Generalizing the GAGA Principle

0.24 MB·

by  Jack Hall

GENERALIZING THE GAGA PRINCIPLE
JACKHALL
1
1
0
ABSTRACT. Thispaper generalizes thefundamental GAGAresults ofSerre[Ser56]in
2
three ways—to the non-separated setting, tostacks, and tofamilies. Asanapplication

n of these results, we show that analytic compactifications of Mg,n possessing modular
a interpretationsarealgebraizable.
J

7
2
1. INTRODUCTION
]
G
A locallyseparatedandlocallyoffinitetypealgebraicC-spaceXmaybefunctorially
A analytified to an analytic space X . More generally, one may functorially analytify a
an
. locallyoffinitetypeC-stackwithlocallyseparateddiagonal. ItwasshownbyB.Conrad
h
and M. Temkin in [CT09, Thm. 2.2.5] that the analytification of a locally of finite type
t
a algebraic C-space is an analytic space only if it is locally separated. In particular, for
m
locally of finite type C-stacks Z and X with locally separated diagonals, one obtains an
[ analytificationfunctor

Hom(Z,X)→Hom(Z ,X ).
2 an an
v Thefirstmainresultofthispaperis
3
2 Theorem1. ForalocallyoffinitetypeDeligne-MumfordC-stackXwith quasi-compact
1 and separated diagonal, and a proper Deligne-Mumford C-stack Z, the analytification
5 functor:
.
1 Hom(Z,X)→Hom(Z ,X )
an an
0
isanequivalence.
1
1 In[Lur04],J.LurieprovesarelatedresulttoTheorem1—thestackXispermittedtobe
:
v algebraic(asopposedtoDeligne-Mumford),butthediagonalisassumedtobeaffine.Thus
i Theorem1isstrongerthanwhatappearsin[loc.cit.]whenappliedtoDeligne-Mumford
X
stacks(e.g. schemesandalgebraicspaces). OurproofofTheorem1willrelyonaversion
r
a ofSerre’sGAGAprinciplefornon-separatedalgebraicstacks.
ForalocallyoffinitetypealgebraicC-stackX,definethecategoryofpseudosheaveson
X,PS (X),tohaveobjectsthepairs(Z −→s X,F), wheresisarepresentable,quasi-finite,
p
andseparatedmorphismofalgebraicstacks,withZproper,andFacoherentO -module.A
Z
morphismofpseudosheaves(Z−→s X,F)→(Z′ −s→′ X,F′)isapair(Z′ −→t Z,F−φ→t∗F′),
wheretisafiniteX-morphismandφisamorphismofcoherentO -modules.Thecategory
Z
of pseudosheaveson an algebraic stack X is a non-abelianrefinementof the categoryof
coherentsheavesonX.
Amorphismofanalyticstacksσ:Z→Xislocallyquasi-finiteifitisrepresentableby
analyticspaces,andforanyx ∈ |X|thefiberσ−1(x)isafinitesetofpoints. Thus,foran
analyticstack X, we maysimilarlydefine a categoryofanalyticpseudosheavesPS (X),
p
2010MathematicsSubjectClassification. Primary14C05;Secondary32C99,14D23.
WewouldliketoexpressoursincerestthankstoBrianConradforhisinsightfulcommentsandsuggestions.
1
2 J.HALL
with objectsgiven by pairs (Z −→σ X,F), where σ is a locally quasi-finite and separated
morphismofanalyticstacks,Zisaproperanalyticstack,andFisacoherentO -module.
Z
Thereisananalytificationfunctorforpseudosheaves:
Ψ :PS (X)→PS (X ).
X p p an
Theorem1willfollowfrom
Theorem2. ForanylocallyoffinitetypeDeligne-MumfordC-stackXwithquasi-compact
andseparateddiagonal,theanalytificationfunctor
Ψ :PS (X)→PS (X )
X p p an
isanequivalence.
Theorem 2 admits a useful variant. For a locally of finite type algebraic C-stack X,
definethefullsubcategoryPS (X)⊂PS (X)toconsistofthosepseudosheaves(Z−→s
p,DM p
X,F),wherethealgebraicstackZisDeligne-Mumford. ForananalyticstackX,onemay
similarlydefineafullsubcategoryPS (X) ⊂ PS (X). Thereisaninducedanalytifi-
p,DM p
cationfunctor:
Ψ :PS (X)→PS (X ).
eX p,DM p,DM an
Theorem 3. For any locally of finite type algebraic C-stack X with quasi-compact and
separateddiagonal,theanalytificationfunctor
Ψ :PS (X)→PS (X )
eX p,DM p,DM an
isanequivalence.
This follows from Theorem 2, since for any locally of finite type algebraic C-stack
X with quasi-compactand separated diagonal, there is an open substack X0 ⊂ X which
isDeligne-MumfordsuchthattheinclusionX0 ֒→ XinducesanequivalencePS (X0) =∼
p
PS (X). Also,ifanalgebraicstackXhasaffinestabilizers,thenPS (X)=∼ PS (X).
p,DM p,DM p
ThusTheorem2canbetriviallystrengthenedtoshowthatthefunctorψ isanequivalence
X
foralgebraicstackswithaffinestabilizers.
What follows will be made precise in §5. Let g > 1 and n ≥ 0. Take M (resp.
g,n
M ) to be thealgebraicC-stack(resp. analyticstack) ofsmooth, n-pointed,algebraic
g,n
(resp. analytic)curvesofgenusg. Analgebraic(resp. analytic)modularcompactification
ofM (resp. M )isaproperalgebraic(resp. analytic)stackN(resp. N),withfinite
g,n g,n
stabilizers, possessinga “modular”interpretation,togetherwitha diagramofdenseopen
immersionsM ←֓V ֒→N(resp. M ←֓V֒→N). Theorem3implies
g,n g,n
Theorem4. AnalyticmodularcompactificationsofM arealgebraizabletoalgebraic
g,n
modularcompactificationsofM .
g,n
Using Moishezon techniques(as in [Art70]) for the coarse modulispaces, previously
existingtechnologydemonstratesthatitispossibletoalgebraizetheanalyticcoarsemoduli
spaceswhicharemodularcompactificationsof M . TheMoishezontechniquesfailto
g,n
algebraizethemodularinterpretation,aswellasthemodulistack,however.Describingthe
modularcompactificationsofM isanactiveareaofresearchinmodulitheory,referred
g,n
toastheHassett-Keelprogram(cf.[HH08],[HH09],and[AFS10]).
We willnowdiscusstheGAGAprincipleforfamilies. TheHilbertfunctor,whichpa-
rameterizesclosedimmersionsintoafixedspace,alwaysfailstobeanalgebraicspacein
thenon-separatedsetting. ThisassertionisduetoC.LundkvistandR.Skjelnes[LS08].In
[HalR10], a generalizationoftheHilbertfunctor,the Hilbertstack, wasshownto bean
GENERALIZINGTHEGAGAPRINCIPLE 3
algebraicstackfornon-separatedalgebraicstacks. ForanalgebraicstackY,andalocally
finitelypresentedmorphismofalgebraicstackswithquasi-compactandseparateddiagonal
X → Y,theHilbertstackHS assignstoeachY-schemeT thegroupoidofquasi-finite
X/Y
andrepresentablemorphismsofalgebraicstacksZ → X× T suchthatthecomposition
Y
Z→X× T →T isproper,finitelypresented,flat,andwithfinitediagonal.
Y
ForamorphismofanalyticstacksX→Y,theanalyticHilbertstackHSan assignsto
X/Y
eachanalyticY-spaceT,thegroupoidoflocallyquasi-finitemorphismsofanalyticstacks
Z → X× T,suchthatthecompositionZ → X× T → T isproperandflatwithfinite
Y Y
diagonal.Hence,givenamorphismX→YoflocallyoffinitetypealgebraicC-stackswith
quasi-compactandseparateddiagonal,thereisthusaninducedmorphismofY -stacks:
an
Φ :(HS ) →HSan .
X/Y X/Y an Xan/Yan
Itis importantto note that thereare no results in the literaturefor analytic spaceswhich
exertthatHSan isan analyticstackin thecasethatthemorphismX → Y isnon-
Xan/Yan an an
separated.WealsoprovethefollowingGAGAresultforfamilies.
Theorem5. LetX→Y beaquasi-separatedmorphismoflocallyoffinitetypealgebraic
C-spaces.Thenthenaturalmap
Φ :(HS ) →HSan
X/Y X/Y an Xan/Yan
isanequivalenceofY -stacks. Inparticular,HSan isananalyticY -stack.
an Xan/Yan an
We wish to emphasizethatTheorems1–3and 5 arecompletelynew, evenin the case
whereXisascheme.
Remark 1.1. Theargumentsin thispaperforTheorem5 generalizereadilytothesetting
ofamorphismofalgebraicstacksX→Y. WhatobstructsafullproofofTheorem5inthis
generalityis a lackof foundationsforanalytic stacks. More precisely, we wouldrequire
ananalogofCorollary3.3foranalyticstacks,forwhichnoprecisereferenceexistsinthe
currentliterature.
CONTENTS
1. Introduction 1
1.1. Notation 3
2. Examples 4
3. Preliminaries 5
4. Non-separatedGAGA 8
5. Applications 15
AppendixA. SeparatedGAGA 17
References 19
1.1. Notation. We introducesome notationhere that will be used throughoutthe paper.
Fora categoryC and X ∈ ObjC, we have the slice categoryC/X, with objectsthe mor-
phisms V → X in C, and morphisms commuting diagrams over X, which are called X-
morphisms. If the category C has finite limits and f : Y → X is a morphism in C, then
for (V → X) ∈ Obj(C/X), define V := V × Y. Given a morphism p : V′ → V,
Y X
there is an induced morphism p : V′ → V . There is usually an induced functor
Y Y Y
4 J.HALL
f∗ : C/X → C/Y : (V → X) 7→ (V → Y). Given a (2,1)-category C′, these notions
Y
readilygeneralize.
GivenaringedspaceU:=(|U|,O ),asheafofidealsI⊳O ,andamorphismofringed
U U
spacesg:V →U,wedefinethepulledbackidealI =im(g∗I→O )⊳O .
V V V
Fix a scheme S, then an algebraic S-space is a sheaf F on the big e´tale site of S,
(Sch/S) ,suchthatthediagonalmorphism∆ : F → F× Fisrepresentedbyschemes,
E´t F S
andthereis a smoothsurjectionU → F froman S-schemeU. An algebraicS-stack is a
stackHon(Sch/S) ,suchthatthediagonalmorphism∆ :H→H× Hisrepresented
E´t H S
byalgebraicS-spacesandthereisasmoothsurjectionU → HfromanalgebraicS-space
U. We make no separation assumptionson our algebraic stacks, but all algebraicstacks
thatareusedinthispaperwillpossessquasi-compactandseparateddiagonals,thusallof
theresultsof[LMB]apply.
LetAndenotethecategoryofanalyticspaces. Thee´taletopologyonAnistheGrothen-
dieck pretopology generated by local analytic isomorphisms. An analytic stack is an
e´tale stack over An, with diagonal representableby analytic spaces, admitting a smooth
surjectionfromananalyticspace.
2. EXAMPLES
We feel that it is instructive to provideexamples illustrating the necessity of some of
thehypothesesoftheresultscontainedinthispaper.
Example 2.1. Let X be the Deligne-Mumford stack BZ, and fix an elliptic C-curve E.
Observe that H1(E ,Z) = 0 and H1(E(C),Z) 6= 0. Hence, there is a map of analytic
e´t
stacks E → BZ which is not algebraizableto a map of algebraicC-stacks E → BZ.
an an
In particular, we concludethat the quasi-compactdiagonalassumptionwhich appearsin
Theorem1isnecessary.
Example2.2. FixanellipticC-curveE.ThenH1(E(C),E(C))=E(C)⊕2andH1(E ,E)=
e´t
E(C)⊕2. Let[γ]∈H1(E(C),E(C))\H1(E ,E)6=∅. Thenthecohomologyclass[γ]cor-
tors e´t
respondstoananalyticmapE →BE ,whichisnotalgebraizabletoamapofalgebraic
an an
C-stacksE→BE. Thus,weseethatastrengtheningofTheorem1tothesettingwhereX
isseparated,butnotDeligne-Mumford,isnotpossible.
Note that given a projectiveC-scheme X and a locally quasi-finite analytic X -space
an
Z, it is easy to produce examples where Z is non-algebraizablewhen Z is not compact.
Indeed,analyticopensubsetsX whichdonotariseasZariskiopensubsetsofXaresuch
an
examples.
Example 2.3. Consider X := P1 and one of its coordinate patches ı : A1 ֒→ X. Let
C C
U ֒→ A1 be the complement of the closed unit disc. Note that U is an analytic space,
C
and the image of U in X is also open. Let Z = X ∐ X , taken in the category
an an U an
of ringed spaces. Then Z is a compact, but non-Hausdorff analytic space. Indeed, Z
is obtained by gluing two analytic spaces along an open analytic subspace, and there is
a continuous surjection X ∐X → Z. There is also a local analytic isomorphism of
an an
analyticspacesZ→X ,butclearlyZisnotalgebraizable.Inparticular,weconcludethat
an
theseparatednessofthemapZ → X isnecessaryinthedefinitionofpseudosheaves,if
an
wewouldlikeTheorem2tohold.
Example2.4. In[Har77,App.B],thereisgivenanexampleofasmoothprojectivesurface
Y overC, with an opensubschemeı : U ֒→ Y whichpossesesa pairof non-isomorphic
linebundlesLandMsuchthattheanalytifiedlinebundlesL andM areisomorphic.
an an
GENERALIZINGTHEGAGAPRINCIPLE 5
NotethatsinceL(resp. M)isO -coherent,thereisacoherentO -moduleL(resp. M)
U X
togetherwithanisomorphism∗L=∼ L(resp. ∗M =∼ M). Definethesmooth,universally
closed,andfinitetypeC-schemeXbygluingtwocopiesofY alongU. Letp,q : Y ⇉ X
denotethetwodifferentinclusionsofYintoX. OnY wehavetwocoherentO -modules
an Yan
L andM . Notethatsincethereisaninducedanalyticisomorphism∗L =∼ ∗M ,
an an an an an an
wemayobtainacoherentO -moduleF suchthatp∗ F =∼ L andq∗ F =∼ M . IfF is
Xan an an an an
algebraizable,thenthereisacoherentO -moduleFtogetherwithananalyticisomorphism
Y
F =∼ F. In particular, we see that there are inducedanalytic isomorphismsof coherent
an
sheavesonY :
an
(p∗F) =∼ p∗F =∼ L and (q∗F) =∼ q∗ F =∼ M .
an an an an an an
SinceYisaprojectiveC-scheme,byGAGA[SGA ,Exp.XII,Thm.4.4],theinducediso-
I
morphism(p∗F) =∼ L (resp. (p∗F) =∼ L )isuniquelyalgebraizabletoanalgebraic
an an an an
isomorphismp∗F=∼ L(resp.q∗F=∼ M). However,thisimpliesanisomorphism:
L=∼ ∗L=∼ ∗p∗F=∼ ∗q∗F=∼ ∗M=∼ M,
which is a contradiction. Hence, F is not algebraizable. From F, we can construct the
finiteandnon-algebraizableO -algebraO [F]. Takingtheanalyticspecofthisalgebra
Xan Xan
producesafinitehomeomorphismZ → X suchthatZisnon-algebraizable. Hence,itis
an
necessaryforZtobeproper,ifwewouldlikeanylocallyquasi-finiteandseparatedmap
Z→X tobealgebraizable.
an
3. PRELIMINARIES
In this section, we will recall and prove some results for separated spaces. The gen-
eralizationsoftheclassicalGAGAresultstoseparatedalgebraicstackswillbedealtwith
in AppendixA. For a morphism of analytic spaces X → Y, the analytic Hilbert functor
Hilban assignstoeachanalyticY-spaceT,thesetofisomorphismclassesofclosedim-
X/Y
mersionsZ ֒→ X such that the compositionZ → T is properand flat. If U is another
T
analyticY-space,thenonemayalsodefinethefunctorHom (U,X),whichassignstoeach
Y
analyticY-spaceT,thesetofT-morphismsU →X .
T T
In the case that the morphismX → Y is separated, then J. Bingener [Bin80] showed
thatthefunctorHilban naturallyhasthestructureofaseparatedanalyticY-space. If,in
X/Y
addition,UisproperandflatoverY,Bingeneralsoproved[loc.cit.] thatHom (U,X)isa
Y
separatedanalyticY-space.
Remark3.1. TheanalyticityoftheabsoluteHilbertfunctor(i.e. whenYisapoint),isdue
to[Dou66].
ThefollowingvariantoftheHom-spacewillbeusefulinthispaper. Again,wefixan
analyticY-space. LetU → VbeamorphismofanalyticY-spaces. DefineSec (U/V)to
Y
bethefunctorwhichassignstoeachY-spaceT,thesetofanalyticsectionstothemorphism
U →V . Standardargumentsinvolvingfiberproductsprovethefollowing
T T
Proposition 3.2. Fix an analytic space Y, and let U → V be a separated morphism of
analytic Y-spaces, where V → Y is proper and flat. Then Sec (U/V) is a separated
Y
analyticY-space.
Wewillrecordforfuturereferencethefollowing
Corollary3.3. LetX → Ybeamorphismofanalyticstackssuchthatthediagonalmap
∆X/Y isamonomorphism. Thenthediagonalmap∆HSan isrepresentablebyseparated
X/Y
analyticspaces.
6 J.HALL
Proof. FixananalyticY-spaceT. LetZ,Z′ belocallyquasi-finiteandseparatedanalytic
X× T-stacks which are both T-proper and flat. The monomorphism condition on the
Y
diagonalofthemapX → Yensuresthatthe diagonalsofthemapsZ → T andZ′ → T
aremonomorphisms.Inparticular,theanalyticstacksZandZ′arethusanalyticspaces. It
remainstoshowthatIsomHSan (Z,Z′)isaseparatedanalyticT-space.Thereis,however,
X/Y
aninclusionofe´taleT-sheaves
IsomHSan (Z,Z′)֒→SecT(Z×XZ′/Z),
X/Y
which,by[Dou66,§10.1,Prop.1],isrepresentablebyopenembeddingsofanalyticspaces.
ApplyingProposition3.2,weconcludetheresult. (cid:3)
Using GAGA for separated and locally of finite type C-stacks (cf. Theorem A.1 and
CorollaryA.2),themethodofproofemployedinCorollary3.3readilyprovesthefollowing
Lemma3.4. ForalocallyoffinitetypealgebraicC-stackXwithlocallyseparateddiago-
nal,theanalytificationfunctor
Ψ :PS (X)→PS (X )
X p p an
isfullyfaithful.
GivenalgebraicC-spacesUandV,itisa non-trivialtasktodeterminehowthesetof
analyticmorphismsbetweenU andV relatestothesetofalgebraicmorphismsbetween
an an
UandV. Sinceweareinterestedinanalytificationsofalgebraicstacks,whicharedefined
bymoduliproblems,sucharelationwillbeessentialforustoproceed,however.
Lemma3.5. FixalocallyoffinitetypealgebraicC-stackV withlocallyseparateddiago-
nalandalocallyoffinitetypealgebraicV-stackUwithlocallyseparateddiagonal.LetS
bealocalartinianV-scheme,thentheanalytificationfunctor
Hom (S,U)→Hom (S ,U )
V Van an an
isanequivalenceofcategories.
Proof. It suffices to treat the case where V = SpecC. Let k > 0 be such thatthe max-
imal ideal m of the artinian C-algebra Γ(S,O ) satisfies mk = 0. If U is a scheme,
S S S
then a morphismS → U is equivalentto specifyinga pointu ∈ U(C) and a morphism
of C-algebras O /mk → Γ(S,O ). Similarly, an analytic morphism S → U
U,u OU,u S an an
is equivalent to specifying a point u ∈ U = U(C), as well as a morphism of C-
an
algebras O /mk → Γ(S,O ). Recalling that there is a bijection of C-algebras
O /mk Uan,u=∼ OOUan,u/mk asSwell as a bijection of sets U → U(C), then we
U,u OU,u Uan,u OUan,u an
conclude that we’re done in the case that U is a scheme. If U is an algebraic stack, let
U′ →UbeasmoothcoverbyaschemeU′. SinceSislocalartinian,thentheliftingcrite-
rionforsmoothnessimpliesthattheanalytificationfunctorHom(S,U) →Hom(S ,U )
an an
isessentiallysurjective.Thattheanalytificationfunctorisfullyfaithfulisclear. (cid:3)
Lemma 3.6. Fix an analytic stack W, and consider a morphism of analytic W-stacks
p:U→V. SupposethatforanylocalartinianW-schemeS,thefunctor
p∗ |San:HomW(San,U)→HomW(San,V)
isfully faithful. Then p isa monomorphismofanalyticstacks, andis, in particular,rep-
resentablebyanalyticspaces. If, inaddition,thefunctorp∗ |San isessentially surjective,
thenpisananalyticisomorphism.
GENERALIZINGTHEGAGAPRINCIPLE 7
Proof. First, we assume that the p is representable by analytic spaces and p∗ |San is an
equivalence. The lifting criteria for analytic e´tale morphismsshows that p is a bijective
e´tale morphism, hence an analytic isomorphism. In the case where the map p is not as-
sumedtoberepresentablebyanalyticspaces,andp∗ |San isfullyfaithful,weobservethat
thediagonalmap∆p isrepresentablebyanalyticspacesand(∆p)∗ |San isanequivalence.
Bywhatwehaveshownalready,thediagonalmap∆ isananalyticisomorphism,andso
p
themappisamonomorphismofanalyticstacks. Notingthatamonomorphismofanalytic
stacksisrepresentablebyanalyticspaces,thenwe’redone. (cid:3)
ToproveTheorem5,thefollowingstrengtheningofLemma3.6willbeuseful.
Corollary3.7. FixananalyticstackY. Consideramorphismofe´taleY-stacksP :F →G.
Supposethefollowingconditionsaresatisfied:
(1) Fisananalyticstack;
(2) thediagonalofGisrepresentablebyanalyticspaces;
(3) foranylocalartinianY-schemeS,thefunctor
F(S )→G(S )
an an
isfullyfaithful(resp. anequivalence).
ThenPisamonomorphism(resp. anequivalence).
Proof. Conditions (1) and (2) imply that the morphism P is representable by analytic
stacks. ThuswemayassumethatGisananalyticspace,andFisananalyticstack. Condi-
tion(3)impliesLemma3.6applies,andwe’redone. (cid:3)
We deferthe proofTheorem2 to§4, andfocuson applyingthis resulttothe proofof
Theorem5. ByLemma3.5andCorollary3.7,toproveTheorem5,thenextresultwillbe
ofuse.
Lemma 3.8. Fix a locallyoffinitetypealgebraicC-spaceY, anda locallyoffinitetype
algebraicY-spaceX. ForanylocalartinianY-schemeS,thefunctor
Φ (S ):HS (S)→HSan (S )
X/Y an X/Y Xan/Yan an
isanequivalence.
Proof. First, we show that Φ (S ) is fully faithful. Given quasi-finite and separated
X/S an
mapsZ, Z′ ⇉ X× S, wherethetwocompositionsZ,Z′ ⇉ Sareproperandflat,then
Y
regarding X× S as a locally of finite type C-space, as well as Z and Z′ as proper C-
Y
spaces,weconcludebythefullfaithfulnessofΨX×YS thatanyanalyticXan×Yan San-map
Z → Z′ arisesfromanalgebraicX× S-mapZ → Z′ overC. ApplyingGAGA(cf.
an an Y
CorollaryA.3),weconcludethatC-mapsZ→Z′whichanalytifytoS -mapsZ →Z′
an an an
areautomaticallyS-maps—hencewe’veshownthatΦ (S )isfullyfaithful.
X/Y an
ToshowthatthefunctorΦ (S )isessentiallysurjective,wenotethatgivenalocally
X/Y an
quasi-finite and separated morphism of analytic spaces Z → X × S such that the
an Yan an
composition Z → S is proper and flat, then again regarding Z and X × S as living
an Y
over C, then Theorem 2 implies that the morphism Z → X × S algebraizes to a
an Yan an
quasi-finiteandseparatedmorphismofalgebraicC-spacesZ→X× SsuchthatZisC-
Y
proper. ApplyingGAGA(cf. CorollaryA.3)again,wemayalgebraizetheflatmorphism
Z → S to a flat morphismZ → S, which provesessential surjectivity of the functor
an an
Φ (S ). (cid:3)
X/S an
8 J.HALL
ProofofTheorem5. We use the criteria of Corollary3.7. For (1), by [HalR10, Thm.2],
we have that (HS ) is an analytic stack. For (2), by Corollary 3.3 we deduce that
X/Y an
HSan hasrepresentablediagonal.For(3),weapplyLemma3.8. (cid:3)
Xan/Yan
4. NON-SEPARATEDGAGA
ToproveTheorem2,wewillutilizetwomethodsofde´vissage,whicharesummarized
bythenexttwopropositions.
Proposition4.1(Birationalde´vissage). SupposethatXisafinitetypeC-scheme.Suppose
thatforanyclosedimmersion V ֒→ X, thereisaproper, schematic, andbirationalmor-
phism V′ → V such that the functor ΨV′ is an equivalence, then the functor ΨX is an
equivalence.
Proposition4.2(Genericallyfinitee´talede´vissage). SupposethatXisafinitetypeDeligne-
MumfordC-stackwithquasi-compactandseparateddiagonal.Supposethatforanyclosed
immersionV ֒→X,thereisafiniteandgenericallye´talemapV′ →V suchthatthefunc-
torsΨV′ andΨV′×VV′ areequivalences,thenthefunctorΨX isanequivalence.
Itwillalsobenecessarytohavearesultonsectionstoe´talemorphisms.
Proposition4.3. Consider a quasi-compactand e´tale map of locally of finite type alge-
braicC-spacesW →Z. Then,themapof sets
Hom (Z,W)→Hom (Z ,W )
Z Zan an an
isbijective.
The proofs of Propositions 4.1 and 4.2 will fill the remainder of this section, and we
willutilizeargumentsverysimilartothosein[HalR10,§4]. Withtheaboveresultsatour
disposal,however,wecanproveTheorem3immediately.
ProofofTheorem3. By Lemma 3.4, we know that the functorΨ is fully faithful, so it
X
remainstoshowthatthefunctorΨ isessentiallysurjective.
eX
BasicCase.First,wesupposethatXisafinitetypeC-scheme,andthestructuremorphism
factorsasthecompositionX−→f Y −→g SpecC,wherethemorphismfisquasi-compacte´tale
andthe morphismg is projective. Fix a locallyquasi-finitemorphismofanalytic spaces
s:Z→X suchthatZisproper.SincetheanalyticspaceY isseparated,oneconcludes
an an
that the composition f ◦ s : Z → Y is a locally quasi-finite and proper morphism of
an
analytic spaces. By [GR84, Thm. XII.4.2] such a morphism is a finite. Since Y is also
projective,by[SGA ,Exp. XII,Thm. 4.6],weconcludethatthereisafinitemorphismof
I
C-schemesZ → Y such that Z is proper, and Z =∼ Z overY . To complete the proof
an an
inthissetting,itsufficestoproducea quasi-finitemorphismofschemest : Z → Xsuch
that t = s. Note that it is equivalent to produce a section to the quasi-compact e´tale
an
morphism h : X× Z → Z which agrees with the analytic section to the morphism of
Y
analyticspacesh inducedbys. ThisispreciselythecontentofProposition4.3. Now,if
an
weattachtoZacoherentO -moduleF,thensinceZisalgebraizable,GAGAforproper
Z
C-schemes[SGA ,Exp. XII,Thm. 4.4]impliesthatF isalgebraizable. Thusthefunctor
I
Ψ isanequivalenceinthiscase.
X
Finite type C-schemes. Next, we just assume that X is a finite type C-scheme. Note
thatanyclosedsubschemeV ֒→ XisalsoafinitetypeC-scheme. Thus,by[RG71, Cor.
5.7.13], there is a schematic and birational morphism V′ → V such that the structure
morphismof V′ over C factorsas V′ −f−→V Y −g−→V SpecC, where f is quasi-compact
V V
GENERALIZINGTHEGAGAPRINCIPLE 9
e´tale and gV is projective. By the basic case considered above, the functor ΨV′ is an
equivalenceandsobyProposition4.1,weconcludethatthefunctorΨ isanequivalence.
X
FinitetypeDeligne-MumfordC-stacks. Here,weassumethatXisafinitetypeDeligne-
MumfordC-stack.NotethatifV ֒→Xisanyclosedimmersion,thenV isalsoafinitetype
Deligne-MumfordC-stack. By [LMB, Thm. 16.6], there is a finite and genericallye´tale
morphismV′ → V, whereV′ is a scheme. Since V′ is thusa finite typeC-scheme, we
mayconcludebythecase previouslyconsideredthatthefunctorsΨV′ andΨV′×VV′ are
equivalences.Hence,byProposition4.2weconcludethatthefunctorΨ isanequivalence.
X
Generalcase.Fixalocallyquasi-finiteandseparatedmorphismofanalyticstackss:Z→
X whereZ isproperwith finite diagonal. LetX0 denotetheopensubstackofXwhich
an
has quasi-finite diagonal. Note that since Z has finite diagonal, then the quasi-finite and
separatedmaps : Z → X factorscanonicallythroughX0. ReplacingXbyX0,wemay
an an
henceforthassumethatXhasquasi-finiteandseparateddiagonal. Next,letO denotethe
X
categoryofquasi-compactopensubsetsofX. Wenotethat{U}U∈OX isanopencoverof
Xandso{s−1(Uan)}U∈OX isanopencoverofZ. SinceZisacompacttopologicalspace,
andtheexhibitedcoverisclosedunderfiniteunions,thereisanopenimmersionU ֒→ X
suchthatthemapZ → X factorsuniquelythroughU andU isafinitetypeC-stack
an an an
with quasi-finite and separated diagonal. By the previous case considered, we conclude
thatthefunctorΨ isessentiallysurjective,thusanequivalence. (cid:3)
X
Givenamorphismofringedspacesf : U → V,thenwesaythatfisSteinifthemap
of sheaves f♯ : OV → f∗OU is an isomorphism. By [GR84, §10.6.1], if the morphism
f is a proper morphism of analytic spaces, then there is a Stein factorization U −A−(−f→)
A(U) −B−(−f→) V, where the morphism A(f) is proper, Stein, surjective, with connected
fibers,andthemorphismB(f)isfinite. Similarly,ifthemorphismfisapropermorphism
of locally noetherian schemes, then by [EGA, III, 4.3.1], the morphism f has a Stein
factorizationU −A−(−f→) A(U) −B−(−f→) V, whoseformationcommuteswithflat basechange
onV. Whatisimportanthereisthatby[SGA ,Exp. XII,Thm. 4.2],theformationofthe
I
Steinfactorizationcommuteswithanalytification.
Lemma4.4. Fix apropermorphismoflocallyoffinitetypeC-schemesπ : Y → X, and
alocallyquasi-finiteandseparatedmorphismofanalyticspacesσ:Z→X suchthatZ
an
isproper, andtheanalyticspace Z is algebraizableto a quasi-finiteandseparatedY-
Yan
schemeW.IntheSteinfactorizationofthepropermorphism(π ) :Z →Zofanalytic
an Z Yan
spaces,Z −A−(−(π−a−n)−Z→) A(Z )−B−(−(π−a−n)−Z→) Z,theanalyticspaceA(Z )isalgebraizable
Yan Yan Yan
toaquasi-finiteandseparatedX-scheme.
Proof. By [EGA, IV, 18.12.13], there is a finite Y-scheme t : W → Y and an open
immersion:W ֒→W. Definethemapρ:=π◦t:W →X,whichisproper,andconsider
theSteinfactorizationW −A−(−ρ→) A(W)−B−(−ρ→) X. WeclaimthatA(ρ)−1A(ρ)(|W|)=|W|.
Inparticular,sincethemapA(ρ)isproperandsurjective,itisuniversallysubmersive,thus
itwillfollowthatA(ρ)(|W|)isanopensubsetof|A(W)|. Sincetheseparatedmorphism
W → A(W)hasproperfibers, it followsthatforanyw ∈ A(W), the morphismonthe
geometric fibers over w: W → W is an open and closed immersion. Since W is
w w w
geometrically connected, we conclude that W = W or ∅, and the claim is proved.
w w
We let the quasi-finite and separated morphism s′ : Z′ → X denote the induced open
subschemeofA(ρ)(W)definedbytheimageof|W|.Itnowremainstoconstructaunique
morphism of analytic spaces α : Z′ → Z which is compatible with the induced map
an
10 J.HALL
β : W → Z′ . Indeed,sinceW → Z′ isproperandStein,thenW → Z′ → Zis
an an an an an an
theSteinfactorizationof(π ) .
an Z
Wefirstdefinethemapαset-theoretically.Forw∈W′ ,thesubsetβ−1(w)⊂W is
an an
closedandconnected,thussince(π ) isproper,wededucethatu := (π ) β−1(w)
an Z w an Z
isaclosedandconnectedsubsetof|Z|. Moreover,u isalsoquasi-finiteovertheimage
w
ofwinX andwededucethatu isasinglepoint.Hence,weobtainawell-definedmap
an w
ofsetsα : |W′ | → |Z|. Clearly,αiscontinuous,asthemaps(π ) andβaresurjective
an an Z
andsubmersive.Weobtainthemaponfunctionsfromthefollowingcomposition:
OZ →((πan)Z)∗OWan =∼ α∗β∗OWan =∼ α∗OWa′n,
withthelastisomorphismbecauseβisStein. (cid:3)
Wehavetwoeasylemmata.
Lemma4.5. FixacompactanalyticstackX,aO -coherentsheafFandacoherentO -
X X
idealI.
(1) If|suppF|⊂|V(I)|,thenthereisak>0suchthatIkF =(0).
(2) Given a coherentsubsheafF′ ⊂ F suchthatIF′ = (0), thenthere is a k > 0)
suchthat(IkF)∩F′ =(0).
Proof. For(1),theRu¨ckertNullstellansatz[GR84,§3.2]impliesthatforanyx∈X,there
is an open neighborhoodUx and a kx such that (IkxF)Ux = (0). Since X is compact,
we concludethe result. For (2), fix x ∈ X and observe that since the local ring O is
X,x
noetherian, by [AM69, Cor. 10.10]∃kx > 0 such that (IkxxFx)∩Fx′ = (0). Note that
forany x ∈ X, the sheaf of OX-modulesGx = (IkxF)∩F′ is coherent,and hasclosed
support. Thus, as (Gx) = (0), then there is an open neighborhoodU of x ∈ X such
x x
thatGx | =(0). SinceXiscompactintheanalytictopology,weconcludethatthereisa
Ux
finitesetofpointsx1,...,xn ∈XsuchthatGxi |Uxi= (0)and{Uxi}ni=1 coversX. Take
k=maxikxi,then(IkF)∩F′ =Tni=1Gxi =(0),asclaimed. (cid:3)
Lemma 4.6. Fix a propermorphism of finite type C-schemesπ : Y → X, anda locally
quasi-finite and separated morphism of analytic spaces σ : Z → X , such that Z is
an
compact. Let U ֒→ X be an open subscheme such that the induced map π−1U → U
is an isomorphism. Fix a coherent ideal sheaf I⊳O such that |V(I)| = |X\U|. The
X
Steinfactorizationofthepropermorphismofanalyticspaces(π ) : Z → Zinduces
an Z Yan
a map of coherentOZ-algebrasB((πan)Z)♯ : OZ → B((πan)Z)∗OA((πan)Z). There is a
k > 0 such that the coherentO -ideal(I )k annihilatesthe kerneland cokernelof the
Z an Z
mapB((π ) )♯.
an Z
Proof. Define the open subset U′ = σ−1(U ), then certainly we have that the map
an
B((π ) )♯ is an isomorphism when restricted to the open subset U′. In particular, it
an Z
followsthatthekernelandcokernelofthemapB((π ) )♯ aresupportedontheanalytic
an Z
set|V((I ) )|. NowapplyLemma4.5(1). (cid:3)
an Z
For a morphismof ringedtopoif : U → V, we denotethe inducedmap on structure
sheaves by f♯ : OV → f∗OU. Define the conductor of the morphism f to be Cf :=
Ann (cokerf♯). Theresultthatfollowsisthemainde´vissagetechniqueusedinproving
OV
Propositions4.1and4.2.