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Springer 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. 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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 determinantal 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