Table Of ContentSpringer Monographs in Mathematics
Winfried Bruns
Aldo Conca
Claudiu Raicu
Matteo Varbaro
Determinants,
Gröbner
Bases and
Cohomology
Springer Monographs in Mathematics
Editors-in-Chief
MinhyongKim,SchoolofMathematics,KoreaInstituteforAdvancedStudy,Seoul,
SouthKorea,InternationalCentreforMathematicalSciences,Edinburgh,UK
KatrinWendland,SchoolofMathematics,TrinityCollegeDublin,Dublin,Ireland
SeriesEditors
SheldonAxler,DepartmentofMathematics,SanFranciscoStateUniversity,San
Francisco,CA,USA
MarkBraverman,DepartmentofMathematics,PrincetonUniversity,Princeton,
NY,USA
MariaChudnovsky,DepartmentofMathematics,PrincetonUniversity,Princeton,
NY,USA
TadahisaFunaki,DepartmentofMathematics,UniversityofTokyo,Tokyo,Japan
IsabelleGallagher,DépartementdeMathématiquesetApplications,EcoleNormale
Supérieure,Paris,France
SinanGüntürk,CourantInstituteofMathematicalSciences,NewYorkUniversity,
NewYork,NY,USA
ClaudeLeBris,CERMICS,EcoledesPontsParisTech,MarnelaVallée,France
PascalMassart,DépartementdeMathématiques,UniversitédeParis-Sud,Orsay,
France
AlbertoA.Pinto,DepartmentofMathematics,UniversityofPorto,Porto,Portugal
GabriellaPinzari,DepartmentofMathematics,UniversityofPadova,Padova,Italy
KenRibet,DepartmentofMathematics,UniversityofCalifornia,Berkeley,CA,
USA
RenéSchilling,InstituteforMathematicalStochastics,TechnicalUniversity
Dresden,Dresden,Germany
PanagiotisSouganidis,DepartmentofMathematics,UniversityofChicago,
Chicago,IL,USA
EndreSüli,MathematicalInstitute,UniversityofOxford,Oxford,UK
ShmuelWeinberger,DepartmentofMathematics,UniversityofChicago,Chicago,
IL,USA
BorisZilber,MathematicalInstitute,UniversityofOxford,Oxford,UK
Thisseriespublishesadvancedmonographsgivingwell-writtenpresentationsofthe
“state-of-the-art”infieldsofmathematicalresearchthathaveacquiredthematurity
neededforsuchatreatment.Theyaresufficientlyself-containedtobeaccessibleto
morethanjusttheintimatespecialistsofthesubject,andsufficientlycomprehensive
toremainvaluablereferencesformanyyears.Besidesthecurrentstateofknowledge
initsfield,anSMMvolumeshouldideallydescribeitsrelevancetoandinteraction
with neighbouring fields of mathematics, and give pointers to future directions of
research.
· · ·
Winfried Bruns Aldo Conca Claudiu Raicu
Matteo Varbaro
Determinants, Gröbner Bases
and Cohomology
WinfriedBruns AldoConca
InstitutfürMathematik DipartimentodiMatematica
UniversitätOsnabrück UniversitàdiGenova
Osnabrück,Germany Genova,Italy
ClaudiuRaicu MatteoVarbaro
DepartmentofMathematics DipartimentodiMatematica
UniversityofNotreDame UniversitàdiGenova
NotreDame,IN,USA Genova,Italy
ISSN 1439-7382 ISSN 2196-9922 (electronic)
SpringerMonographsinMathematics
ISBN 978-3-031-05479-2 ISBN 978-3-031-05480-8 (eBook)
https://doi.org/10.1007/978-3-031-05480-8
MathematicsSubjectClassification: 13A35, 13A50, 13C40, 13C70, 13D02, 13D07, 13D10, 13D40,
13D45,13F50,13F65,13H10,13H15,13P10,14B15,14L30,14M12,14M15,14M25,20G05,20G15
©SpringerNatureSwitzerlandAG2022
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,expressedorimplied,withrespecttothematerialcontainedhereinorforany
errorsoromissionsthatmayhavebeenmade.Thepublisherremainsneutralwithregardtojurisdictional
claimsinpublishedmapsandinstitutionalaffiliations.
ThisSpringerimprintispublishedbytheregisteredcompanySpringerNatureSwitzerlandAG
Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland
Winfried
ForPhilipp,HeleneandJakob
Aldo
ForMariaGrazia,FabrizioandCarolina
Claudiu
Formyfamily
Matteo
ForGiulia, Annaluisa,RoccoandRoberto
Preface
Determinantalringsandvarietieshavebeenacentraltopicofcommutativealgebra
andalgebraicgeometryforseveraldecades.Numerousresearchershavecontributed
to the development of their theory, and they have served as a testing ground for
newnotionsandtools.Amongtheareasofmathematicsthathaveintertwinedwith
determinantalringsandvarieties,onefindsalgebraicgeometry,commutativealgebra,
combinatorics,invariantandrepresentationtheoryofthegenerallineargroup.They
allplaymajorrolesinthisbook,andtheirinteractionisaubiquitousmotive.
WhentheseniorauthorwasaPh.D.studentexactlyhalfacenturyago,theEagon–
Northcottcomplexhadbeenconstructed10yearsearlier,andthepaperofHochster
andEagon,whichestablishedtheCohen–Macaulaypropertyofdeterminantalrings
ingeneral,hadjustappeared.Meanwhile,theliteratureondeterminantalringsand
idealshasgrowntoacollectionofseveralhundredsofarticlesofwhichthereferences
citedinthisbookrepresentonlyaselection.
The monograph Determinantal Rings by Bruns and Vetter appeared in 1988.
It gave a comprehensive overview of the algebraic theory available at the time of
its writing. Now, more than 30 years later, this area has seen a number of new
developments,andgeometrictechniqueshavecontributedmany,verypreciseresults.
Oneofthemajorthemesthatcameattheendofthe80sisGröbnerbasesofideals
in polynomial rings and Sagbi bases of subalgebras. Chapter 1 gives a compact,
butaquitecompleteintroductiontoGröbnerbasesandSagbibasesingeneral.The
focusisonthestructuralaspects,namelytheuseofGröbnerandSagbidegenerations
in the transfer of homological and enumerative information from Stanley–Reisner
and/or toric rings to those objects that degenerate to them. This has been a major
themeinthelast30years,notonlyfordeterminantalideals.Chapter2pushesthis
developmentforwardinseveraldirections:finercontrolofminimalandassociated
primes,connectedness,tightconnectionsbetweenanidealanditssquarefreeinitial
ideals,shouldthelatterexist.Someoftheseresultswereveryrecentlyestablished
byConcaandVarbaro.
Chapter3givesashortintroductiontostandardbitableauxandthestraightening
law,followingDeConcini,EisenbudandProcesi.Thispowerfultechniquehasbeen
vii
viii Preface
dealtwithextensivelyinthemonographofBrunsandVetter,butisofcentralimpor-
tanceinthisbookaswell:forthecomputationofGröbnerandSagbibasesonthe
one hand and for the representation theoretic approach on the other. The straight-
eninglawisalsothebasisfortheprimarydecompositionofpowersofdeterminantal
ideals. Nevertheless, we keep the chapter to a minimum, and for some proofs we
onlyindicatethebasicideas.
Chapter 4 presents the computation of Gröbner bases, based on standard
bitableaux and the Robinson–Schensted–Knuth correspondence to which we give
ashortintroduction.ThisdevelopmentwasinitiatedbySturmfels.ThenHerzogand
Trungderivedseveralstructuralandenumerativetheoremsondeterminantalideals
fromtheshellabilityofthesimplicialcomplexdefinedbytheinitialideal.Among
themisGiambelli’sformulaforthedegreeofadeterminantalvariety,publishedin
1903.TheconnectionbetweentheRobinson–Schensted–Knuthcorrespondenceand
determinantalidealswascarriedonbyBrunsandConcawhosearticleGröbnerbases
anddeterminantalidealscanbeconsideredthenucleusofthisbook.Wealsoinclude
thesimpleapproachviasecantsbySturmfelsandSullivant.
ItisatheoremofBernstein,SturmfelsandZelevinskythatthemaximalminors
areauniversalGröbnerbasis.AsanaddendumtoChapter4,wepresentthesurpris-
ingly simple approach to this result by Conca, De Negri and Gorla in Chapter 5.
A second, simpler case, in which the universal Gröbner basis can be described, is
thatof2-minorsasaspecialinstanceofbinomialideals.Thatmaximalminorsand
2-minorshavespecialpropertieshadalreadybeenforeshadowedbythelastsection
ofChapter4:exactlyforthemthedeterminantalringhassquarefreemultidegreesin
the“gradingbycolumns”,athemegoingbacktovanderWaerden’sworkin1929.
ThehomogeneouscoordinateringoftheGrassmannianisanalgebrageneratedby
minors,anditsinvestigationisaparadigmfortheexploitationoftoricdeformations.
SincetheresultsonGröbnerbasesofsymbolicandordinarypowersofdeterminantal
idealsallowustofindtheinitialalgebrasofthecorrespondingReesalgebras,thebasic
arithmeticalandhomologicalpropertiesbecomeaccessibleaswellinChapter6.
Algebras generated by lower order minors are coordinate rings of varieties
parametrizedbyexteriorpowersoflinearmaps.Theyaremuchmorecomplicated
thanGrassmanniansandnotyetfullyexplored.Theirinitialalgebrasturnouttobe
normalmonoiddomains,andthisallowsonetoshowthattheyarenormalCohen–
Macaulaydomains,andeventheGorensteinonesamongthemcanbedetermined.
Wealsodescribetheirorbitstructure.However,theirdefiningidealsareonlypartially
understood,andthebestgeneralinformationhasbeenconvertedtoaconjectureby
Bruns,ConcaandVarbaro.Onlyforthecaseof2-minorsonenowhasacomplete
systemofrelationsbytherecentworkofHuang,Perlman,Polini,RaicuandSammar-
tano.Forreasonsofspace,wemustcontentourselveswithasurveyonthisdifficult
topic. At the end of the chapter, we return to firm ground and analyze the deter-
minantal rings via their initial algebras in a suitable embedding. This helps us to
understandtheirCohen–Macaulayrank1modules.
Chapter 7 treats ring theoretic properties derived from the Frobenius functor in
positivecharacteristicsthatwerepioneeredbyHochsterandHunekeinconnection
withtightclosure.Wedevelopthemfarenoughtoprovethatdeterminantalringsare
Preface ix
stronglyF-regular.Inordertoapplydeformationarguments,oneneedsF-rationality.
Itiscloselyrelatedtotherationalityofsingularitiesincharacteristic0byatheorem
ofHaraandSmith,sothatwecanatleastbrieflydiscussthispropertyforourrings.
The F-rationality of normal monoid domains makes it a very handy tool for the
exploitation of toric deformations to such algebras, and helps to apply equivariant
deformationbasedonso-calledU-invariants.Weconcludethechapterbycomputing
F-purethresholdsfordeterminantalringsandideals.
For standard graded algebras over fields, Castelnuovo–Mumford regularity has
becomeanindispensableinvariant.Chapter8developsthisnotionfromscratch,but
inamoregeneralversionforstandardgradedalgebrasoverNoetherianbaserings.In
thisgenerality,theJanus-facednatureofregularitystillsurvives:itisdefinedinterms
of graded local cohomology, but can also be computed from minimal graded free
resolutionsifthesearesuitablydefined.AthirdfacetinthegeneralcaseisKoszul
homology. The main advantage of the general version is the significantly larger
flexibility,forexampleinapplicationstoReesalgebras.ThetheoremofCutkosky–
Herzog–TrungandKodiyalamontheregularityofpowersofidealsremainstrue.In
thecontextofdeterminantalrings,wearemainlyinterestedinlinearfreeresolutions
ofpowersofidealsofmaximalminorsinthenon-genericcase,exemplifiedbyideals
ofrationalnormalscrolls.
With Chapter 9, the scenery changes significantly from mainly algebraic to
predominantlygeometricmethods.Thischapterdevelopsthebasictoolstobeapplied
inthechaptersthatfollow:multilinearalgebra,Schurfunctors,tautologicalvector
bundles on flag varieties, their cohomology and the celebrated Kempf vanishing
theorem.Thetheoryofalgebraicgroupschemesisoutlinedtothenecessaryextent.
Most of the material in Chapter 9 is classical, but the vanishing theorems from
Sections 9.5 and 9.6 are new, and they play a crucial role in obtaining bounds for
Castelnuovo–MumfordregularityinChapter10.
Chapter 10 applies the methods of Chapter 9 to study the ideals “defined by
shape” that were introduced already in Chapter 3. Many of the ideas were present
in the work of Raicu in characteristic zero, but here we develop the theory in a
characteristic-freefashion.Foridealsdefinedbyshape,weexhibitnaturalfiltrations
where the composition factors arise as global sections of vector bundles on desin-
gularizations of determinantal varieties, which leads via Grothendieck duality to a
description of the corresponding Ext modules as sheaf cohomology groups. The
filtrationstakeaparticularlyniceformforsymbolicpowersofdeterminantalideals,
wherethevanishingtheoremsfromChapter9,combinedwiththecharacterization
from Chapter 8 of Castelnuovo–Mumford regularity via Ext modules, allow us to
determineanexplicitformulafortheasymptoticregularity.WeendChapter10witha
briefsurveyofseveralotherhomologicalandarithmeticpropertiesofdeterminantal
idealsthatcanbederivedinacompactwayviageometricarguments.
ThegoalofChapter11istoextendandprovesharperversionsoftheresultsof
theprecedingchapter,whenworkingoverafieldofcharacteristiczero.Twocrucial
advantagesinthissettingarethelinearreductivityofthegenerallineargroup,and
the Borel–Weil–Bott theorem describing the cohomology of line bundles on flag
varieties.TheclassofidealsdefinedbyshapecanbeenlargedtothatofGL-invariant
x Preface
ideals,towhichthetheoryoffiltrationsfromChapter10canbeextended,andfor
which all the Ext modules can be calculated exactly as GL-representations. The
consequencesoftheBorel–Weil–Botttheoremarevastlysuperiortothevanishing
theoremsusedinthecharacteristic-freesetting,andinparticulartheyleadtoeffective
versions of the earlier asymptotic results on Castelnuovo–Mumford regularity. As
an application of the calculation of Ext modules, we explain how to describe the
GL-structure for the cohomology with support in determinantal ideals, and briefly
discusshowthisfitsinwiththeD-modulestructureofthesaidcohomologygroups.
Finally, we conclude the book with a quick survey of the important topic of free
resolutions of determinantal ideals, which was pioneered by Kempf and Lascoux,
andistreatedatlengthinWeyman’sCohomologyofvectorbundlesandsyzygies.
Wearegratefultoallourfriendsandcolleagueswhohavehelpedusbyproviding
valuableinformationandsuggestions:AldoBrigaglia,MarcChardin,DaleCutkosky,
MicheleD’Adderio,ChristianKrattenthaler,AndrásLörincz,MikePerlman,Peter
Schenzel,LisaSeccia,AnuragSingh,KellerVandeBogertandJerzyWeyman.
Wecordiallythank DinhVan Lewho readlargepartsofthebook.Henotonly
corrected uncountably many typos but also helped to improve the text and the
mathematics.Also,ManolisTsakiris,AlessandroDeStefaniandAlessioCaminata
supportedusbyreadingselectedchaptersandsuggestingvaluablecomments.Our
thanksgotoUlrichvonderOheforhisTEXnicaladvice.WethankAlessioD’Alì
whohasassistedusincheckingthegalleyproofs.
TheauthorsthankRemiLodhofSpringerLondonfortheexcellentcooperation.
Osnabrück,Germany WinfriedBruns
Genova,Italy AldoConca
NotreDame,USA ClaudiuRaicu
Genova,Italy MatteoVarbaro
January2022
