Alexandre Grothendieck’s EGA V - James Milne

1 AlexandreGrothendieck'sEGAVT ranslationandEditingof his`prenotes'byPiotrBlassandJosephBlass1 SummaryThisformulationgives p^elem^elea detailedsummaryof thesetof resultsthatshouldappearin a arrive at thelatterwe needto reorganizethoroughlythepresent rststepshouldprobablybe to make a newplan(inwhich withouta doubtthepresent sections11,12,14,15 willcomemuch earlier).I have noteven writtensection16which shouldneitherin principlecauseany di culty nordoes it in uenceinany way involvedis a simplematterof a shouldappearin a wouldlike to tellyou in thisconnectionthatI have severalotherresultsquitediversebutalldea lingwithbirationalmappingsthatI wouldlove to seemsto methatthereis notenoughto make a have a suggestionwhereto placethem?

2 I planto sendthemto yousoonas wellas section16of thepresent notes.*In addition,thepresent paragraph20 willprobablyblow upinto two paragraphs,oneconsistingof resultsof thetype \elementarygeometry" needbe, couldoneincludetherealso(lackinga betterplace)thesupplements thatI toldyouaboutdealingwithbirationaltransfo rmations?*AskAG if hasever beenwritten.[Tr]2 HyperplaneSectionsandConicProjections1) )Generichyperplanesection, )Generichyperplanesectiongeometricirredu cibility )Variablehyperplanesection:\su cientlygeneral" )Theoremsof )Connectednessofany )Applicationto theconstructionsof )Dimensionof thesetof )Changeof projective )Pencilsof hyperplanesectionsand brationsof ) ) Generalizationof thepreviouslymentionedresultsto )Elementarymorphismsanda thoeremof ) )Axiomatizationof someof )Translationinto thelanguageof PreliminariesandNotationLetSbe a prescheme,letEbe a locallyfreemoduleof nitetype overS, denotebyP=P(E) theprojective brationde nedbyEandbyPvtheprojective brationde nedbyEv.

3 We shallcallPvtheschemeof hyperplanesofp. Thisterminologycanbe justi edas be a sectionofPvoverSwhich is threforedeterminedby aninvertiblelquotient moduleLofEv. Fromit we obtainaninvertiblequotient moduleLPof (Ev)P= (EP)v, ontheotherhand,we have theinvertiblequotientmoduleOp(1)ofEp. Passingto dualswe may takeLP ( 1) to be invertilesubmodules(locallydirectfactors ) ofEP( (EP)v) andthepairingEP EPv!OPdeifnesthereforea naturalpairing( )OP( 1) LP 1 !OPor alsothetransposehomomorphism( )OP !OP(1) LP=LP(1) sectionofLP(1)canonicallyde nedby . The\divisor"of thatsection, ofPde nedby theimageidealof ( ), is calledthehyperplaneinPde nedby theelement 2PV(S). We coulddescribe it by notingthatlocallyoverS, is givenby a section ofEsuch that (s)6= 0 foralls( is determinedby uptomultiplicationby aninvertiblesectionofOS); sinceE=p (Op(1)),(p:p!)

4 Sbeingtheprojection), canbe consideredas a sectionofOp(1),thedivisorof which is nothingelsebutH .Ofcourse,if we considerL 1as aninvertiblesubmoduleofElocallya directfactorinEthenthecorrespondencebetw een ( 1 E) and is obtainedby takingfor a sectionofL 1which does notvanishat any point, a trivializationofL 1(which existsin everycaselocally).LetusnotethatH is nothingelsebyP(E=L 1)(canonicalisomorphism)thatis a thirdway of describingH ( (E=L 1) is indeedcanonicallyembeddedinP=P(E) which hastheadvantageof provingin additionthatH is a projective brationoverSandis a fortiorismoothoverS. (Againit wouldbenecessaryto say in EGAIV thata projective brationis smooth.).Withoutadoubtit wouldbe betterto is compatiblewithbasechange,inotherwordsone ndsa homomorphismof functors(Sch=S0)!

5 (Ens),Pv!Div(P=S) wherethesec-ondtermdenotesthefunctorof \relative divisors"ofP=SwhosevaluesatS0(anarbi-4tr arySprescheme)is thesetof closedsubschemesofPS0which arecompleteintersectionstransversalandof codimension1 relative toS0( )[ofEGAIVTr.].1It is easyto show thatthishomomorphismof functorsis a monomorphism,in otherwordsthat is determinedofH . (Thislastfactjusti estheterminology\schemeofhyperplanes"use dabove.)We shallseethatthefunctorDiv(P=S) is representablebytheprescheme(direct)sumof P(Symmk(Ev)) so thatPvcanbe identi edto anopenandclosedsubschemeof Div(P=S):::2( tellthetruth,thedeterminationof therelative divisorsofP=Scouldbe donewiththemeansavailableright now, andcouldbe addedas anexampleto EGAIV[Tr.)]

6 ].)Letusnow make thebasechangeS0=Pv!Sandletusconsiderthed iagonalsection(or\genericsection")ofPvS0 =P(EvS0) overS0: we nda closedsubschemeHSofPS0=P SPvwhich is calledsometimestheincidence schemebetweenPandPvde nedby theimageidealof thecanonicalhomomorphismOP( 1) SOPv( 1) !OP xSPvfromwhich oneseesthatit is a projective brationoverPv, andby symmetryit is alsoa projective brationoverP. Letus notethatonerecoversthe\special"hyperplan esH (for a sectionofPvoverS) by startingoutfromthe\universalhyperplane"H andbytakingitsinverseimageforthebasechan geS ! ofPvwithvaluesin anarbitraryS-preschemeS0which (consideredas a sectionofPS0overS0) allowsusto de neanH PS0; thelatteris nothingelsebuttheinverseimageofHby thebasechangeS0 !

7 Whatfollowswe assumea preschemeXof nitetype overP[Tr]3andanSmor-phismf:X!P. Oneof themainobjectives of thisparagraphis tostudyforeveryhyperplaneH ofPitsinverseimageY =f 1(H ) =XXPH andespeciallyto relatethepropertiesofXandY . Asusualonealsohasto considertheP(S0),S0anarbitrarySscheme,in thiscaseH is a hyperplaneinPS0andwe putagainY =f 1S0(H ) =XS0 pS0H =XXPH wherethesubscriptS0denotesas usualthee ectof thebasechangeS0!Sandwherein thelastexpressionwe considerH as aPschemeviathecombinedmorphismH !PS0!P. It is thereforeagainconvenient to considerthecasewhere is \universal" neweditionof EGAI [Tr.]2 ComparewithMumford's:`Lecturesoncurves onanalgebraicsurface.'[Tr]3or overS, I amnotsure[Tr]5whereS0=Pvand is thediagonalsectionso thatH =H, in thiscaseoneobserves (upto betternotationsto be suggestedby Dieudonn e) thatY=Y.

8 In thegeneralcaseof a :S0!Pv, onehasthereforealsoY =Y xvPS0. FinallyifFis asheaf of modules4overXwe denotebyG itsinverseimageoverY byGitsinverseimageoverHso thatonealsohasG =G summarizein a smalldiagramtheessentialsof X XxSPv Y Y ??y??y??y??yP PxSPv H H ??y?? Pv S0(Thesquaresanddiamondsappearingin thisdiagramareCartesian).In thenextsectionwe willstudysystematicallythefollowingcase: S0is thespectrumof a eldKanditsimageinPvis genericin thecorresponding berPvs. AftermakingthebasechangeSpeck(s)!Swe arereducedto thecasewhereSis thespectrumof a eldk, which iswhatas assumein of propertiesstudiedforXandY areof \geometricnature"andthereforeinvariant underbasechange,which allowsus also(withoutlossof generality) to limitourselves to thecasewhereKis algebraicallyclosedor to thecasewhereK=k( ), beingthegenericpoint ofPvand: : Spec(K)!

9 Pvisof alsonotethatforgeometricquestionsconcern ingX,Y we can(aftermakinga basechange)restrictourselves to animmersionwe usuallycallY a hyperplanesectionofX(relative to theprojective immersionfandthehyperplaneH [Tr.]).Thereis noreasonnotto extendthisterminologyto thecaseof Studyof a generichyperplanesection:localproperties LetusrecallthatnowS=Spec(k),kis a is a point ofPvandif : Speck( )!Pvis thecanonicalmorphismwe alsowriteH ,Y ,G in placeofH ,Y ,G .In thissection(numero) denotesalways thegenericpoint modulealways meanscoherent or quasi-coherent sheafof assumethatXis is irreducibleor emptyandin the rstcaseit dominatesX[illegible,askAG]5Y[Tr.]is ,sinceH!Pis aprojective brationthatis alsotrueforY!

10 XwhichimpliesthatYis irreducibleifXis berY [Tr.]ofYoverPvis irreducibleor empty in the rstcaseitsgenericpoint is thegenericpoint ofYwhichthereforeliesover thegenericpoint a subsetofP. ThenitsinverseimageZ inH is empty ifandonlyif everypoint ofZis particularifZis constructiblethenZ = ifandonlyifZis may supposethatZis reducedto a singlepointzandwe onlyhave to provethattheimageofH inPconsistsexactlyof thenon-closedpoints ofP. onlyhave to prove thatZ = if andonlyifXis nite(Xbeinga closedsubschemeofP).ReplacingXbyXk( ),!Pk( )the`onlyif'[French `il faut']partresultsfromthefollowingfactfor which we have to have a referenceandwhich factdeserves to be restatedhereas a lemma:ifYisanyhyperplanesectionofXandifY = thenXis nite(indeedX P His a neandprojective:::).