Derived categories and Grothendieck duality

17 Pages·2014·0.14 MB·English

Grothendieck duality: lecture 3
Derived categories and Grothendieck duality
Pieter Belmans
January 22, 2014
Abstract
These are the notes for my third lecture on Grothendieck duality in the
ANAGRAMSseminar.We(finally)cometoastatementofGrothendieckduality.
In order to do so we first review derived categories, from the viewpoint of
someonewhohasalreadytouchedhomologicalalgebraintheusualsense[9].
Afterthisquickremindersomemotivationforconsideringapossiblegen-
eralisation of Serre duality is discussed, after which the full statement of
Grothendieck duality (in various incarnations) is given. To conclude some
applicationsofGrothendieckdualityarediscussed,frommypointofviewon
thesubject.IhopetheseservebothasamotivationforGrothendieckduality
andasamotivationtostudytheseinterestingsubjects,regardlessfromtheir
relationshipwithGrothendieckduality.
Contents
0 Reminderonderivedcategories 2
0.1 Derivedfunctors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
0.2 Derivedcategories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1 Grothendieckduality 3
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Theidealtheorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3 Whataboutdualisingcomplexes? . . . . . . . . . . . . . . . . . . . . . . 8
2 ApplicationsofGrothendieckduality 10
2.1 Theyogaofsixfunctors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.2 Fourier–Mukaitransforms . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.3 Themoduliofcurves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.4 Otherapplications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1
0 Reminder on derived categories
0.1 Derived functors
Themainideabehindderivedcategoriesistomakeworkingwithderivedfunctors
more natural. Recall that given a left (or right) exact functor between abelian
categoriesonecandetermineitsderivedfunctors,whichformafamilyoffunctors.
Thesefunctorsmeasuretheextenttowhichtheoriginalfunctorisnotexact,and
they can give interesting algebraic or geometric information (depending on the
originalchoiceoffunctor).
Example 1.So far we have used just one derived functor, which was sheaf co-
homology.Theleft-exactfunctorunderconsiderationisΓ(X,−),itsderivedfunc-
torsHi(X,−).
Thereareotherexamples.
Example2.Globalsectionsareaspecialcaseofpushforward:if f :X →Speckis
thestructuralmorphismthen f∗(−)=Γ(X,−).WeobtainasheafonSpeck,which
is nothing but a vectorspace, the only non-empty open set has Γ(f−1(Speck),−)
asitssections.Wecanconcludethat f∗ willnotberight-exactingeneralasitisa
generalisationofglobalsections.
Example3.Anotherwell-knownleftexactfunctorisHom (M,−),whichisanend-
A
ofunctorontheabeliancategoryA-ModforAacommutativeringand M anA-mod-
ule),whoseright-derivedfunctorsaretheExt-functors.Thesehaveadown-to-earth
interpretationasextensions,bytheYonedaExt-construction.
Example4.AdjointtoHom (M,−)wehave−⊗ M,whoseleft-derivedfunctors
A A
aretheTor-functors.
0.2 Derived categories
Thegoalistocapturealloftheseinonesingletotalderivedfunctor.Soinsteadof
workingwiththefamily(RnF)n∈(cid:78) onewantstoconstructafunctorRF replacing
thewholefamily.
Tocalculate(co)homologyoneusesinjective(orprojective,orflat,orflabby,or...
depending on the context) resolutions. So instead of using a single object, it is
naturaltoconsiderawhole(co)chaincomplexofobjects.Thatiswhy,insteadof
usinganabeliancategoryA(takeforexampleA=Coh/X theabeliancategoryof
coherentsheavesonascheme),oneusesCh(A):theabeliancategoryof(co)chain
complexesoverA.
Becausethecalculationof(co)homologyisinvariantuptohomotopyequivalence,
we construct the category K(A) by identifying morphisms in Ch(A) which are
homotopyequivalent.Thisisanintermediatestepwhichcanbeskipped,butithelps
inprovingthemainpropertiesoftheresultingobject.
The final step in the construction is the most technical one, and consists of in-
vertingthequasi-isomorphismstoobtainthederivedcategory.Recallthataquasi-
isomorphismisamorphismwhichinducesisomorphismsinthe(co)homology,i.e.
if f :A• → B• is a morphism such that Hn(A•)∼=Hn(B•) for all n then we would
• •
likeA andB tobeisomorphicinourdesiredderivedcategory.Thisway,anobject
becomes isomorphic to its resolution. The way to obtain this is analogous to the
2
localisation of a ring: we formally add inverses. That this construction works as
intendedfollowsfromtheGabriel–Zismantheorem.
Tosummarise,theconstructiongoesthroughthefollowingsteps
1. pickanabeliancategoryA(Coh/X,Qcoh/X orjustA-Modifyoulike);
2. considertheabeliancategoryof(co)chaincomplexesoverA;
3. constructthecategoryof(co)chaincomplexesK(A)overAbyidentifyingthe
homotopyequivalencesinCh(A);
4. constructthederivedcategoryD(A)ofAbyinvertingthequasi-isomorphisms
inK(A).
Instead of considering all (co)chain complexes, we can also consider complexes
whichsatisfyacertainboundednessassumption.Wecouldaskfor(thecohomology
of)thecomplexestobe
1. boundedonbothsides(denotedDb(A)),
2. boundedbeloworabove(denotedD+(A)resp.D−(A)),
3. concentratedinpositiveornegativedegrees(denotedD≥0(A)resp.D≤0(A)).
OneinterestingpropertyofthederivedcategoryisthattheHom-functorforAturns
intoadevicethatknowsallExti.
Example5.WehaveacanonicalinclusionofAintoD(A),byconsideringanobject
asacochaincomplexindegree0.Nowwecanshiftobjects:theobjectA[i]inD(A)
isthecochaincomplexsuchthatAlivesindegreei.Thenweobtaintheformula
∼
(1) HomD(A)(A,B[i])=ExtAi (A,B).
To see why this is true: the object B[i] is isomorphic to a shift of an injective
resolutioninD(A),hencetheHominD(A)isnothingbutawayofcomputingthe
derivedfunctorsofHom.
Sometimeswe’lldenoteHomD(A)(A•,B•)byRHom•(A•,B•)tosavealittleonthe
notation.
1 Grothendieck duality
1.1 Motivation
ThereareseveralwaysofmotivatingGrothendieckduality,andthedesiretogen-
eraliseSerreduality1.Ofcourse,therestrictionontheclassicalSerredualityare
rathersevere:wewantasmooth(ormildlysingular)projectivevarietyoverafield,
andavectorbundle.Canwedosimilarthings:
1. formoregeneralschemes?
2. overmoregeneralbaseschemes?
3. formoregeneralsheaves?
Theanswerwillbeyes,otherwisewewouldn’tbediscussingGrothendieckduality.
1IfunfamiliarwithSerreduality,oneiseitherinvitedtoreadthenotestothefirstlecture,orglanceat
thesummaryofSerredualitylateron.
3
Adjointfunctors Amoredown-to-earth(orlesscategorical)motivationforthe
formthatGrothendieckdualityoftentakesisgivenin[6,chapter6].Recallfrom
thefirstlecturethestatementofSerreduality,precededbytherequireddefinition
ofadualisingsheaf,asgivenin[4].
Definition6.LetX/kbeapropern-dimensionalvariety.Adualisingsheaf forX isa
coherentsheafω◦ togetherwithatracemorphismtr: Hn(X,ω◦)→k,suchthatfor
X X
allF∈Coh/X thenaturalpairing
(2) Hom(F,ω◦)×Hn(X,F)→Hn(X,ω◦)
X X
composedwithtrgivesanisomorphism
(3) Hom(F,ω◦)∼=Hn(X,F)∨.
X
Thenthestatement,whereX admitsadualisingsheaf,reads:
Theorem7(Serreduality).LetX/kbeaprojectiven-dimensionalvariety.Letω◦
X
beitsdualisingsheaf.Thenforalli≥0andF∈Coh/X wehavefunctorialmaps
(4) θi: Exti(F,ω◦)→Hn−i(X,F)∨
X
such that θ0 corresponds to tr. Moreover, if X is Cohen–Macaulay2 the θi are
isomorphismsforalli≥0andF∈Coh/X.
Wealsoobtainedacorollary,whichexplainsthename“duality”:itrelatesHi toHn−i,
anddoessobyusingadualityofvectorspaces.
Corollary 8.Let X be projective Cohen–Macaulay of (equi-)dimension n over k.
LetFbealocallyfreesheafonX.Thenwehaveisomorphisms
(5) Hi(X,F)∼=Hn−i(X,F∨⊗ω◦)∨.
X
Thisisjustoneofthemanywaysofwritingtheisomorphism.Anotherwouldbe
(6) Hom (cid:128)Hi(X,F),k(cid:138)∼=Hn−i(cid:128)X,Hom(F,ω◦)(cid:138).
k X
Wearealmostwherewewanttobe.Thelaststeptotakeis“gorelative”.Which
of course in this case is not that spectacular. So let’s look at the structural mor-
phism f : X → Speck. We are working for a vector bundle F on X, so we could
alsolookatavectorbundleonSpeck,whichisnothingbutafinite-dimensional
vectorspace V.Generalisingthepreviousequation,andapplyingthetensor-Hom
adjunctionweobtain
(7) Hom (cid:128)Hi(X,F),V(cid:138)∼=Hn−i(cid:128)X,Hom(F,V ⊗ ω◦)(cid:138)∼=Extn−i(cid:128)F,V ⊗ ω◦(cid:138).
k k X k X
Withalittleimaginationthislookslikeanadjunction:
1. thecohomologygroupsHi(X,F)canbetakentogether3 toRf∗(F);
2Atechnicalconditionthatsaysthat“mildsingularities”areallowed.Itmeansthateachlocalringhas
Krulldimensionequaltothedepth(wealwayshavethatdepthisboundedabovebyKrulldimension),
wheredepthcorrespondstothelengthofamaximalregularsequenceforthelocalringitself.Onecan
justreadnon-singular,whichisthecasewewillneedinlaterapplications.
3The functor f∗ in this case is nothing but global sections, as f∗(F) evaluated on Speck
isΓ(f−1(Speck),F),seeexample2.
4
2. by the properties of the derived category we can take Ext’s together into
aHominthederivedcategory,see(1).
HencewecanwriteSerredualityas
(8) HomDb(Speck)(cid:0)Rf∗(F),V(cid:1)∼=HomDb(X)(cid:128)F,f!(V)(cid:138)
where Db(Speck) is the bounded derived category of finite-dimensional k-vector
spacesandDb(X)istheboundedderivedcategoryofcoherentsheaveson X (as-
sumeX issmooth).
HenceSerredualityassertstheexistenceofadual
(9) f!:Db(Speck)→Db(X)
which in this case is explicitly given by −⊗ ω◦. But in this statement we could
k X
easilyreplace f :X →Speck byamoregeneral f :X →Y,andtheexistenceofa
rightadjoint f! wouldstillmakesense!
Awordonthenotation f!:
1. Itisoftenpronounced“f uppershriek”,andit’snamed“exceptionalinverse
image”;
∗
2. It only lives on the level of derived categories, unlike f∗ and f , which get
derivedintoRf∗ andLf∗,soinlinewith[SGA43,§3.1,éxposéXVIII]which
says(inaslightlydifferentcontext,butstillvalid)
N.B.LanotationRf! estabusiveencequeRf! n’estengénéralpas
ledérivéd’unfoncteur f!.
ForthisreasonIwilljustdenoteitby f!,asisalreadydoneforinstancein
[5].
Dualising complexes So we know that there is some virtue in looking at the
relative context, and that we will obtain an adjoint pair encoding Grothendieck
duality. Another thing we could do is look at the dualising sheaf. Then we can
motivateGrothendieckdualitybyconsideringanotherrathertrivialsituation:Spec(cid:90),
aswasdonein[5,§V.1].
Therearetwowaysoftakingthedualofanabeliangroup4:
1. the Pontryagin dual of a finite abelian group, which is given by the func-
torHom (−,(cid:81)/(cid:90));
Ab
2. the dual of a finitely generated free group, which is given by the func-
torHom (−,(cid:90)).
Ab
Eachappliedtwicetothecorrectsituationgivesaabeliangroupisomorphictothe
one you started with. We can consider these two dualising functors at the same
time5,byconsideringthecomplex
(10) ...→0→(cid:81)→(cid:81)/(cid:90)→0→...
4Remarkthatthereisatypoin[5,§V.1],dualisingfinitelygeneratedfreegroupsrequires(cid:90),not(cid:81).
Thismakestheexpositionlessmiraculous.
5Thisiswherethelackofamiracleoccurs.
5
inDb(Ab),thederivedcategoryofboundedcomplexesinsideD+(Ab)whosecoho-
fg
mology is finitely generated. This 2-term complex is an injective resolution of (cid:90),
as both groups are divisible! So it is isomorphic to (cid:90) in Db(Ab), and to perform
fg
computationsinthederivedcategorywecaninterchangethemfreely.
Thisyieldsthefollowingproposition[5,proposition:V.1-1].
Proposition9.Thefunctor
(11) D:Db →Db :M•→RHom•(M•,(cid:90))
fg fg
isacontravariantendofunctor,andthereisanaturalequivalence
(12) η:id ⇒D◦D.
Db(Ab)
fg
Henceonthe“small”categoryDb(Ab)sittinginsidethebiggerD+(Ab)thisduality
fg
functoristrulyaduality.Thesmallcategorycorrespondstotheboundedderived
categoryofcoherentsheaves,asbytheusualmantra“coherent=finitelygenerated.”
Proofofproposition9. Wehave
(13) Hi(D(M•))=Hi(RHom•(M•,(cid:90)))=Exti(M•,(cid:90))
hence if M• has finitely generated and bounded cohomology, so has D(M•). We
obtainthatDisawell-definedendofunctor.
Thenaturalequivalenceisdefinedin[5,lemmaV.1.2]inanobviousway.Tocheck
•
thatitisanaturalequivalence:takeafreeresolutionof M .Thismeansfindinga
•
surjectionofafreeabeliangrouponto M andrepeatingthisprocessforthekernel
ofthismap,andsoon.
By[5,lemmaI.7.1]itsufficestocheckitfor M• =(cid:90)r forsome r ≥1asweonly
care about finitely generated cohomology. But things are additive, hence we can
take r=1.Nowitsufficestoobservethat
(cid:168)(cid:90) i=0
(14) Exti((cid:90),(cid:90))=
0 i(cid:54)=0
whichgivesthedesirednaturalequivalence.
1.2 The ideal theorem
The first thing we can consider as a form of Grothendieck duality is [5, Ideal
theoremonpage6].Thissummariseswhatonetriestoprovetobeabletospeakof
a“Grothendieckdualityresult”.Afterthestatementwewillcollectsomecontextsin
whichwecanprovethis.
Theorem10(Idealtheorem).
1. Foreverymorphism f :X →Y offinitetype6ofpreschemes7thereisafunctor
6Recallthatamorphismoffinitetypemeansthatthereexistsanopenaffinecoveringofthecodomain,
suchthattheinverseimagesoftheseopensetsadmitafiniteopenaffinecovering,suchthateachof
theseringsisfinitelygeneratedovertheopenaffineinthecodomain.
7Ihaveleftthishistoricterminologyin:whatnowadaysarecalledschemeswerecalledpreschemesin
theearlydaysofschemetheory.Atfirstthephilosophywasthatwe’dbemostlyinterestedinseparated
schemes,whichwerecalledschemes,andnotnecessarilyseparatedschemeswerepreschemes.Incase
youdidn’tknowthislittlefactfromthehistoryofschemetheory,youknowunderstandreferencesto
preschemes.
6
(15) f!:D(Y)→D(X)
suchthat
(a) if g:Y →Z isasecondmorphismoffinitetype,then(g◦f)!= f!◦g!;
(b) if f isasmoothmorphism,then
(16) f!(G)= f∗(G)⊗ω,
whereω=Ωn isthesheafofhighestorderdifferentials;
X/Y
(c) if f isafinite8 morphism,then
(17) f!(G)=HomO (f∗OX,G).
Y
2. Foreveryproper9 morphism f :X →Y ofpreschemes,thereisatracemor-
phism
(18) Trf:Rf∗◦f!⇒id
offunctorsfromD(Y)toD(Y)suchthat
(a) if g:Y →Z isasecondpropermorphism,thenTrg◦f =Trg◦Trf;
(b) ifX =(cid:80)n,thenTr isthemapdeducedfromthecanonicalisomorphism
Rnf∗(ω)Y∼=OY; f
(c) if f is a finite morphism, then Tr is obtained from the natural map
f
“evaluationatone”
(19) HomO (f∗OX,G)→G.
Y
3. If f :X →Y isapropermorphism,thenthedualitymorphism
(20) Θf: RHomX(F,f!(G))→RHomY(Rf∗F,G)
obtainedbycomposingthenaturalmap10 abovewithTr ,isanisomorphism
f
forF∈D(X)andG∈D(Y).
Sotheidealtheoremhas3mainthemes:
1. theexistenceoftheadjoint f!;
2. thetracemorphismforpropermorphisms;
8Recallthatamorphismisfiniteifthereexistsanopenaffinecoveringofthecodomainsuchthatthe
inverseimageofeachopenaffineisagainaffine,andmoreoverfiniteasamoduleovertheoriginalring.
9Recallthatapropermorphismbetweenschemesislikeapropermapoftopologicalspaces,where
inverseimagesofcompactsetsareagaincompact.Asbeingcompactinanon-Hausdorffcontextdoesn’t
makemuchsense,algebraicgeometersuseadifferentdefinitionofproper:themap f:X→Y between
schemesissaidtobeproperifitisuniversallyclosed(i.e.forallY→ZisX× Z→Zclosedonthelevel
Y
ofunderlyingtopologicalspaces)andseparated(i.e.∆ :X→X× X isclosed,whichisananalogueof
f Y
beingHausdorff).
10ObtainedastheYonedapairing,see[5,page5].
7
3. thedualityforpropermorphisms.
Thefirstthemedescribesthefunctor f! inthecaseswhereweknowwhatitshould
be.Thesecondthemedescribeswhatthecounitadjunctionshouldlooklike,while
thethirdthemeassertsthatitactuallyisanadjunction.Remarkthat
1. Wehaven’tspecifiedD(X).
2. TheroleofD(X)willchangedependingon
(a) thetypeofschemesweareconsidering;
(b) thetypeofmapsweareconsidering;
(c) thethemeweareconsidering.
Thequestionis:inwhichsituationscanweprovethisidealtheorem?Theanswer
(asper[5],nowadaysitismoregeneral,thiswillbediscussedinthenextlecture)
is:
1. noetherianschemesoffiniteKrulldimensionsandmorphismswhichfactor
throughasuitableprojectivespace[5,§III.8,§III.10,§III.11]:allstatements
areappliedforD+(−)exceptthelastinwhichcaseFlivesinD−(X);
qc qc
2. noetherian schemes which admit dualizing complexes (see [5, §V.10], it
impliesfiniteKrulldimension)andmorphismswhosefibresareofbounded
dimension[5,§VII.3]:allstatementsareappliedtoD+ (−)exceptthelastin
coh
whichcaseFlivesinD−(X);
qc
3. noetherian schemes of finite Krull dimension and smooth morphisms [5,
§VII.4]:theresultsin1areappliedtoD+(−),in2toDb (−)andin3toFin
qc qc
D−(X)andGinDb (Y);
qc qc
4. noetherianschemesandarbitrarymorphisms[5,appendix],butonlystate-
ments1a,2aand3:theresultsareappliedtoD(Qcoh(−)).
Intables1and2wehavefittedthisinformationinaniceoverview.
Remark 11.The duality morphism (20) also has a relative version [2, p. 3.4.4],
whichreads
(21) Θf:Rf∗RHom•X(F•,f!(G•))→RHom•Y(Rf∗(F•),G•).
Takingglobalsectionsyieldstheresultmentionedintheidealtheorem.
1.3 What about dualising complexes?
The ideal theorem as stated here does not mention dualising objects. But in the
case of Serre duality we really phrased things in terms of our “magic object” ω◦
X
whichmadethetheorywork.IntheapproachtoGrothendieckdualityofHartshorne
thesedualisingobjectsstillplayanimportantrole(inthenextlecturewewillsee
approachesinwhichtheroleofthisexplicitobjectisgreatlydiminished),andtrying
to get a hold on them is the main difficulty. The problem in handling dualising
complexes is that they are objects in the derived category, and there is the usual
8
situation X andY f
1 noetherian,finiteKrulldimension factorthrough(cid:80)n
Y
2 noetherianwithdualizingcomplex
3 noetherian,finiteKrulldimension smooth
4 noetherian
Table1:Descriptionofthe4situationsforGrothendieckdualityin[5]
property situation D(X) D(Y)
existenceof f! 1 D+(Y)
qc
2 D+ (Y)
coh
3 Db (Y)
qc
4 D+(Qcoh/Y)
tracemorphism 1 D+(Y)
qc
2 D+ (Y)
coh
3 Db (Y)
qc
4 D+(Qcoh/Y)
duality 1 D−(X) D+(Y)
qc qc
2 D−(X) D+ (Y)
qc coh
3 D−(X) Db (Y)
qc qc
4 D(Qcoh/X) D+(Qcoh/Y)
Table2:Overviewoftheconfigurationofthederivedcategoriesforeachofthethree
partsofaGrothendieckdualitycontext
9
mantrathattriangulatedcategoriesdonotallowgluing,soastraightforwardlocal
approachdoesn’twork.
IntheproofofGrothendieckdualityonetriestokeeptrackofwhatthedualising
object looks like. Recall that in the case of Riemann–Roch this dualising object
wasΩ1,andinthecontextofSerredualityweusedΩn forprojectivespace,and
C (cid:80)n/k
aExtconstructionforthegeneralcaseofaprojectivevkariety.
In the situation of Grothendieck duality where we have used derived categories
everywhereonecanaskwhatω◦ lookslike.Theanswerisgivenintable3,based
X
on[5,§V.9].
Weobservethatthephilosophyis“thenicerX,thenicerω◦”.
X
2 Applications of Grothendieck duality
Duetotimeconstraints,bothinpreparingthesenotesandactuallylecturingabout
them,thefollowinglistofapplicationsisnotasworkedoutasIwantittobe.
2.1 The yoga of six functors
Coherentduality ThenotionofGrothendieckdualitythatwehaveseensofaris
inthefollowingsituation:
1. (quasi)coherentsheaves;
2. Zariskitopologyforschemes;
Étalecohomology Butonecanconsiderothercontextstoo.Inthestudyofétale
cohomologywehave:
1. torsionsheaves;
2. étaletopologyforschemes.
Poincaré–Verdierduality Inthecaseofmanifoldsandlocallycompactspaceswe
havePoincaré–Verdierduality:
1. sheavesofabeliangroups;
2. locallycompactspaces.
Formalism Informalisingthepropertiesthataresimilarineachofthesecontexts
weseethat
• weareconsideringimagefunctorsofsheaves;
• weareusingtheclosedmonoidalstructureofthecategoryofsheaves.
Inthegeneralsituationofa“sixfunctorsformalism”wecanidentifythefollowing
functors[1],for f :X →Y amorphisminsomecategory,andC(X)somecategory
ofsheavesassociatedtoX.
10