Table Of ContentSpringer 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.
Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication
doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant
protectivelawsandregulationsandthereforefreeforgeneraluse.
Thepublisher,theauthorsandtheeditorsaresafetoassumethattheadviceandinformationinthisbook
arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor
theeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinorforany
errorsoromissionsthatmayhavebeenmade.Thepublisherremainsneutralwithregardtojurisdictional
claimsinpublishedmapsandinstitutionalaffiliations.
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
