Homological and Computational Methods in Commutative Algebra: Dedicated to Winfried Bruns on the Occasion of his 70th Birthday PDF
Preview Homological and Computational Methods in Commutative Algebra: Dedicated to Winfried Bruns on the Occasion of his 70th Birthday
Springer INdAM Series 20 Aldo Conca Joseph Gubeladze Tim Römer Editors Homological and Computational Methods in Commutative Algebra Dedicated to Winfried Bruns on the Occasion of his 70th Birthday Springer INdAM Series Volume 20 Editor-in-Chief G.Patrizio SeriesEditors C.Canuto G.Coletti G.Gentili A.Malchiodi P.Marcellini E.Mezzetti G.Moscariello T.Ruggeri Moreinformationaboutthisseriesathttp://www.springer.com/series/10283 Aldo Conca • Joseph Gubeladze (cid:129) Tim RoRmer Editors Homological and Computational Methods in Commutative Algebra Dedicated to Winfried Bruns on the Occasion of his 70th Birthday 123 Editors AldoConca JosephGubeladze DipartimentodiMatematica DepartmentofMathematics UniversitaJdiGenova SanFranciscoStateUniversity Genova,Italy SanFrancisco,CA,USA TimRoRmer FBMathematik/Informatik UniversitaRtOsnabruRck OsnabruRck,Germany ISSN2281-518X ISSN2281-5198 (electronic) SpringerINdAMSeries ISBN978-3-319-61942-2 ISBN978-3-319-61943-9 (eBook) DOI10.1007/978-3-319-61943-9 LibraryofCongressControlNumber:2017951038 ©SpringerInternationalPublishingAG2017 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped. 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Printedonacid-freepaper ThisSpringerimprintispublishedbySpringerNature TheregisteredcompanyisSpringerInternationalPublishingAG Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland Foreword WinfriedBrunswasbornonMay5,1946,atOkernearGoslarin Germany.From 1956 to 1965, he attended the classical gymnasium at Goslar whose focus is on oldlanguages—LatinandGreek.Afterhisgraduationfromschool,heservedinthe armyfortwoyears,andthenstartedstudyingmathematicsatTUHannoverin1967. Duringhisstudiesanduntilhisdiploma,hewassupportedbyadistinguishedgrant forexcellentstudentsfromtheStudienstiftungdesDeutschenVolkes. After his graduation, he became scientific assistant at TU Clausthal where, in 1972hereceivedhisPhDforhisdissertationBeispielereflexiverDifferentialmoduln withUdoVetterashisadvisor.Largepartsofhisdissertationwerelaterpublished inJournalfürreineundangewandteMathematik. At that time basic elementtheory had become verypopular,as it turnedout to beanimportanttooltostudygeneratingsetsofmodules.Byextendingtechniques on basic elements, as developed by Eisenbud and Evans in their influential paper “Generating modules efficiently: Theorems from algebraic K-theory” Winfried proved a beautiful and surprising theorem in his paper “Jede” endliche freie Auflösungist freie AuflösungeinesvondreiElementen erzeugtenIdeals.Thiswas the beginningof an outstanding scientific career. The methods used in this paper, namely, those of basic element theory, were also the subject of his habilitation dissertationfrom1977. In 1979, Winfried became a full professor at the University of Osnabrück, AbteilungVechta,andin1995movedtotheMathematicalDepartmentofthesame universityin1995.BeforehistimeinVechta,however,hespent1yearasVisiting Lecturer at The University of Illinois at Urbana-Champaign,mainly to work with GrahamEvans.Thiscooperationhadaformativeeffectonhisworkandresultedin alifelongfriendshipwithGraham.AsoneoftheachievementsofhisvisittoUrbana Champaign,in1980Winfried,jointlywithEvansandGriffith,publishedthepaper Syzygies,idealsofheighttwoandvectorbundles.Inthispaper,amongotherresults, theso-calledsyzygyproblem,insomespecialcaseswasansweredintheaffirmative. One year later, Evans and Griffith succeeded in proving the syzygy theorem in general for any Noetherian local domain containing a field. In the following this theoremhasbeengeneralizedinvariousdirections,alsobyWinfried. v vi Foreword During his time in Vechta, Winfried published numerous influential papers, covering many key aspects of modern Commutative Algebra, including generic maps,genericresolutions,divisorsonvarietiesofcomplexes,straighteninglawson modules,straighteningclosedideals.Thetheoryofalgebraswithstraighteninglaws had been developedby De Concini, Eisenbud and Procesi at the beginningof the 1980s, for which determinantalrings are prominent examples. Winfried extended thescopeofthistheorybyshowingthatitprovidesupperboundsforthearithmetical rank of ideals generatedby a poset ideal of the poset underlyingthe algebra with straighteninglaw,andfurtherintroducedmoduleswithstraighteninglawsinorder tostudythesymmetricalgebraofgenericmodules. Afewyearslater,in1988,WinfriedandVetterpublishedtheirfamousSpringer Lecture Notes Determinantal Rings, Vol. 1327. These lecture notes are to date the standard reference on this subject. Besides the classical theory developed by Hochster and Eagon, powers and symbolic powers of determinantal ideals are considered,ASL theoryis employed,the canonicalclass of determinantalringsis determined,andmuchmoreispresented. Gröbner basis entered the theory of determinantal ideals with the work of Sturmfelsin1990.ThisaspectofthetheoryhadahighimpactonWinfried’slater work and his future scientific cooperations. In 2003, jointly with Aldo Conca, he wrote the article Gröbner bases and determinantal ideals, in which they follow the line of investigations started by Sturmfels. Ever since their first joint paper KRSandpowersofdeterminantalideal,in1998,AldoandWinfriedhaveenjoyed an extremely fruitful cooperation, which continues today. They wrote a series of brilliant papers on the theory of determinantal rings in which they studied Gröbner bases and powers of determinantal ideals. In other papers jointly with Tim RömertheyconsideredKoszulcycles,and Koszulhomologyandsyzygiesof Veronesesubalgebras.Morerecently,MatteoVarbarojoinedtheteam,andin2013 Winfried,togetherwithAldoandMatteo,publishedinAdvancesinMathematicsa fundamentalpaperdealingwith the verydifficultproblemof better understanding therelationsofthealgebrageneratedbythet-minorsofamatrixofindeterminates. Winfried has written several monographs. Besides his Springer Lecture Notes with Vetter, his book Cohen–Macaulay rings (with Jürgen Herzog as coauthor) became extremely popular. Its first edition was published in 1993, with a revised editionin1998.TogetherwiththebookCommutativealgebrawithaviewtowards algebraic geometry by David Eisenbud, this monograph is a must-read for any advancedstudentinCommutativeAlgebra. His most recent monograph is his book Polytopes, Rings and K-theory, with Joseph Gubeladze. This book treats the interaction between discrete convex geo- metry, commutative ring theory, algebraic K-theory, and algebraic geometry, and is the culmination of years of scientific cooperation with Joseph which began 1996.So farWinfriedandJoseph havepublished20 jointpaperscontainingsome spectacular results. Among these, one should mention first, the paper Normality and covering properties of affine semigroups which appeared 1999 in Journal für die reine and angewandte Mathematik. In that paper they succeeded in finding a six-dimensional counterexample to a conjecture of Sebö, who conjectured that a Foreword vii finite rationalcone admits a unimodularcoveringby simplicial cones spanned by elementsoftheHilbertbasis.ItwasknownpreviouslybySeböandothersthatthree- dimensional rational cones even admit unimodular triangulations, and that such triangulations in general no longer exist in dimension 4. In their paper, Winfried and Joseph also presented an algorithm to decide whether a finite rational cone admits a unimodular covering. The computational and algorithmic aspects of the theoryofpolytopesandaffinemonoidshavealwaysbeenoneofWinfried’sespecial concerns. The same year, in the same journal, Winfried together with Gubeladze, Henk, Martin and Weismantel showed that the Bruns-Gubeladze counterexample to unimodular coverings turned out also to be a counterexample to the integral Carathéodory property of cones which requires that any integral vector of the n-dimensional integral polyhedral pointed cone can be written as a nonnegative integral combination of at most n elements of the Hilbert basis of the cone. This result showed that Carathéodory’s theorem for convex cones does not have an integeranalogue. The discovery of the counterexamples certainly promoted experimental math- ematics a lot. Progress has been made possible by the huge capacity of modern computers and powerful computer algebra programs, such as Normaliz. It is not going to far to say that the software Normaliz, which was created by Winfried together with his former PhD student, Koch, is one of his passions. Normaliz is regularly updated by Winfried and his team consisting at the moment of Ichim, Sieg,RömerandSöger.Theprogamis freelydownloadableandalso integratedin otheralgebraicsoftware,suchasMacaulay2,Singularandpolymake.Ithasbecome anindispensabletoolinthestudyofpolytopesandaffinemonoids. Winfried has six PhD students, but has supported many more young students, helping them to find their way into mathematics. In 2002 he was awarded the Osnabrück University prize for excellence in teaching. Together with Holger Brenner,TimRömerandothercolleaguesinthedepartment,Winfriedsuccessfully appliedfortheGraduiertenkollegatDFG(theGermannationalsciencefoundation). The Graduiertenkolleg was established 2013 and has since been running since thenextremelysuccessfullyprovidingPhDpositionsandPostdocsformanyyoung researchers.ItishismerittohavemadethemathematicsdepartmentofOsnabrück oneofthemostimportantcentersofCommutativeAlgebrainEurope. Winfried’s international contacts are countless. So far he has published more than 100 papers, and since 1975 he has presented invited lectures at almost all major international conferences on Commutative Algebra. He has also organized severalimportantinternationalconferencesatOberwolfach,Luminy,Cortona,and Berkeley, thereby promoting the exchange of ideas and giving young mathemati- ciansthechancetopresenttheirresultstolargeaudiencesofexperts.Overtheyears hehasmaintainedaparticularlyclosescientificrelationshipwiththeCommutative Algebra research team in Genoa, and also the algebraists in Messina. This was partly made possible by DAAD funded programs. In recognition of his scientific meritshe was elected as a correspondingmemberof the AcademiaPeloritanadei Pericolanti di Messina. He also served on the editorial board of Communications inAlgebra,andheisstillamemberoftheeditorialboardofHomology,Homotopy viii Foreword andApplications.Furthermore,hehasrefereedresearchprojectsfortheDFG, the DAAD,theAlexandervonHumboldt-Stiftung,theNSFandothers. The authorsand editorsof this volume wish Winfried goodhealth and joy and happinesswithhisfamily.CommutativeAlgebraasweknowittodaywouldnotbe thesamewithouthiswork,whichhassogreatlyinfluencedandshapedourfield.So wewishhimhappyresearchandexpectmanymoregreattheorems. Essen,Germany Prof.JürgenHerzog 3January2017 Preface The present volume is an outcome of the INdAM Conference “Homological and Computational Methods in Commutative Algebra”, which took place at the PalazzoneinCortona,Italy,fromMay30toJune3,2016. The volume reflects much of the spirit of the INdAM Conference. It provides a snapshot of the current developments in Commutative Algebra, with special emphasisonhomologicalnotions,combinatorics,andsymboliccomputations.The fifteenchaptersaddressawiderangeoftopics,fromthemesthatareattheheartof thedisciplinetothemesthatestablishconnectionswithrelatedareas. Both the conference and this volume are dedicated to Prof. Winfried Bruns on theoccasionofhis70thbirthday. Wethanktheparticipantsandthespeakersatthemeetingaswellastheauthors ofthechaptersinthisvolumefortheirwonderfulcollaboration. In particular, we wish to thank Prof. Jürgen Herzog for writing the Foreword, whichprovidesaveryniceoverviewofProfessorBruns’contributiontothesubject. Genova,Italy AldoConca SanFrancisco,CA,USA JosephGubeladze Osnabrück,Germany TimRömer 3January2017 ix
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