic Methods in Commutative Algebra Á. Granja et al., Ring Theory and Algebraic Geometry A. K. Katsaras et al., p-adic Functional Analysis R. Salvi, The Navier-Stokes Equations F. U. Coelho and H. A. Merklen, Representations of Algebras S. Aizicovici and N. H. Pavel, Differential Equations and Control Theory G. Lyubeznik, Local Cohomology and Its Applications G. Da Prato and L. Tubaro, Stochastic Partial Differential Equations and Applications W. A. Carnielli et al., Paraconsistency A. Benkirane and A. Touzani, Partial Differential Equations A. Illanes et al., Continuum Theory M. Fontana et al., Commutative Ring Theory and Applications D. Mond and M. J. Saia, Real and Complex Singularities V. Ancona and J. Vaillant, Hyperbolic Differential Operators and Related Problems G. R. Goldstein et al., Evolution Equations A. Giambruno et al., Polynomial Identities and Combinatorial Methods A. Facchini et al., Rings, Modules, Algebras, and Abelian Groups J. Bergen et al., Hopf Algebras A. C. Krinik and R. J. Swift, Stochastic Processes and Functional Analysis: A Volume of Recent Advances in Honor of M. M. Rao S. Caenepeel and F. van Oystaeyen, Hopf Algebras in Noncommutative Geometry and Physics J. Cagnol and J.-P. Zolésio, Control and Boundary Analysis S. T. Chapman, Arithmetical Properties of Commutative Rings and Monoids O. Imanuvilov, et al., Control Theory of Partial Differential Equations Corrado De Concini, et al., Noncommutative Algebra and Geometry Alberto Corso, et al., Commutative Algebra: Geometric, Homological, Combinatorial and Computational Aspects Giuseppe Da Prato and Luciano Tubaro, Stochastic Partial Differential Equations and Applications – VII Thiscollectivevolumeisdedicatedto WOLMERV. VASCONCELOS forbeingsuchawonderfulmathematician,mentor,colleagueandfriend. iii Commutative Algebra Geometric, Homological, Combinatorial and Computational Aspects edited by Alberto Corso University of Kentucky Lexington, Kentucky Philippe Gimenez Universidad de Valladolid Valladolid, Spain Maria Vaz Pinto Instituto Superior Tecnico Lisbon, Portugal Santiago Zarzuela Universitat de Barcelona Barcelona, Spain Boca Raton London New York Singapore DK2960_Discl.fm Page 1 Wednesday, June 22, 2005 8:19 AM Published in 2006 by CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2006 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group No claim to original U.S. Government works Printed in the United States of America on acid-free paper 10 9 8 7 6 5 4 3 2 1 International Standard Book Number-10: 0-8247-2335-X (Softcover) International Standard Book Number-13: 978-0-8247-2335-4 (Softcover) Library of Congress Card Number 2005049112 This book contains information obtained from authentic and highly regarded sources. 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For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Library of Congress Cataloging-in-Publication Data Commutative algebra : geometric, homological, combinatorial, and computational aspects / edited by Alberto Corso …[et al.]. p. cm. – (Lecture notes in pure and applied mathematics ; v.244) Includes bibliographical references and index. ISBN 0-8247-2335-X (alk. Paper) 1. Commutative algebra—Congresses. I. Corso, Alberto. II. Series. QA251.3.C6537 2005 512'.44—dc22 2005049112 Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com Taylor & Francis Group and the CRC Press Web site at is the Academic Division of T&F Informa plc. http://www.crcpress.com Preface Thelastdecadeshaveseenagreatdealofactivitiesanddevelopmentsinseveral areas at the interface of commutative algebra and algebraic geometry. Excit- ingnewresultshavebeenachieved“borrowing”methodspropertoneighboring fields such as combinatorics, homological algebra, polyhedral geometry, sym- bolic computation, and topology. This volume – Commutative Algebra: Geo- metric, Homological, Combinatorial and Computational Aspects – would like to highlight some of these proficuous interactions by gathering a collection of refereedresearchpaperswhichencompassseveralofthoseaspects. Blowupal- gebras,Castelnuovo-Mumfordregularity,integralclosureandnormality,Koszul homology, liaison theory, multiplicities, polarization, and reductions of ideals are some of the topicsfeaturedin the fifteen originalresearch articles included inthisvolume. Threesurveyarticlesontopicsofcurrentinterestalsoenrichthe publication: Poincare´ series of singularities, uniformArtin-Rees theorems, and Gorenstein rings. Most of the results published in this volume were presented duringtwomeetingsthatwereheldduringthemonthofJune2003,inSpainand Portugal.Namely, CommutativeAlgebra:Geometric,Homological,CombinatorialandCom- (cid:0) putationalAspects,oneoftheSpecialSessionsheldduringtheFirstJoint Meeting of the American Mathematical Society and the Real Sociedad Matema´ticaEspan˜olainSevilla,Spain,intheperiodJune18-21,2003; LisbonConferenceonCommutativeAlgebra,heldattheInstitutoSuperior (cid:0) Te´cnicoinLisbon,Portugal,intheperiodJune23-27,2003. These conferencesare partofa long-establishedtraditionofWestern European summer meetingsin commutativealgebraand algebraicgeometry. Besides the traditionalopportunityfor expertsfrom the U.S. and Western Europe to gather v vi Preface and exchange the most recent discoveries, special attention was paid to allow fortheparticipationofscholarsfromunderrepresentedcountries. Overall,more thanonehundredparticipantscametogetherfromnumerouspartsoftheWorld: Brazil,Canada,France,Germany,Greece,India,Iran,Italy,Japan,Mexico,The Netherlands,Norway,Portugal,Romania,Spain,Switzerland,UnitedKingdom, UnitedStatesofAmerica,andVietnam. Thiscertainlyenhancedthebreathand scopeofthetwoevents. Needlesstosay,wewouldliketoexpressourgratitudetothecontributorsofthis volumefortheirenthusiasmintheproject.Theanonymousreferees,whoworked verycloselywithus,alsodeserveaspecialcreditforalltheirtimespentinread- ingandcorrectingtheoriginalmanuscripts: Weareawareofthemanydemands on ourtime thatthe academicprofessionrequiresfromeach of us. Finally, we would also like to take the opportunitythat this Preface offersus to personally acknowledgethemanycolleaguesandinstitutionsthatcontributedtothesuccess ofthesetwoconferences.First,wewouldliketoexpressourthankstoJoanElias (Universitatde Barcelona), John Greenlees (University of Sheffield), Giuseppe Valla(Universita`diGenova),WolmerV.Vasconcelos,andCharlesWeibel(Rut- gersUniversity)forservingasmembersoftheScientificCommitteeoftheLis- bonConference. AspecialmentiongoestoPedroFerreiradosSantos,Gustavo Granja,MichaelPaluchandtheentirestaffoftheInstitutoSuperiorTe´cnicofor theirendlesshelpandsupportinplanningandorganizingtheLisbonConference from the very first stage. Finally, we wish to express our heartfelt gratitude to thefollowinginstitutionsfortheirgenerousfinancialsupport: BancoBPI,Cen- tro de Ana´lise Matema´tica, Geometria e Sistemas Dinaˆmicos (Instituto Supe- rior Te´cnico), Fundac¸a˜o Calouste Gulbenkian, Fundac¸a˜o Luso-Americanapara oDesenvolvimento,Fundac¸a˜oOriente,Fundac¸a˜oparaaCieˆnciaeaTecnologia, UniversitatdeBarcelonaandUniversidaddeValladolid. TheEditors, ALBERTO CORSO Lexington USA [email protected] (cid:1) (cid:1) (cid:1) PHILIPPE GIMENEZ Valladolid Spain [email protected] (cid:1) (cid:1) (cid:1) MARIA VAZ PINTO Lisbon Portugal [email protected] (cid:1) (cid:1) (cid:1) SANTIAGO ZARZUELA Barcelona Spain [email protected] (cid:1) (cid:1) (cid:1) March15,2005 Contents Dedication iii Preface v Contents vii Participants ix ListofSevilla’sTalks xv ListofLisbon’sTalks xvii AtheoremofEakinandSathayeandGreen’shyperplane restrictiontheorem GIULIO CAVIGLIA 1 Liaisonofvarietiesofsmalldimensionanddeficiencymodules MARC CHARDIN 7 Regularityjumpsforpowersofideals ALDO CONCA 21 Integralclosureofidealsandannihilatorsofhomology ALBERTO CORSO, CRAIG HUNEKE, DANIEL KATZandWOLMER V. VASCONCELOS 33 Poincare´seriesofsurfacesingularities STEVENDALECUTKOSKY 49 vii viii Contents Multi-gradedHilbertcoefficients CLARE D’CRUZandANNA GUERRIERI 59 Torsionfreenessandnormalityofblowupringsofmonomialideals CESARA. ESCOBAR, RAFAEL H.VILLARREALandYUJIYOSHINO 69 Monomialidealsviasquare-freemonomialideals SARA FARIDI 85 WhendoestheequalityI2 QIholdtrueinBuchsbaumrings? (cid:0) SHIRO GOTOandHIDETOSAKURAI 115 Integralclosuresofidealsincompletionsofregularlocaldomains WILLIAMHEINZER, CHRISTEL ROTTHAUSandSYLVIAWIEGAND 141 Thecomponentsofavarietyofmatriceswithsquarezero andsubmaximalrank DIKRAN KARAGUEUZIAN, BOB OLIVERandJOANA VENTURA 151 Thesupportoftopgradedlocalcohomologymodules MORDECHAI KATZMAN 165 AnapproachtotheuniformArtin-Reestheoremsfrom thenotionofrelationtype FRANCESC PLANAS-VILANOVA 175 Castelnuovo-MumfordregularityandfinitenessofHilbertfunctions MARIA EVELINA ROSSI, NGOˆ VIEˆTTRUNGandGIUSEPPEVALLA 193 Differentialidealizersandalgebraicfreedivisors ARON SIMIS 211 Gorensteinringscallthetune ANNE-MARIE SIMONandJAN R. STROOKER 227 Onfreeintegralextensionsgeneratedbyoneelement IRENASWANSONandORLANDO VILLAMAYOR 239 DegenerationandG-dimensionofmodules YUJIYOSHINO 259