Commutative Algebra : Expository Papers Dedicated to David Eisenbud on the Occasion of His 65th Birthday PDF
Preview Commutative Algebra : Expository Papers Dedicated to David Eisenbud on the Occasion of His 65th Birthday
Commutative Algebra Irena Peeva Editor Commutative Algebra Expository Papers Dedicated to David Eisenbud on the Occasion of His 65th Birthday 123 Editor IrenaPeeva CornellUniversity MalottHall Ithaca,NewYork USA ISBN978-1-4614-5291-1 ISBN978-1-4614-5292-8(eBook) DOI10.1007/978-1-4614-5292-8 SpringerNewYorkHeidelbergDordrechtLondon LibraryofCongressControlNumber:2012955368 MathematicsSubjectClassification(2010):13-02 ©SpringerScience+BusinessMediaNewYork2013 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped.Exemptedfromthislegalreservationarebriefexcerptsinconnection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’slocation,initscurrentversion,andpermissionforusemustalwaysbeobtainedfromSpringer. PermissionsforusemaybeobtainedthroughRightsLinkattheCopyrightClearanceCenter.Violations areliabletoprosecutionundertherespectiveCopyrightLaw. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. While the advice and information in this book are believed to be true and accurate at the date of publication,neithertheauthorsnortheeditorsnorthepublishercanacceptanylegalresponsibilityfor anyerrorsoromissionsthatmaybemade.Thepublishermakesnowarranty,expressorimplied,with respecttothematerialcontainedherein. Printedonacid-freepaper SpringerispartofSpringerScience+BusinessMedia(www.springer.com) Preface Commutative algebra is a vibrant field with activity on many fronts and lively interactionswith otherfieldssuchas algebraicgeometry,algebraiccombinatorics, computational algebra, invariant theory, mathematical physics, noncommutative algebra,representationtheory,singularitytheory,andsubspacearrangements.There havebeentrulyexcitingrecentdevelopmentsbothincorecommutativealgebraand attheinterfacewiththeabovelistedfields. The main goalof this book is to showcase the field of commutativealgebra in expository papers, especially for the benefit of young mathematicians. This book willaidthereaderstobroadentheirbackgroundandgaindeeperunderstandingof thecurrentresearchinthearea. AllpapersarededicatedtoDavidEisenbudincelebrationofhismanyandinspiring contributionstoabroadrangeoftopics. Currently, David Eisenbud is a professor of mathematics at the University of California,Berkeley.HereceivedhisPh.D.inmathematicsin1970attheUniversity of Chicago under Saunders MacLane and Chris Robson. He was director of the Mathematical Sciences Research Institute (MSRI) from 1997 to 2007 and vice presidentformathematicsandthephysicalsciencesattheSimonsFoundationfrom 2009to 2012.From2003to 2005David Eisenbudwas presidentofthe American MathematicalSociety.In 2006he waselecteda fellow ofthe AmericanAcademy ofArtsandSciences. Ithaca,NY,USA IrenaPeeva PeevawaspartiallysupportedbyNSFgrantDMS-1100046. v Contents Lazarsfeld–MukaiBundlesandApplications................................. 1 MarianAprodu SomeApplicationsofCommutativeAlgebratoStringTheory............. 25 PaulS.Aspinwall MeasuringSingularitieswithFrobenius:TheBasics ........................ 57 Ange´licaBenito,EleonoreFaber,andKarenE.Smith ThreeFlavorsofExtremalBettiTables........................................ 99 ChristineBerkesch,DanielErman,andManojKummini p(cid:2)1-LinearMapsinAlgebraandGeometry.................................. 123 ManuelBlickleandKarlSchwede Castelnuovo–MumfordRegularityofAnnihilators, ExtandTorModules ............................................................ 207 MarkusBrodmann,CaoHuyLinh,andMaria-HelenaSeiler SelectionsfromtheLetter-PlacePanoply ..................................... 237 DavidA.Buchsbaum KoszulAlgebrasandRegularity................................................ 285 AldoConca,EmanuelaDeNegri,andMariaEvelinaRossi PowersofIdeals:BettiNumbers,CohomologyandRegularity............. 317 MarcChardin SomeHomologicalPropertiesofModulesoveraComplete Intersection,withApplications................................................. 335 HailongDao PowersofSquare-FreeMonomialIdealsandCombinatorics............... 373 ChristopherA.Francisco,HuyTa`iHa`,andJeffreyMermin vii viii Contents ABriefHistoryofOrderIdeals ................................................ 393 E.GrahamEvansandPhillipGriffith ModuliofAbelianVarieties,Vinberg(cid:2)-Groups,andFreeResolutions ... 419 LaurentGruson,StevenV.Sam,andJerzyWeyman F-Purity,FrobeniusSplitting,andTightClosure ............................ 471 MelvinHochster Hilbert–KunzMultiplicityandtheF-Signature.............................. 485 CraigHuneke PureO-Sequences:KnownResults,Applications,andOpenProblems ... 527 JuanMigliore,UweNagel,andFabrizioZanello BoundingProjectiveDimension................................................ 551 JasonMcCulloughandAlexandraSeceleanu Brauer–ThrallTheoryforMaximalCohen–MacaulayModules........... 577 GrahamJ.LeuschkeandRogerWiegand TightClosure’sFailuretoLocalize-aSelf-ContainedExposition......... 593 PaulMonsky IntroductiontotheHyperdeterminantandtotheRank ofMultidimensionalMatrices .................................................. 609 GiorgioOttaviani CommutativeAlgebraofSubspaceandHyperplaneArrangements....... 639 HalSchenckandJessicaSidman CohomologicalDegreesandApplications..................................... 667 WolmerV.Vasconcelos Lazarsfeld–Mukai Bundles and Applications MarianAprodu Introduction Lazarsfeld–Mukai bundles appeared naturally in connection with two completely different important problems in algebraic geometry from the 1980s. The first problem, solved by Lazarsfeld, was to find explicit examples of smooth curves which are generic in the sense of Brill–Noether–Petri [18]. The second problem was the classification of prime Fano manifolds of coindex 3 [23]. More recently, Lazarsfeld–Mukai bundles have found applications to syzygies and higher-rank Brill–Noethertheory. The common feature of all these research topics is the central role played by K3surfacesandtheir hyperplanesections.For the Brill–Noether–Petrigenericity, Lazarsfeldprovesthata generalcurvein a linear system thatgeneratesthe Picard groupofaK3surfacesatisfies thiscondition.FortheclassificationofprimeFano manifoldsof coindex3, after having provedthe existence of smooth fundamental divisors,oneusesthegeometryofatwo-dimensionallinearsectionwhichisavery generalK3surface. The idea behind this definition is that the Brill–Noether theory of smooth curvesonaK3surface,alsocalledK3sections,isgovernedbyhigher-rankvector bundles on the surface. To be more precise, consider S a K3 surface (considered alwaystobe smooth,complex,projective),C a smoothcurveonS ofgenus(cid:2) 2, and jAj a base-point-free pencil on C. If we attempt to lift the linear system jAj to the surface S, in most cases, we will fail. For instance, jAj cannot lift to a pencil on S if C generates Pic.S/ or if S does not contain any elliptic curve at M.Aprodu((cid:2)) RomanianAcademy,InstituteofMathematics“SimionStoilow”P.O.Box1-764, RO014700,Bucharest,Romania S¸coalaNormala˘Superioara˘Bucures¸ti,CaleaGrivi¸tei21,RO-010702,Bucharest,Romania e-mail:[email protected] I.Peeva(ed.),CommutativeAlgebra:ExpositoryPapersDedicatedtoDavidEisenbud 1 ontheOccasionofHis65thBirthday,DOI10.1007/978-1-4614-5292-8 1, ©SpringerScience+BusinessMediaNewYork2013
The list of books you might like
