introduction - kth › ~dary › hilbertstacksnonsep20120626.pdf · unlike schemes, the norm for...

THE HILBERT STACK

JACK HALL AND DAVID RYDH

Abstract. We show that for any locally finitely presented morphism of al-

gebraic stacks X → S with quasi-compact and separated diagonal, there isan algebraic stack HSX/S , the Hilbert stack, parameterizing proper algebraic

stacks with finite diagonal mapping quasi-finitely to X. The technical heart

of this is a generalization of formal GAGA to a non-separated morphism of

algebraic stacks, something that was previously unknown for a morphism ofschemes. The Hilbert stack, for a morphism of algebraic spaces, was claimed

to exist in [Art74, Appendix §1], but was left unproved due to a lack of foun-dational results for non-separated algebraic spaces.

Introduction

Fix a scheme S and a morphism of algebraic stacks π : X → S. Define theHilbert stack, HSX/S , to be the S-stack which sends an S-scheme T to the

groupoid of quasi-finite and representable morphisms (Zs−→ X ×S T ), such that

the composition Zs−→ X ×S T

πT−−→ T is proper, flat, and finitely presented with fi-nite diagonal. There is a substack HSmono

X/S ⊂ HSX/S , described by monomorphisms

(Zs−→ X ×S T ). The main results of this paper are

Theorem 1. Fix a scheme S and a non-separated morphism of noetherian alge-braic stacks π : X → S, then HSmono

X/S is never an algebraic stack.

Theorem 2. Fix a scheme S and a morphism of algebraic stacks π : X → S whichis locally of finite presentation, with quasi-compact and separated diagonal. Then,HSX/S is an algebraic stack, locally finitely presented over S, with quasi-affinediagonal over S.

Theorem 3. Fix a scheme S, a locally finitely presented morphism of algebraicstacks X → S with quasi-finite and separated diagonal; and a morphism of algebraicstacks Z → S which is proper, flat, and finitely presented with finite diagonal. Then,the S-stack T 7→ HOMT (Z×S T,X×S T ) is algebraic, locally finitely presented overS, with quasi-affine diagonal over S.

Fix an algebraic stack U , and a morphism of algebraic stacks Y → U . Foranother morphism of algebraic stacks p : U → V , define the fibered category p∗Y ,the restriction of scalars of Y along p, by (p∗Y )(T ) = Y (T ×V U).

Theorem 4. Fix an algebraic stack Z, and a locally finitely presented morphism ofalgebraic stacks Y → Z with quasi-finite and separated diagonal. For any proper,flat, finitely presented morphism of algebraic stacks p : Z →W with finite diagonal,the restriction of scalars of Y along p, p∗Y , is an algebraic stack, locally finitelypresented over W , with quasi-affine diagonal over W .

Date: 2012-06-26.2010 Mathematics Subject Classification. Primary 14C05; Secondary 14A20, 14D15, 14D23.We would like to sincerely thank Brian Conrad, Jacob Lurie, Martin Olsson, Jason Starr, and

Ravi Vakil for their comments and suggestions.

1

2 J. HALL AND D. RYDH

Theorem 1 is similar to the main conclusion of [LS08], and is included for com-pleteness. In the case that the morphism π : X → S is separated, the Hilbert stack,HSX/S , is equivalent to the stack of properly supported algebras on X, which was

shown to be algebraic in [Lie06]. Thus the new content of this paper is in the removalof separatedness assumptions from similar theorems in the existing literature. Thestatement of Theorem 2 for algebraic spaces appeared in [Art74, Appendix §1], butwas left unproved due to a lack of foundational results. It is important to note thatTheorems 2, 3, and 4 are completely new, even for schemes and algebraic spaces.We wish to point out that if X0 ⊂ X denotes the open locus where the inertia stackIX/S is quasi-finite, then there is an isomorphism of S-stacks HSX0/S

→ HSX/S .

Theorems 3 and 4 generalize [Ols06, Thm. 1.1 & 1.5] and [Aok06] to the non-separated setting, and follow easily from Theorem 2. In the case where π is notflat, as was remarked in [Hal12b], Artin’s Criterion is insufficient. Thus, to proveTheorem 2 we use the algebraicity criterion [op. cit., Thm. A]. The results of [op.cit., §9.2] show that it is sufficient to understand how infinitesimal deformationscan be extended to global deformations (i.e. the effectivity of formal deformations).

Extending infinitesimal deformations of the Hilbert stack is the dearth of “for-mal GAGA” type results (cf. [EGA, III, §5]) for non-separated schemes, algebraicspaces, and algebraic stacks. In this paper, we will prove a generalization of formalGAGA to non-separated morphisms of algebraic stacks. The proof of our version ofnon-separated formal GAGA requires the development of a number of foundationalresults on non-separated spaces, and forms the bulk of the paper.

Contents

Introduction 10.1. Background 20.2. Outline 50.3. Notation 61. Quasi-finite Pushouts 71.1. Algebraic Stacks 71.2. Formal Schemes 122. Adjunctions for Quasi-finite Spaces 142.1. The adjoint pair (π!, π

∗) 142.2. The adjoint pair ($!, $

∗) 192.3. The Theorem on Formal Functions 253. The Existence Theorem 264. Proof of Theorem 2 33References 33

0.1. Background. The most fundamental moduli problem in algebraic geometry isthe Hilbert moduli problem for PN : find a scheme which parameterizes flat familiesof closed subschemes of PN . It was proven by Grothendieck, in [FGA], that thismoduli problem has a solution which is a disjoint union of projective schemes.

In general, given a morphism of schemes X → S, one may consider the Hilbertmoduli problem: find a scheme HilbX/S parameterizing flat families of closedsubschemes of X. It is more precisely described by its functor of points: for any

THE HILBERT STACK 3

scheme T , a map of schemes T → HilbX/S is equivalent to a diagram

Z� � //

##

X ×S T

��T,

where the morphism Z → X×ST is a closed immersion, and the composition Z → Tis proper, flat, and finitely presented. Grothendieck, using projective methods,constructed the scheme HilbPNZ /Z

.

In [Art69], M. Artin developed a new approach to constructing moduli spaces. Itwas proved, by M. Artin in [Art69, Cor. 6.2] and [Art74, Appendix], that the functorHilbX/S had the structure of an algebraic space for any separated and locally finitelypresented morphism of algebraic spaces X → S. The algebraic space HilbX/S isnot, in general, a scheme—even if X → S is a proper morphism of smooth complexvarieties. In more recent work, Olsson–Starr [OS03] and Olsson [Ols05] showedthat the functor HilbX/S is an algebraic space in the case of a separated and locallyfinitely presented morphism of algebraic stacks X → S.

A separatedness assumption on a scheme is rarely restrictive to an algebraicgeometer. Indeed, most schemes algebraic geometers are interested in are quasi-projective or proper. Let us examine some spaces that arise in the theory of moduli.

Example 0.1 (Picard Schemes). Let C → A1 be the family of curves correspond-ing to a conic degenerating to a node. Then the Picard scheme PicC/A1 , which

parameterizes families of line bundles on C/A1 modulo pullbacks from the base, isnot separated. This is worked out in detail in [FGI+05, Ex. 9.4.14].

Example 0.2 (Curves). Let U be the stack of all curves. That is, a morphismT → U from a scheme T , is equivalent to a morphism of algebraic spaces C → Twhich is proper, flat, finitely presented, with one-dimensional fibers. In particular,U parameterizes all singular curves, which could be non-reduced and have manyirreducible and connected components. In [Smy09, Appendix B], it was shown thatU is an algebraic stack, locally of finite presentation over Z. The stack U is inter-esting, as Hassett [Has03], Schubert [Sch91], and Smyth [Smy09] have constructedmodular compactifications of Mg, different from the classical Deligne–Mumfordcompactification [DM69], that are open substacks of U. The algebraic stack U isnot separated.

Unlike schemes, the norm for interesting moduli spaces is that they are non-separated. Indeed, families of interesting geometric objects tend not to have uniquelimits. This is precisely the reason why compactifying moduli spaces is an active,and very difficult, area of research.

Lundkvist and Skjelnes showed in [LS08] that for a non-separated morphismof noetherian algebraic spaces X → S, the functor HilbX/S is never an algebraicspace. We will provide an illustrative example of this phenomenom.

Example 0.3. Consider the simplest non-separated scheme: the line with thedoubled origin. Let us write it as A1

s qA1s=t−(0) A1

t . Now, for a y-line, A1y, we have

a Spec k[[x]]-morphism Ty : Spec k[[x]]→ A1y ×k Speck[[x]] given by

1⊗ x, y ⊗ 1 7→ x.

Thus, we have an induced map over Spec k[[x]]:

Speck[[x]]Ts−→ (A1

s)×k Speck[[x]]→ (A1s qA1s=t−(0)

A1t )×k Speck[[x]].


Now, where x = 0, this becomes the inclusion of one of the two origins, which is aclosed immersion. Where x 6= 0, this becomes a non-closed immersion.

Note that if X → S is separated, any monomorphism Z → X ×S T , such thatZ → T is proper, is automatically a closed immersion. Thus, for a separatedmorphism X → S, the stack HSmono

X/S is equivalent to the Hilbert functor HilbX/S .In the case that the morphism X → S is non-separated, they are different. Notethat in Example 0.3, the deformed object was still a monomorphism, so will notprove Theorem 1 for the line with the doubled-origin. Let us consider anotherexample.

Example 0.4. Consider the line with doubled-origin again, and retain the notationand conventions of Example 0.3. Thus, we have an induced map over Speck[[x]]:

Spec k[[x]]q Spec k[[x]] TsqTt−−−−→ (A1s q A1

t )×k Spec k[[x]]→ (A1s q

A1s=t−(0)

A1t )×k Spec k[[x]].

Where x = 0, this becomes the inclusion of the doubled point, which is a closedimmersion. Where x 6= 0, this becomes non-monomorphic.

Combining this last example, with the proof of [LS08, Thm. 2.6], one readilyobtains the proof of Theorem 1. Thus, for non-separated morphisms of schemes, theobstruction to the existence of a Hilbert scheme is that a monomorphism Z ↪→ Xcould be deformed to a non-monomorphism. So, one is forced to consider maps Z →X. There exist other versions of Hilbert type moduli problems (in the separatedcase) in the literature, all of which were shown to be algebraic stacks:

• Vistoli’s Hilbert stack, described in [Vis91], parameterizes families of finiteand unramified morphisms to a separated stack.• Lieblich, in [Lie06], showed that the stack of properly supported coherent

algebras on a separated and locally finitely presented algebraic stack wasalgebraic.• The stack of branchvarieties—parameterizes varieties mapping finitely toPN . It was shown that this was an algebraic stack with proper componentsin [AK10].• Lieblich, in [Lie06], also considered a generalization of the stack of branchva-

rieties. It was shown that for certain separated stacks (those with projectivecoarse moduli or those admitting a proper flat cover by a quasi-projectivescheme), the stack of brachvarieties has proper components.• Hønsen, in [Høn04], constructed a proper algebraic space parameterizing

Cohen–Macaulay curves with fixed Hilbert polynomial mapping finitely,and birationally onto its image, to projective space.• The second author, in [Ryd11], constructed the Hilbert stack of points for

any morphism of algebraic stacks.

It would be optimal to subsume all of these problems. Hence, the natural class ofmaps to parameterize is quasi-finite maps Z → X. If we can understand quasi-finitemaps, then we can understand unramified morphisms (a natural generalization ofimmersions), and branchvarieties using standard techniques.

Example 0.5. Closed immersions, quasi-compact open immersions, quasi-compactunramified morphisms, and finite morphisms are all examples of quasi-finite mor-phisms. By Zariski’s Main Theorem [EGA, IV, 18.12.13], any quasi-finite andseparated map of schemes Z → X factors as Z → Z → X where Z → Z is an openimmersion and Z → X is finite.

We now define the Generalized Hilbert moduli problem: for a morphism ofalgebraic stacks π : X → S, find an algebraic stack HSX/S such that a map T →HSX/S is equivalent to the data of a quasi-finite and representable map Z → X×ST ,

THE HILBERT STACK 5

with the composition Z → T proper, flat, and finitely presented. This is the fiberedcategory which appears in Theorem 2. The main result in this paper, Theorem 2,is that this stack is algebraic.

Example 0.6. If X → S is separated, then any quasi-finite and separated mapZ → X which has Z → S proper is automatically finite. Hence, HSX/S is the

stack of properly supported coherent algebras on X. Lieblich, in [Lie06], showedthat HSX/S is an algebraic stack whenever the morphism X → S is locally of finitepresentation and separated.

0.2. Outline. In §4, we will prove Theorem 2 using the algebraicity criterion[Hal12b, Thm. A]. To apply this criterion, like Artin’s Criterion [Art74, Thm. 5.3],it is necessary to know that formally versal deformations of objects in the Hilbertstack can be effectivized. Note that effectivity results for moduli problems relatedto separated objects usually follow from the formal GAGA results of [EGA, III, §5],and the relevant generalizations to algebraic stacks, which appear in M. Olsson’spapers [OS03] and [Ols05]. Since we are concerned with non-separated objects, nopreviously published effectivity result applies.

In §3, we prove a generalization of formal GAGA for non-separated algebraicstacks, which is the main technical result of this paper. That is, fix an I-adicnoetherian ring R, and set S = SpecR and Sn = SpecR/In+1. For a morphism ofalgebraic stacks π : X → S, locally of finite type with quasi-compact and separateddiagonal, let πn : Xn → Sn denote the pullback of the map π along the closedimmersion Sn ↪→ S. Suppose that for each n ≥ 0, we have compatible quasi-finiteSn-morphisms sn : Zn → Xn such that the composition πn ◦sn : Zn → Sn is properwith finite diagonal. We show that there exists a unique, quasi-finite S-morphisms : Z → X, such that the composition π ◦ s : Z → S is proper with finite diagonal,and compatible Xn-isomorphisms Z ×X Xn → Zn.

In §§1 and 2, we will develop techniques to prove the effectivity result of §3. Tomotivate these techniques, it is instructive to explain part of Grothendieck’s proofof [EGA, III, 5.1.4]. So, given an I-adic noetherian ring R, let S = SpecR, andSn = SpecR/In+1. For a proper morphism of schemes f : Y → S, let fn : Yn →Sn denote the pullback of the morphism f along the closed immersion Sn ↪→ S.Suppose that for each n ≥ 0, we have a coherent Yn-sheaf Fn and isomorphismsFn+1|Yn ∼= Fn, then [EGA, III, 5.1.4] states that there is a coherent Y -sheaf F

such that F|Yn ∼= Fn. It is better to think of these adic systems of coherent sheavesas coherent sheaves on a formal scheme. We say that a coherent sheaf F on the

formal scheme Y is effectivizable, if there exists a coherent sheaf F on the scheme

Y and an isomorphism of coherent sheaves F ∼= F on the formal scheme Y . Theeffectivity problem is thus recast as: any coherent sheaf F on the formal scheme

Y is effectivizable. This is proven using the method of devissage on the categoryof coherent sheaves Coh (Y ) on the scheme Y . The proof consists of the followingsteps:

(1) given coherent sheaves H, H′ on Y , the natural map of R-modules:

ExtiY (H,H′)→ ExtiY

(H, H′)

is an isomorphism for all i ≥ 0.

(2) Show that if we have an exact sequence of coherent sheaves on Y :

0 // H′ // H // H′′ // 0

and two of H′, H′′, H are effectivizable, then the third is. This follows fromthe exactness of completion and the i = 0,1 statements of (1).

(3) Prove the result for all projective morphisms Y → S.


(4) Use Chow’s Lemma [EGA, II, 5.6.1] to construct a projective S-morphismp : Y ′ → Y that is an isomorphism on a dense open subset of Y and suchthat Y ′ is S-projective;

(5) Use (3) for the projective morphism Y ′ → S to show that p∗F ∼= G for someY ′-coherent G;

(6) Use the Theorem on Formal Functions [EGA, III, 4.1.5] to show that(p∗G)∧ ∼= p∗p

∗F,(7) Combine (5) and (6) to see that p∗p

∗F is effectivizable and note that wehave an adjunction morphism η : F→ p∗p

∗F.(8) Form the two exact sequences:

0 // ker η // Fη // im η // 0

0 // im η // p∗p∗F // coker η // 0.

By noetherian induction on the topological space |Y |, we may assume thatker η and coker η are effectivizable. Since p∗p

∗F is effectivizable, then ap-plying (2) twice, we conclude that F is effectivizable.

The proof of the non-separated effectivity result in §3 will be very similar to thetechnique outlined above, once the steps are appropriately reinterpreted. For a non-separated morphism of schemes X → S, instead of the abelian category Coh (X),we consider the (non-abelian) category QF(X) which consists of quasi-finite andseparated morphisms Z → X. In §1, we will reinterpret (1) and (2) in termsof the existence of pushouts of quasi-finite and separated morphisms along fi-nite morphisms in the category QF(X). The exactness of the completion functor

(·) : Coh (Y )→ Coh (Y ) is reinterpreted as the preservation of these coequalizersunder completion. The analog of projective morphisms Y → S in (3) and (5) aremorphisms which factor as Y → Y ′ → S, where Y → Y ′ is etale, and Y ′ → S isprojective. The analogue of the Chow Lemma used in (4), is a generalization dueto Raynaud–Gruson [RG71, Cor. 5.7.13]. In §2, for a proper morphism q : Y ′ → Y ,we construct an adjoint pair (q!, q

∗) : QF(Y ′) � QF(Y ) which takes the role ofthe adjoint pair (q∗, q∗) : Coh (Y ) � Coh (Y ′). We also show that the adjoint paircan be constructed for locally noetherian formal schemes, and prove an analogue of(6) and (7) in this setting. In §3, we will combine the results of §§1 and 2 to obtainan analogue of (8).

0.3. Notation. We introduce some notation here that will be used throughout thepaper. For a category C and X ∈ ObjC, we have the slice category C/X, withobjects the morphisms 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 → Xis a morphism in C, then for (V → X) ∈ Obj(C/X), define VY := V ×X Y . Givena morphism p : V ′ → V , there is an induced morphism pY : V ′Y → VY . There isan induced functor f∗ : C/X → C/Y : (V → X) 7→ (VY → Y ). Given a highercategory C′, these notions readily generalize.

Given a ringed space U := (|U |,OU ), a sheaf of ideals I / OU , and a morphismof ringed spaces g : V → U , we define the pulled back ideal IV = im(g∗I →OV ) / OV .

Fix a scheme S, then an algebraic S-space is a sheaf F on the big etale site(Sch/S)Et, such that the diagonal morphism ∆F : F → F ×S F is representedby schemes, and there is a smooth surjection U → F from an S-scheme U . Analgebraic S-stack is a stack H on (Sch/S)Et, such that the diagonal morphism∆H : H → H ×S H is represented by algebraic S-spaces and there is a smooth

THE HILBERT STACK 7

surjection U → H from an algebraic S-space U . Note, we make no separationassumptions on our algebraic stacks. We do show, however, that all algebraicstacks figuring in this paper possess quasi-compact and separated diagonals. Thusall of the results of [LMB] apply. We denote the (2, 1)-category of algebraic stacksby AlgStk.

1. Quasi-finite Pushouts

Fix an algebraic stack X, and let the 2-category of algebraic stacks over the stackX be denoted by AlgStk/X. Define the full 2-subcategory RSch/X ⊂ AlgStk/X

to have those objects Ys−→ X, where the morphism s is schematic. We have three

observations here:

(1) given a morphism f : (Zs−→ X) → (Z ′

s′−→ X) in RSch/X, then thecorresponding X-map f : Z → Z ′ is schematic;

(2) since an X-morphism Z → Z ′ is the data of a morphism of stacks Z → Z ′,together with a 2-morphism α : s ⇒ s′ ◦ f , and since s and s′ are repre-sentable, it follows that the 1-morphisms in RSch/X have trivial automor-phisms. Thus RSch/X is naturally 2-equivalent to a 1-category;

(3) if the algebraic stack X is a scheme, then the natural functor Sch/X →RSch/X is an equivalence of categories.

Definition 1.1. LetX be an algebraic stack. Define the following full subcategoriesof RSch/X:

(1) RAff/X has objects the affine morphisms to X;(2) QF(X) has objects the quasi-finite and separated maps to X.

1.1. Algebraic Stacks. For the moment, we will be principally concerned withthe existence of pushouts in the category QF(X), where X is a locally noetherianalgebraic stack. Pushouts in algebraic geometry are usually subtle, so we restrict ourattention to the types that will be useful in §3—pushouts of two finite morphisms.We will, however, pay attention to some more general types of colimits in the2-category of algebraic stacks, as the added flexibility will be useful.

Remark 1.2. Note that because we have a fully faithful embedding of categoriesQF(X) ⊂ RSch/X, a categorical colimit in RSch/X, which lies in QF(X), isautomatically a categorical colimit in QF(X). This will happen frequently in thissection.

Remark 1.3. It is important to observe that a categorical colimit in AlgStk/X is,in general, different to a categorical colimit in RSch/X. Indeed, let X = Spec k,and let G be a finite group, then the X-stack BG is the colimit in AlgStk/X ofthe diagram [GX ⇒ X]. The colimit in the category RSch/X is just X.

The definitions that follow are closely related to those given in [Ryd07, §2].

Definition 1.4. Fix a directed system of algebraic stacks {Zi}i∈I and consider analgebraic stack Z, together with compatible morphisms φi : Zi → Z for every i ∈ I.Then we say that the data (Z, {φi}i∈I) is a

(1) Zariski colimit if the induced map on topological spaces φ : lim−→i|Zi| →

|Z| is a homeomorphism (equivalently, the map qi∈Iφi : qi∈IZi → Z issubmersive and the map φ is a bijection of sets);

(2) weak geometric colimit if it is a Zariski colimit of the directed system{Zi}i∈I , and the canonical map of sheaves of rings OZ → lim←−i(φi)∗OZi is

an isomorphism (in the case of schemes, this is equivalent to the scheme Zbeing the colimit of the directed system of schemes {Zi}i∈I in the categoryof ringed spaces);


(3) universal Zariski colimit if for any algebraic Z-stack Y , the data (Y, {(φi)Y }i∈I)is a Zariski colimit of the directed system {(Zi)Y }i∈I ;

(4) geometric colimit if it is a universal Zariski and weak geometric colimitof the directed system {Zi}i∈I ;

(5) uniform geometric colimit if for any flat, algebraic Z-stack Y , the data(Y, {(φi)Y }i∈I) is a geometric colimit of the directed system {(Zi)Y }i∈I .

The following criterion will be useful for verifying when a colimit is a universalZariski colimit.

Lemma 1.5. Suppose that we have a directed system of algebraic stacks {Zi}i∈I ,an algebraic stack Z, and compatible morphisms φi : Zi → Z for all i ∈ I. If themap qiφi : qiZi → Z is surjective and universally submersive and also

(1) for any geometric point SpecK → Z, the map φK : lim−→i|(Zi)K | → |SpecK|

is an injection of sets; or(2) there is an algebraic stack X such that Z and Zi ∈ QF(X) for all i ∈ I, the

maps φi are X-maps, and for any geometric point SpecL → X, the mapψL : lim−→i

|(Zi)L| → |ZL| is an injection of sets,

then (Z, {φi}i∈I) is a universal Zariski colimit of the directed system {Zi}i∈I .

Proof. To show (1), we observe that the universal submersiveness hypothesis onthe map qiφi : qiZi → Z reduces the statement to showing that for any morphismof algebraic stacks Y → Z, the map φY : lim−→i

|(Zi)Y | → |Y | is a bijection of

sets, which will follow if the map φK : lim−→i|(Zi)K | → | SpecK| is bijective for any

geometric point SpecK → Y . By assumption, we know that φK is injective, forthe surjectivity, we note that we have a commutative diagram of sets:∐

i |(Zi)K |αK //

qiφi ''

lim−→i|(Zi)K |

φK

��|SpecK|

,

where αK and qiφi are surjective, thus φK is also surjective. For (2), given ageometric point SpecL → X, then Z(SpecL) = |ZL| and Zi(SpecL) = |(Zi)L|,thus we may apply the criterion of (1) to obtain the claim. �

The exactness of flat pullback of sheaves easily proves

Lemma 1.6. Suppose that we have a finite directed system of algebraic stacks{Zi}i∈I , then a geometric colimit (Z, {φi}i∈I) is a uniform geometric colimit.

Also, note that weak geometric colimits in the category of algebraic stacks are notunique and thus are not categorical colimits—the example in Remark 1.3 demon-strates this. In the setting of schemes, however, we have

Lemma 1.7. Given a directed system of schemes {Zi}i∈I , with a weak geometriccolimit (Z, {φi}i∈I), then it is a colimit in the category of locally ringed spaces, thusis a colimit in the category of schemes.

To obtain a similarly useful result for stacks, we will need some relative notions,unlike the previous definitions which were all absolute.

Definition 1.8. Fix an algebraic stack X and let {Zi}i∈I be a directed systemof objects in RSch/X. Suppose that Z ∈ RSch/X, and we have compatibleX-morphisms φi : Zi → Z for all i ∈ I. We say that the data (Z, {φi}i∈I) is auniform categorical colimit in RSch/X if for any flat, algebraic X-stack Y , thedata (ZY , {(φi)Y }i∈I) is the categorical colimit of the directed system {(Zi)Y }i∈Iin RSch/Y .

THE HILBERT STACK 9

Combining Lemmata 1.6 and 1.7 we obtain

Proposition 1.9. Let X be an algebraic stack, and suppose that {Zi}i∈I is a finitedirected system of objects in RSch/X. Then a geometric colimit (Z, {φi}i∈I) inRSch/X is also a uniform geometric and uniform categorical colimit in RSch/X.

Proof. By Lemma 1.6, it suffices to show that Z is a categorical colimit. Let U → Xbe a smooth surjection from a scheme. By Lemmata 1.6 and 1.7, we see that(ZU , {(φi)U}i∈I) is a categorical colimit of the system {(Zi)U}i∈I in RSch/U , andremains so after flat schematic base change on U . Suppose that we have compatibleX-morphisms αi : Zi → W . Then we want to show that there is a unique X-morphism α : Z → W which is compatible with this data. Let R = U ×X U , andlet s, t : R → U denote the two projections. Since ZU is the categorical colimit inRSch/U , there is a unique X-morphism βU : ZU → WU which is compatible with(αi)U : (Zi)U →WU .

Suppose for the moment that X is an algebraic space. Then R is a schemeand the maps s, t : R → U are morphisms of schemes. In particular, we knowthat ZR is a categorical colimit in RSch/R and so there is a unique morphismβR : ZR → WR which is compatible with the morphisms (αi)R : (Zi)R → WR.Noting that (βU )×U,sR and (βU )×U,tR are also such maps, we conclude that thesemaps are actually equal (as maps of algebraic spaces) and so by smooth descent,we conclude that there is a unique X-morphism α : Z →W , and so (Z, {φi}i∈I) isa categorical colimit if X is an algebraic space. Applying Lemma 1.6, one is ableto conclude that (Z, {φi}i∈I) remains a categorical colimit after flat representablebase change on X, for all algebraic spaces X.

Now, returning to the case that X is an algebraic stack, we know that s, t : R→U are flat representable morphisms, repeating the descent argument given aboveshows that (Z, {φi})i∈I is a categorical colimit in RSch/X. �

The following result is a generalization of [Kol11, Lem. 17], which treats the caseof a finite equivalence relation of schemes.

Theorem 1.10. Let X be a locally noetherian algebraic stack, and suppose that

[Z ′t′←− Z t′′−→ Z ′′] is a diagram in RSch/X. Assume that t′ and t′′ are finite, and

that Z, Z ′, Z ′′ ∈ QF(X). Then this diagram has a uniform categorical colimit Z ′′′

in RSch/X, which is a uniform geometric colimit and Z ′′′ ∈ QF(X). Moreover,the structure morphisms m′ : Z ′ → Z ′′′ and m′′ : Z ′′ → Z ′′′ are finite. If, in addi-tion, t′ (resp. t′′) is a closed immersion, then the cocartesian diagram in RSch/X:

Zt′ //

t′′

��

Z ′

m′

��Z ′′

m′′ // Z ′′′

is also cartesian and m′′ (resp. m′) is a closed immersion.

We will need some lemmata to prove Theorem 1.10.

Lemma 1.11. Let X be an algebraic stack, and suppose that {Zi}i∈I is a finitedirected system in RAff/X. Then this system has a categorical colimit in RAff/X,whose formation commutes with flat base change on X. If the system is of the form

[Z ′t′←− Z t′′−→ Z ′′] and t′ is a closed immersion, then the induced cocartesian square

in RAff/X is also cartesian and the morphism to the colimit m′′ : Z ′′ → Z ′′′ is aclosed immersion.

Proof. By [LMB, Prop. 14.2.4], there is an anti-equivalence of categories betweenRAff/X and the category of quasicoherent sheaves of OX -algebras, which com-

mutes with arbitrary change of base, and is given by (Zs−→ X) 7→ (OX

s]−→ s∗OZ).


Since the category of quasicoherent OX -algebras has finite limits, it follows thatif si : Zi → X denotes the structure map of Zi, then the categorical colimit isZ = SpecX lim←−i(si)∗OZi . Since flat pullback of sheaves is exact, the formation of

this colimit commutes with flat base change on X. If t′ is a closed immersion, theisomorphism of rings A ⊗A×A/JB B ∼= A/J for ring maps [A → A/J ← B] showsthat the pushout diagram is also cartesian. �

Lemma 1.12. Let X be a locally noetherian algebraic stack, and suppose that

[Z ′t′←− Z

t′′−→ Z ′′] is a diagram in RSch/X, where Z, Z ′, Z ′′ are all finite overX, then this diagram has a uniform categorical and uniform geometric colimit Z ′′′

in RSch/X, which is finite over X. If t′ is a closed immersion, then the inducedcocartesian square in RSch/X is also cartesian and m′′ : Z ′′ → Z ′′′ is a closedimmersion.

Proof. By Lemma 1.11, the diagram has a categorical colimit Z ′′′ in RAff/X, andsince X is locally noetherian, one readily deduces that Z ′′′ is finite over X. ByProposition 1.9, it remains to show that Z ′′′ is a geometric colimit. Note that sinceZ ′ q Z ′′ → Z ′′′ is dominant and finite, it is surjective, universally closed and thusuniversally submersive. First, we show that Z ′′′ is a universal Zariski colimit usingthe criterion of Lemma 1.5(2).

We now follow [Kol11, Lem. 17]. Let x : Speck → X be a geometric point.

By [EGA, 0III, 10.3.1] this map factors as Spec k x1

−→ X1 p−→ X, where p is flatand X1 is the spectrum of a maximal-adically complete, local noetherian ring withresidue field k. Since the algebraic stack Z ′′′ is a uniform categorical colimit inRAff/X, we may replace X by X1, and we denote the unique closed point ofX by x. For a finite X-scheme U , let π0(U) be its set of connected components.The assumptions on X guarantee that there is a unique, universal homeomorphismhU : Ux → qm∈π0(U){x} which is functorial with respect to U . In particular, there

is a unique factorization UsU−−→ qm∈π0(U)X → X such that (sU )x = hU . Since

π0(−) is a functor, we obtain a diagram:[ ∐m∈π0(Z′)X

∐m∈π0(Z)X

π0(t′)oo π0(t′′) // ∐m∈π0(Z′′)X

]in RAff/X, and we let X ′′′ be the categorical colimit of this diagram in RAff/X.There is thus a canonical map µ : Z ′′′ → X ′′′ and a functorial bijection of setsπ0(Z ′)qπ0(Z) π0(Z ′′)→ π0(X ′′′). In particular, the bijection νx : |Z ′x| q|Zx| |Z ′′x | →|X ′′′x | factors as |Z ′x| q|Zx| |Z ′′x |

ψx−−→ |Z ′′′x |µx−−→ |X ′′′x | and thus ψx : |Z ′x| q|Zx| |Z ′′x | →

|Z ′′′x | is injective. Hence, we have shown that Z ′′′ is a universal Zariski colimit andit remains to show that Z ′′′ has the correct functions.

Let m′ : Z ′ → Z ′′′, m′′ : Z ′′ → Z ′′′ denote the canonical maps and let m : Z →Z ′′′ denote the induced map from Z. Now there is a canonical morphism of sheavesof OX -algebras ε : OZ′′′ → m′∗OZ′ ×m∗OZ m′′∗OZ′′ , which we have to show is anisomorphism. Let h′′′ : Z ′′′ → X denote the structure map, then by functoriality,we have an induced morphism of sheaves of OX -algebras:

ε2 : h′′′∗ OZ′′′h′′′∗ ε−−−→ h′′′∗ (m′∗OZ′ ×

m∗OZm′′∗OZ′′)

ε1−→ h′′′∗ m′∗OZ′ ×

h′′′∗ m∗OZ

h′′′∗ m′′∗OZ′′ .

Since the functor h′′′∗ is left exact, ε1 is an isomorphism; by construction of Z ′′′ themap ε2 is an isomorphism and so h′′′∗ ε is an isomorphism. Since h′′′ is affine, thefunctor h′′′∗ is faithfully exact and we conclude that the map ε is an isomorphismof sheaves. �

Proof of Theorem 1.10. By Proposition 1.9, it suffices to construct a geometric col-imit. By hypothesis, the morphisms s : Z → X, s′ : Z ′ → X, s′′ : Z ′′ → X are

THE HILBERT STACK 11

quasi-finite, separated, and representable. Using Zariski’s Main Theorem [LMB,Thm. 16.5(ii)], there are finite X-morphisms s′ : W ′ → X, s′ : W ′′ → X and open,dense immersions ı′ : Z ′ ↪→ W ′ and ı′′ : Z ′′ ↪→ W ′′. Let W0 = W ′ ×X W ′′, andapply Zariski’s Main Theorem again [loc. cit.] to Z →W0 and let W be finite over

W0 with ı : Z ↪→ W open and dense. We thus obtain finite maps t′

: W → W ′,

t′′

: W → W ′′. Note that if t′ (resp. t′′) is a closed immersion, then we can choose

t′

(resp. t′′) to be a closed immersion. To fix notation, we let s : W → X denote

the induced structure map.

We observe by Lemma 1.12, that the diagram [W ′t′

←−W t′′

−→W ′′] has a uniformgeometric and uniform categorical colimit in QF(X), s′′′ : W ′′′ → X. We take|Z ′′′| to be the set-theoretic image of |Z ′|q |Z ′′| in |W ′′′|. We claim that |Z ′′′| is anopen subset of |W ′′′|. Since W ′ qW ′′ → W ′′′ is universally submersive, to checkthat |Z ′′′| is an open subset of |W ′′′|, it suffices to show that the preimage of |Z ′′′|under W ′ qW ′′ → W ′′′ is precisely |Z ′| q |Z ′′| (which is open). This last claimwill follow if we show that the pullbacks of ı′ : Z ′ ↪→ W ′ and ı′′ : Z ′′ ↪→ W ′′ byW → W ′ and W → W ′′, respectively, both are Z. Indeed, this is nothing otherthan the definition of the colimit of topological spaces. By symmetry, it suffices

to prove this claim for Z ′. Now, we have canonical maps Zα−→ Z ′ ×W ′ W

β−→ Wand since the maps β ◦ α and β are open immersions, α : Z → Z ′ ×W ′ W is an

open immersion. We also see that Zα−→ Z ′ ×W ′ W → Z ′ is a composition of finite

morphisms thus α : Z → Z ′ ×W ′ W is open and closed. Since β ◦ α : Z → W isdense, we conclude that Z = Z ′ ×W ′ W . In particular, we have shown that thecontinuous map ψX : |Z ′| q|Z| |Z ′′| → |Z ′′′| is a homeomorphism.

Hence, we let ı′′′ : Z ′′′ ↪→ W ′′′ be the open immersion associated to the open

subset |Z ′′′| ⊂ |W ′′′| and it remains to show that (Z ′′′s′′′−−→ X) ∈ QF(X) is a

geometric colimit. We obtain a commutative diagram in QF(X):

(1.1)

Zı��

t′ //

t′′

��

Z ′

m′

��

ı′

��W

t′′

��

t′

// W ′

m′

��Z ′′

m′′//

ı′′��Z ′′′

ı′′′��W ′′

m′′// W ′′′,

where all sides are cartesian, except for the front and back. In particular, we havethat m′ : Z ′ → Z ′′′ and m′′ : Z ′′ → Z ′′′ are finite and if t′ (resp. t′′) is a closedimmersion, then the front and back squares are also cartesian and m′′ (resp. m′)is a closed immersion (Lemma 1.12). Moreover, for every morphism of algebraicstacks Y → X, we have a bijection ψY : |W ′Y | q|WY | |W ′′Y | → |W ′′′Y | and a mapψY : |Z ′Y | q|ZY | |Z ′′Y | → |Z ′′′Y |. By Lemma 1.5(2), it suffices to show that ψY isinjective when Y = SpecK and K is an algebraically closed field. Note that wehave the commutative diagram of sets:

|Z ′Y | q|ZY | |Z ′′Y |

δY��

ψY // |Z ′′′Y |

ı′′′Y��

|W ′Y | q|WY | |W ′′Y |ψY

// |W ′′′Y |,

and from the injectivity of the map ψY , it suffices to prove that the map δY isinjective, which is obvious from the injectivity of ıY , ı′Y , ı′′Y ; thus Z ′′′ is a universalZariski colimit. To show that Z ′′′ has the correct functions, we let m : Z → Z ′′′


and m : W →W ′′′ denote the canonical maps. We observe that there are canonicalisomorphisms:

OZ′′′ ∼= ı′′′−1OW ′′′ ∼= ı′′′−1(m′∗OW ′) ×ı′′′−1(m∗OW )

ı′′′−1(m′′∗OW ′′).

Hence, it suffices to show that ı′′′−1m′∗OW ′ = m′∗OZ′ (and similarly for the otherobjects), but this is obvious from the definitions and the cartesian squares in dia-gram (1.1). �

1.2. Formal Schemes. Here, we will extend Theorem 1.10 to locally noetherianformal schemes. Denote the category of formal schemes by FSch. We require somemore definitions that are analogous to those given in §1.1.

Definition 1.13. Consider a directed system of formal schemes {Zi}i∈I , a formalscheme Z, and suppose that we have compatible maps ϕi : Zi → Z for every i ∈ I.Then we say that the data (Z, {ϕi}i∈I) is a

(1) formal Zariski colimit if the induced map on topological spaces lim−→i|Zi| →

|Z| is a homeomorphism;(2) formal weak geometric colimit if it is a formal Zariski colimit and

the canonical map of sheaves of rings OZ → lim←−i(ϕi)∗OZi is a topological

isomorphism, where we give the latter sheaf of rings the limit topology (thisis nothing other than Z being the colimit in the category of topologicallyringed spaces).

If, in addition, the formal scheme Z is locally noetherian, and the maps ϕi : Zi → Zare topologically of finite type, then we say that the data (Z, {ϕi}i∈I) is a

(3) universal formal Zariski colimit if for any adic formal Z-scheme Y, thedata (Y, {(ϕi)Y}i∈I) is the formal Zariski colimit of the directed system{(Zi)Y}i∈I ;

(4) formal geometric colimit if it is a universal formal Zariski and a formalweak geometric colimit of the directed system {Zi}i∈i;

(5) uniform formal geometric colimit if for any adic flat formal Z-schemeY, the data (Y, {(ϕi)Y}i∈I) is a formal geometric colimit of the directedsystem {(Zi)Y}i∈I .

We have two lemmata which are formal analogues of Lemmata 1.6 and 1.7.

Lemma 1.14. Given a directed system of formal schemes {Zi}i∈I , then a weakgeometric colimit (Z, {ϕi}i∈I) is a colimit in the category of topologically locallyringed spaces, thus is a colimit in the category FSch.

Lemma 1.15. Consider a finite directed system of locally noetherian formal schemes{Zi}i∈I , and a locally noetherian formal scheme Z, together with finite morphismsϕi : Zi → Z. If the data (Z, {ϕi}i∈I) is a formal geometric colimit of the directedsystem {Zi}i∈I , then it is a uniform formal geometric colimit.

Proof. Let $ : Y → Z be an adic flat morphism of locally noetherian formalschemes. It remains to show that the canonical map ϑY : OY → lim←−i[(ϕi)Y]∗O(Zi)Y

is a topological isomorphism. Since ϕi is finite for all i, and we are taking a finitelimit, we conclude that it suffices to show that ϑY is an isomorphism of coherent OY-modules. By hypothesis, the map ϑZ : OZ → lim−→i

(ϕi)∗OZi is an isomorphism, and

since $ is adic flat, $∗ is an exact functor from coherent OZ-modules to coherentOY-modules, thus commutes with finite limits. Hence, we see that ϑY factors asthe following sequence of isomorphisms:

OY∼= $∗OZ

∼= lim←−i

$∗[(ϕi)∗OZi ]∼= lim←−

i

[(ϕi)Y]∗O(Zi)Y . �


Definition 1.16. Fix a locally noetherian formal scheme X and let {Zi}i∈I bea directed system of objects in FSch/X, where each formal scheme Zi is locallynoetherian. Let Z ∈ FSch/X also be locally noetherian, and suppose that we havecompatible X-morphisms ϕi : Zi → Z that are topologically of finite type. Thenwe say that the data (Z, {ϕi}i∈I) is a uniform categorical colimit in FSch/Xif for any adic flat formal X-scheme Y, the data (ZY, {(ϕi)Y}i∈I) is the categoricalcolimit of the directed system {(Zi)Y}i∈I in FSch/Y.

Definition 1.17. Let X be a formal scheme. Define QF(X) to be the category

whose objects are adic, quasi-finite, and separated maps (Zσ−→ X). A morphism in

QF(X) is an X-morphism f : (Zσ−→ X)→ (Z′

σ′−→ X).

For a scheme X, and a closed subset |V | ⊂ |X|, we define the completionfunctor

cX,|V | : Sch/X → FSch/X/|V | : (Zs−→ X) 7→ (Z/|s|−1|V | → X/|V |).

Note that restricting cX,|V | to QF(X) has essential image contained in QF(X/|V |).

Theorem 1.18. Let X be a locally noetherian scheme, suppose that |V | ⊂ |X|is a closed subset, and let [Z ′

t′←− Zt′′−→ Z ′′] be a diagram in QF(X) with t′, t′′

finite. If the scheme Z ′′′ denotes the categorical colimit of the diagram in Sch/X,which exists by Theorem 1.10, then cX,|V |(Z

′′′) is a uniform categorical and uniformformal geometric colimit of the diagram

[cX,|V |(Z′)

t′←− cX,|V |(Z)t′′−→ cX,|V |(Z

′′)]

in FSch/X/|V |.

For a locally noetherian scheme X, and a closed subset |V | ⊂ |X|, if a formal

X/|V |-scheme Z belongs to the essential image of the functor cX,|V |, we will say thatZ is effectivizable. Theorem 1.18 trivially implies

Corollary 1.19. If X is a locally noetherian scheme, and Z ∈ QF(X/|V |) is ob-

tained by pushing out effectivizable objects of QF(X/|V |) along finite morphisms,then Z is effectivizable.

Proof of Theorem 1.18. By Lemmata 1.14 and 1.15, it suffices to show that cX,|V |(Z′′′)

is a formal geometric colimit of the diagram [cX,|V |(Z′)← cX,|V |(Z)→ cX,|V |(Z

′′)].We first show that cX,|V |(Z

′′′) is a universal formal Zariski colimit. Let $ : Y→ Xbe an adic morphism of locally noetherian formal schemes. Let I be a coher-ent sheaf of radical ideals defining |V | ⊂ |X| so that ($−1I)OY is an ideal ofdefinition of Y. Let Xn = V (In) ⊂ X, Yn = Y ×X Xn, Zn = Z ×X Xn,Z ′n = Z ′×XXn, and Z ′′n = Z ′′×XXn. Then by Theorem 1.10 the map of topologi-cal spaces |(Z ′0)Y0

|∐|(Z0)Y0 |

|(Z ′′0 )Y0| → |(Z ′′′0 )Y0

| is a homeomorphism. Noting that

|ZY| = |(Z0)Y0| (and similarly for all of the other objects appearing), we conclude

that cX,|V |(Z′′′) is a universal formal Zariski colimit and it remains to show that

cX,|V |(Z′′′) has the correct functions.

Let m : Z → Z ′′′, m′ : Z ′ → Z ′′′, and m′′ : Z ′′ → Z ′′′ denote the canon-ical morphisms. By Theorem 1.10 we have an isomorphism of sheaves of ringsφ : OZ′′′ → m′∗OZ′ ×m∗OZ m′′∗OZ′′ . Hence, since m, m′ and m′′ are finite, and com-pletion is exact on coherent modules, by [EGA, I, 10.8.8(i)] we have an isomorphismof coherent OcX,|V |(Z′′′)-algebras:

OcX,|V |(Z′′′) = (m′∗OZ′ ×m∗OZ

m′′∗OZ′′)∧ ∼= (m′∗OZ′)

∧ ×(m∗OZ)∧

(m′′∗OZ′′)∧.


Noting that (m′∗OZ′)∧ ∼= m′∗OcX,|V |(Z′) (and similarly for the others), we deduce

that there is an isomorphism of coherent OcX,|V |(Z′′′)-algebras:

OcX,|V |(Z′′′)∼= m′∗OcX,|V |(Z′) ×

m∗OcX,|V |(Z)

m′′∗OcX,|V |(Z′′).

It remains to show that this isomorphism is topological (where we topologize theright side with the limit topology). A general fact here is that the topology on

the right is the subspace topology of the product topology on m′∗OcX,|V |(Z′) ×m′′∗OcX,|V |(Z′′). Since m′ and m′′ are finite, considering the Artin–Rees Lemma

[AM69, Thm. 10.11], one concludes that this is an equivalent topology to the re-spective adic topologies, thus we have the topological isomorphism as required. �

2. Adjunctions for Quasi-finite Spaces

If π : X ′ → X is a morphism of schemes, there is a pullback functor π∗ :QF(X)→ QF(X ′) given by (Z → X) 7→ (Z×XX ′ → X ′). Similarly, if $ : X′ → Xis a morphism of formal schemes, there is a pullback $∗ : QF(X) → QF(X′). Inthis section, we will be concerned with the construction of left adjoints to thesefunctors.

2.1. The adjoint pair (π!, π∗). To motivate the construction of π!, we emphasize

that what we are constructing is an entirely natural thing: for a morphism ofschemes π : X → Y , we want to be able to take a quasi-finite and separated mapZ → X to a quasi-finite and separated map π!Z → Y in a functorial way. Twosituations present themselves immediately that are worth paying closer attentionto:

(1) if π is quasi-finite and separated, then define π! : QF(X) → QF(Y ) as

π!(Z → X) = (Z → Xπ−→ Y ). It is immediate that (π!, π

∗) is an adjointpair.

(2) If π is proper and s : Z → X is finite, then we define π!(Zs−→ X) as

(SpecY (π∗s∗OZ)→ Y ). It is clear that (π!, π∗) forms an adjoint pair, when

we restrict π∗ to the full subcategory of QF(Y ) consisting of (Z ′s′−→ Y ),

where s′ is finite.

For a general, quasi-finite and separated Z → X, one can use Zariski’s Main The-orem to compactify it to Z ↪→ W → X, where W → X is finite. One then pushesW forward to get a finite Y -scheme W ′ and hopes that the image of Z in W ′ isreasonable enough to endow it with an algebraic structure. We would then liketo take π!Z to be the afforementioned image. To get the image of Z in W ′ to besufficiently well-behaved, we need to make the following definition.

Definition 2.1. For a morphism of schemes π : X → Y , let QFp.f.(X/Y ) denote

the full subcategory of QF(X) consisting of the objects (Zs−→ X) such that the

map π ◦ s : Z → Y has proper fibers.

The next result gives a method of producing lots of examples of objects ofQFp.f.(X/Y ).

Lemma 2.2. For a proper morphism of schemes π : X → Y , and (Zs−→ Y ) ∈

QF(Y ), then (π∗Z → X) ∈ QFp.f.(X/Y ).


Proof. Let Specκ(y)→ Y be a point, then we form the following cartesian diagram:

(π∗Z)y //

��**

π∗Z%%

��Xy

//

��

X

��Zy **

// Z%%

Specκ(y) // Y.

Since Z → Y is quasi-finite, then Zy → Specκ(y) is finite, thus (π∗Z)y → Xy

is finite and since Xy → Specκ(y) is proper, we conclude that the compositionπ∗Z → X → Y has proper fibers. �

Recall, that a morphism of schemes f : X → Y is Stein if the induced mapf ] : OY → f∗OX is an isomorphism. We now state the main result of this section.

Theorem 2.3. Let π : X → Y be a proper morphism of locally noetherian schemes,then there is a functor π! : QFp.f.(X/Y ) → QF(Y ), which is left adjoint to π∗.Moreover,

(1) if (Zs−→ X) ∈ QFp.f.(X/Y ), then:

(a) the unit of the adjunction, ηZ : Z → π∗π!Z, is finite;

(b) the canonical map of schemes πZ : ZηZ−−→ π∗π!Z → π!Z is proper and

Stein;

(c) if, in addition, s is finite, then π!(Zs−→ X) = (SpecY (π∗s∗OZ)→ Y );

(d) if π is finite, then π!(Zs−→ X) = (Z

π◦s−−→ Y ).(2) if (Z ′ → Y ) ∈ QF(Y ), then the counit of the adjunction, εZ′ : π!π

∗Z ′ → Z ′,is finite.

We defer the proof of Theorem 2.3 until the end of this section, and will presentlyconcern ourselves with its corollaries.

The statements in Theorem 2.3 are already consistent with our motivation, butit is possible to strengthen this.

Corollary 2.4. Consider a commutative diagram of locally noetherian schemes:

Ug //

ρ��

X

π��V

f// Y.

Suppose π, ρ are proper, then

(1) if f and g are proper, there are natural isomorphisms of functors fromQFp.f.(U/Y ) to QF(Y ):

f!ρ! =⇒ (f ◦ ρ)! ⇐⇒ (π ◦ g)! ⇐= π!g!;

(2) if f and g are instead quasi-finite and separated, there is a natural isomor-phism of functors from QFp.f.(U/Y ) to QF(Y ):

π!g! =⇒ f!ρ!.

Given a quasi-compact and quasi-separated morphism of schemes g : U → V ,

then it has a Stein factorization: Ug′−→ SpecV g∗OU → V , where g′ is a Stein

morphism. One obtains immediately from Corollary 2.4(2):

Corollary 2.5. Given a proper morphism π : X → Y of locally noetherian schemesand (Z ′ → Y ) ∈ QF(Y ), set Z := π∗Z ′ and let πZ′ : Z → Z ′ denote the induced

map. Then there is a natural Z ′-isomorphism (πZ′)!Z → π!Z. Thus, ZπZ−−→

π!π∗Z ′

εZ′−−→ Z ′ is canonically isomorphic to the Stein factorization of the morphismπZ′ : Z → Z ′.


The next result is important for technical reasons, as it allows one to work flatlocally.

Corollary 2.6. Consider a commutative diagram of locally noetherian schemes,

X ′p′ //

π′ ��

X

π��Y ′

p// Y.

If π and π′ are proper, or quasi-finite and separated, then there is a natural trans-formation:

∆ : π′!p′∗ =⇒ p∗π!.

If, in addition, the diagram is cartesian, and

(1) if π is proper, then ∆ induces a finite, universal homeomorphism; or(2) if p is flat, then ∆ is an isomorphism of functors; or(3) if π is quasi-finite and separated, then ∆ is an isomorphism of functors.

Combined with the etale-local structure theorem for quasi-finite morphisms [EGA,IV, 18.12.1] and Corollary 2.6, the following result allows one to compute the ad-junction map explicitly.

Corollary 2.7. Let π : X → Y be a proper map of locally noetherian schemes, let

(Zs−→ Y ) ∈ QF(Y ), and let U → Z be a quasi-finite, separated, and flat morphism

(e.g., an open immersion). Then there is a cartesian diagram:

π!π∗U

ηU //

��

U

��π!π∗Z

η// Z.

In the case that the induced sU : U → Y is finite, we obtain a canonical isomorphism

π!π∗U ∼= SpecY (π∗π

∗(sU )∗OU ).

In this case, the map ηU : π!π∗U → U is given by the adjunction map on coherent

sheaves of OY -algebras: (sU )∗OU → π∗π∗(sU )∗OU .

Before we prove Theorem 2.3, we include here for future reference a technicallemma that will be used frequently. First, we observe that if a quasi-compact andquasi-separated morphism of schemes f : X → Y is Stein, then it remains soafter flat base change on Y . If f : X → Y is a proper Stein morphism of locallynoetherian schemes, then it is surjective and has geometrically connected fibers.Note that even though the property of being Stein is not preserved under arbitrarybase change, the properties of surjectivity and geometric connectivity of the fibersare preserved.

Lemma 2.8. Consider a commutative diagram of schemes

Xf

��g

��Y

h// Z,

where f and g are quasi-compact, quasi-separated, surjective and have geometricallyconnected fibers. If

(1) h is integral; or(2) f and g are universally closed, and h has discrete fibers,

then h is an integral, universal homeomorphism. If, in addition, f and g are Stein,then h is an isomorphism.


Proof. Under both assumptions, we obtain that h is surjective, and has geometri-cally connected fibers. Since in either case, the fibers of h are discrete, h is radiciel.Furthermore, h is universally closed. By [EGA, IV, 18.12.10–11], h is an integraluniversal homeomorphism, which proves the first claim. To prove the latter claim,it suffices to check that the map h] : OZ → h∗OY is an isomorphism of sheaves on|Z|. We may conclude the proof by observing that since the maps f , g are Stein,the map h] factors as the following composition of isomorphisms:

OZ ∼= g∗OX ∼= (h ◦ f)∗OX ∼= h∗f∗OX ∼= h∗OY . �

With this result at our disposal, we return to the task at hand.

Proof of Theorem 2.3. Let s : Z → X belong to QFp.f.(X/Y ) and let BZ be the

integral closure of OX in s∗OZ . We set Z = SpecX BZ , and call this the universalX-compactification of Z, for reasons that we will make clear now. Zariski’s MainTheorem [EGA, IV, 18.12.13] shows that : Z → Z is an open immersion and,moreover, that there is a finite X-scheme t : W → X with ı : Z → W such thatı ◦ : Z →W is an open immersion. We may further assume that t∗OW ⊂ BZ .

Let Z′

= SpecY π∗BZ , W ′ = SpecY π∗t∗OW . Note that there is a canonical mapı′ : Z ′ → W ′ given by the inclusion of sheaves π∗t∗OW ⊂ π∗BZ and we have thefollowing commutative diagram:

Z // Z

ı //

p ��

Wt //

q��

X

π��

Z′

ı′// W ′

t′// Y.

Note that by construction, p and q are Stein morphisms. Since π is proper, by [EGA,III, 3.2.1], t′ is finite, and q is proper with geometrically connected fibers. One alsoconcludes that p is quasi-compact, separated, universally closed, and surjective. Inparticular, both p and q are universally submersive. Furthermore, it follows that ı′

is universally closed and affine, thus by [EGA, IV, 18.12.8], it is integral.Let |Z ′| ⊂ |W ′| denote the the set-theoretic image of |Z| ⊂ |W | under q. We

claim that |Z ′| is open. Since q is universally submersive, it suffices to show thatq−1|Z ′| is open in |W |. Let w′ ∈ |W ′|, then Zw′ → Ww′ is an open immersion byconstruction (where Ww′ is the fiber of W → W ′). Also, since Zw′ is proper over{w′}, then Zw′ → Ww′ is closed (Ww′ is separated). Hence, Zw′ → Ww′ is openand closed, but Ww′ is connected and hence we conclude that either Zw′ = ∅ orZw′ = Ww′ . That is, q−1|Z ′| = |Z| which shows that |Z ′| is an open subset of |W ′|.We define Z ′ to be the open subscheme of W ′ corresponding to the open set |Z ′|.In particular, we observe that Z ′ ×W ′ W = q−1(Z ′) = Z ⊂W and so q|Z : Z → Z ′

is proper and Stein. Moreover, since is a dense open immersion and ı is separated,then, |Z| = ı−1ı(|Z|). Thus, as p, q, ı, ı′ are surjective, we have equalities of subsetsof |Z|:

|Z| = ı−1ı(|Z|) = ı−1q−1qı(|Z|) = p−1ı′−1ı′p(|Z|).Hence,

p(|Z|) = ı′−1ı′p(|Z|) =⇒ p−1p(|Z|) = |Z|.Thus, since p is universally submersive, we conclude that p(|Z|) is an open sub-

scheme of |Z ′|. We define π!Z to be the open subscheme of Z′

corresponding tothe open subset p(|Z|). Note, that there is a natural map πZ : Z → π!Z whichis universally closed, separated, surjective, and Stein. Moreover, since the mapπ!Z → Z ′ is integral, we may apply Lemma 2.8 to conclude that the map π!Z → Z ′

is an isomorphism of Y -schemes. Thus, (π!Z → Y ) ∈ QF(Y ) and πZ : Z → π!Z isproper and Stein.


To define a functor π!, we must next prove that for any map f : Z → Z0 inQFp.f.(X/Y ), there is a map π!f : π!Z → π!Z

0 such that if g : Z0 → Z1 is anyother map in QFp.f.(X/Y ), then π!(g ◦ f) = (π!g) ◦ (π!f) as maps in QF(Y ).

So, let s0 : Z0 → X belong to QFp.f.(X/Y ) and suppose that we have a mor-

phism f : Z → Z0 in QFp.f.(X/Y ). Retaining similar notation used at the begin-

ning of the proof, we have an open immersion Z0 ↪→ Z0, where Z

0= SpecX BZ0

and BZ0 is the integral closure of OX in (s0)∗OZ0 . First of all, we observe that

the morphism f : Z → Z0 provides a map of sheaves f : (s0)∗OZ0 → s∗OZ . Inparticular, since BZ0 is the integral closure of OX in (s0)∗OZ0 , then the image of

any section of BZ0 in s∗OZ under f is integral over OX and a fortiori lies in BZ .

Consequently, the map f |BZ0 : BZ0 → s∗OZ factors through BZ . Hence, associated

to any f : Z → Z0 in QFp.f.(X/Y ), there is a natural map f : Z → Z0, which is

integral.Pushing all of this forward to Y by the morphism π, we obtain a natural map

f′

: Z′ → Z0

′and the claim is that this restricts to a map π!f : π!Z → π!Z

0.

But considering that π!Z (resp. π!Z0) is the set-theoretic image of Z in Z

′(resp.

Z0′), this is certainly true on the level of sets, and since π!Z (resp. π!Z

0) are open

subschemes of Z′

(resp. Z0′), this is clear. In particular, if g : Z0 → Z1 is another

morphism in QFp.f.(X/Y ), then π!(g ◦ f) = (π!g) ◦ (π!f) by construction. Hence,we have defined a functor π! : QFp.f.(X/Y )→ QF(Y ).

We now proceed to prove that (π!, π∗) is an adjoint pair. For this, we construct

a pair of natural transformations η : idQFp.f.(X/Y ) ⇒ π∗π!, ε : π!π∗ ⇒ idQF(Y ) such

that ε(π!) ◦ π!(η) = idπ!and (π∗ε) ◦ η(π∗) = idπ∗ .

Let Z → X belong to QFp.f.(X/Y ). By construction there is a map πZ : Z →π!Z, and thus a map ηZ : Z → π∗π!Z. In particular, this construction is natural inZ and so we have defined η : idQFp.f.(X/Y ) ⇒ π∗π!. Note that since Z → π!Z and

π∗π!Z → π!Z are proper, and ηZ : Z → π∗π!Z is quasi-finite and separated, thenby [EGA, IV, 18.12.4], it is finite.

Now suppose that Z ′ → Y belongs to QF(Y ), and let Z′ → Y be the universal

Y -compactification of Z ′, then if (π∗Z ′) is the universal X-compactification of π∗Z ′,

there is a unique map (π∗Z ′) → Z′ ×Y X. Pushing this all back down to Y and

using adjunction for sheaves again, we obtain a natural map π!π∗Z ′ → Z

′, but this

map factors through Z ′ ↪→ Z′

and so we have constructed a natural transformationε : π!π

∗ ⇒ idQF(Y ). Similarly, we observe that the map εZ′ : π!π∗Z ′ → Z ′ is

quasi-finite and separated, and that the maps π∗Z ′ → π!π∗Z ′ and π∗Z ′ → Z ′ are

proper. Again, by [EGA, IV, 18.12.4], we conclude that the map εZ′ : π!π∗Z ′ → Z ′

is finite.It remains to check the unit/counit equations. Note that the first equation

amounts to checking that

π!Zπ!(ηZ) // π!(π

∗π!Z)επ!Z // π!Z

is the identity map. Since ε and η are constructed via adjunction of sheaves andthen restricted to open subschemes, it is clear that this equation holds. One mayargue similarly for the other equation. �

Proof of Corollary 2.4. The proof of (1) is immediate from the functoriality of

pullbacks and the universal properties defining adjoints. For (2), we fix (Zs−→

U) ∈ QFp.f.(U/V ) = QFp.f.(U/Y ). By Theorem 2.3(1b), there is a natural map

ρZ : Z → ρ!Z. As f is quasi-finite and separated, we obtain a natural mapφ : Z → f!ρ!Z, which is proper and Stein. Also, since the composition of φ with


the structure map of ρ!Z over Y agrees with the composition Zg−→ X

π−→ Y , thereis a naturally induced morphism Z → g∗π∗f!ρ!Z. By adjunction, there is thus anatural map ψ : π!g!Z → f!ρ!Z, which is quasi-finite, and separated. In particular,we have the commutative triangle:

Zπg!Z

��φ

��π!g!Z

ψ// f!ρ!Z.

By Theorem 2.3(1b), the map πg!Z is proper and Stein; thus, by Lemma 2.8, ψ isan isomorphism of schemes. �

Proof of Corollary 2.6. The existence of the natural transformation is a standardproperty of adjoints, so we omit the proof. Since (3) is obvious, we concentrate on

(1) and (2). So, we let (Zs−→ X) and obtain the commutative diagram:

p′∗Z

''

//

yy ��

Z

��

s%%

X ′ p′ //

��

X

��π′!p′∗Z //

//

p∗π!Z //

&&

π!Z

$$Y ′ p // Y,

where the bottom, top, front and back squares are cartesian. The map π′!p′∗Z →

p∗π!Z is quasi-finite (thus has discrete fibers), and by Theorem 2.3(1b), the mapp′∗Z → π′!p

′∗Z is proper and Stein. Also, by Theorem 2.3(1b), p′∗Z → p∗π!Z isproper and has geometrically connected fibers (if p is flat, this map is in additionStein), as it is the base change (resp. flat base change) of the proper and Steinmap Z → π!Z. By Lemma 2.8, the map π′!p

′∗Z → p∗π!Z is a finite, universalhomeomorphism (resp. isomorphism). �

Remark 2.9. Theorem 2.3 and its corollaries readily extend to representable mor-phisms of algebraic stacks. Using the results of Faltings [Fal03], Olsson [Ols05],and Conrad [Con] it is also possible to extend these results to non-representablemorphisms of algebraic stacks.

2.2. The adjoint pair ($!, $∗). Let $ : X → Y be a quasi-finite, separated,

and adic morphism of locally noetherian formal schemes. As in §2.1, the functor

$! : QF(X) → QF(Y), which sends (Zσ−→ X) to (Z

$◦σ−−−→ Y) ∈ QF(Y), is leftadjoint to $∗. The purpose of this section is to construct, for a proper and adicmorphism of locally noetherian formal schemes $ : X→ Y, a left adjoint to $∗, andextend all of the results of §2.1 to locally noetherian formal schemes. In particular,we start with the following

Definition 2.10. Let X be a locally noetherian formal scheme. For an adic mor-phism of locally noetherian formal schemes $ : X → Y, let QFp.f.(X/Y) denote

the full subcategory of QF(X) consisting of the objects (Zσ−→ X) such that the

composition Zσ−→ X

$−→ Y has proper fibers.

There is a formal analog of Lemma 2.2, which says that if $ : X → Y is adicand proper, and (Z′ → Y) ∈ QF(Y), then $∗(Z′ → Y) ∈ QFp.f.(X/Y).

We say that a map of locally noetherian formal schemes f : X → Y is Stein ifthe induced map OY → f∗OX is a topological isomorphism.

Example 2.11. Suppose that Y is a noetherian formal scheme and let I be anideal of definition. Consider an adic and proper morphism $ : X → Y and let


$n : Xn → Yn denote the proper morphism of schemes induced by the pullbackof f by the morphism Yn := V (In) → Y. If $n is Stein for all n, then $ is Stein.Indeed, by [EGA, III, 3.4.4] there is a topological isomorphism

$∗OX∼= lim←−

n

($n)∗OXn∼= lim←−

n

OYn∼= OY.

Note that it is does not follow that if $ : X → Y is an adic, proper, and Steinmorphism, then $n : Xn → Yn is Stein. We now have the formal analog of Theorem2.3.

Theorem 2.12. Let $ : X→ Y be a proper, adic, morphism of locally noetherianformal schemes. There is a functor $! : QFp.f.(X/Y) → QF(Y), which is leftadjoint to $∗. Moreover,

(1) if (Zσ−→ X) ∈ QFp.f.(X/Y), then:

(a) the unit of the adjunction, ηZ : Z→ $∗$!Z, is finite;

(b) the canonical map of locally noetherian formal schemes $Z : ZηZ−−→

$∗$!Z→ $!Z is proper, adic, and Stein;

(c) if, in addition, σ is finite, then $!(Zσ−→ X) = (SpfY($∗σ∗OZ)→ Y);

(d) if $ is finite, then $!(Zσ−→ X) = (Z

$◦σ−−−→ Y).(2) if (Z′ → Y) ∈ QF(Y), then the counit of the adjunction, εZ′ : $!$

∗Z′ → Z′,is finite.

As in §2.1, we will defer the proof of Theorem 2.12 until the end of this section,and focus on the implications.

Corollary 2.13. Consider a commutative diagram of locally noetherian formalschemes:

Ug //

ρ��

X

$��V

f// Y.

Suppose $ and ρ are adic and proper, then

(1) if f and g are adic and proper, there are natural isomorphisms of functorsfrom QFp.f.(U/Y) to QF(Y):

f!ρ! =⇒ (f ◦ ρ)! ⇐⇒ ($ ◦ g)! ⇐= $!g!;

(2) if f and g are instead adic, quasi-finite, and separated, there is a naturalisomorphism of functors from QFp.f.(U/Y) to QF(Y):

$!g! =⇒ f!ρ!.

Corollary 2.14. Given a proper morphism $ : X→ Y of locally noetherian formalschemes and (Z′ → Y) ∈ QF(Y), set Z := $∗Z′ and let $Z′ : Z → Z′ denote theinduced map. Then there is a natural Z′-isomorphism ($Z′)!Z→ $!Z. Thus,

Z$Z

−−→ $!$∗Z′

εZ′−−→ Z′

is canonically isomorphic to the Stein factorization of the morphism $Z′ : Z→ Z′.

Corollary 2.15. Consider a commutative diagram of locally noetherian formalschemes:

X′p′ //

$′ ��

X

$��

Y′p// Y,


then if $ and $′ are adic and proper, or adic, quasi-finite, and separated, there isa natural transformation:

∆ : $′!p′∗ =⇒ p∗$!.

If the diagram is cartesian, and

(1) if $ is proper, then ∆ induces an adic, finite, universal homeomorphism;or

(2) if p is adic and flat, then ∆ is an isomorphism of functors; or(3) if $ is adic, quasi-finite, and separated, then ∆ is an isomorphism of func-

tors.

Corollary 2.16. Let $ : X → Y be a proper and adic map of locally noetherian

formal schemes, let (Zσ−→ Y) ∈ QF(Y), and let U→ Z be an adic, quasi-finite flat

morphism. Then we have a cartesian diagram:

$!$∗U

ηU //

��

U

��$!$

∗Zη// Z.

If the composition σU : U→ Y is finite, there is a canonical isomorphism

$!$∗U ∼= SpfY($∗$

∗(σU)∗OU)

such that the map ηU corresponds to the adjunction map on coherent sheaves ofOY-algebras: (σU)∗OU → $∗$

∗(σU)∗OU.

We now return to the task of proving Theorem 2.12.

Lemma 2.17. Let X be a locally noetherian formal scheme and let U → X be anadic, locally quasi-finite, and separated morphism. Suppose that [R ⇒ U] is an adicfppf equivalence relation over X such that R→ U×XU is a closed immersion. Thenthis equivalence relation has a uniform formal geometric quotient Z which is adic,locally quasi-finite, and separated over X.

Proof. Let I be an ideal of definition for X and denote XI = V (I), UI = U×X XI,and RI = R ×X XI. Then, the hypotheses ensure that [RI ⇒ UI] is an fppfequivalence relation over XI. We observe that the quotient of this fppf equivalencerelation in the category of algebraic spaces is a scheme ZI, as it is locally quasi-finite and separated over XI. This quotient is also a uniform geometric quotient,and regarding the objects as locally noetherian formal schemes one sees that itis, in fact, a uniform formal geometric quotient. Note that for any other ideal ofdefinition J ⊃ I there is a natural isomorphism ZI ×X XJ

∼= ZJ, thus the directedsystem {ZI}I is adic over X. Taking the colimit of this system in the category oftopologically ringed spaces, produces a locally noetherian formal scheme Z, whichis adic, locally quasi-finite, and separated over X, and is a uniform formal geometricquotient of the given equivalence relation. �

Proof of Theorem 2.12. Let I be an ideal of definition for Y, let YI = V (I), andlet XI = X ×Y YI. Then YI is a locally noetherian scheme and since $ is adic,$I : XI → YI is a proper morphism of schemes. If I ⊂ J is another ideal ofdefinition we obtain a commutative square:

XJ� � ı′JI //

$J��

XI

$I��YJ� � ıJI // YI.


Let (Z → X) belong to QFp.f.(X). We observe that, by Corollary 2.6, there is anatural, finite universal homeomorphism:

JI : ($J)!ZJ −→ ($I)!ZI.

Let K denote the largest ideal of definition of Y and for any other ideal of defini-tion I of Y, let I = KI be the homeomorphism above. Define the topologicallyringed space $!Z to be the weak formal geometric colimit of the directed system{($I)!ZI}I, and note that we are yet to show that $!Z is a formal scheme. Explic-itly:

$!Z =(|($K)!ZK|, lim←−

I

−1I O($I)!ZI

),

with the topology on the sheaf of rings given by the induced limit topology. Thereis also a unique morphism of topologically ringed spaces t : $!Z → Y. We willshow that $!Z ∈ QF(Y) for any Z. To prove that $!Z is a formal scheme, it doesnot suffice to apply [EGA, I, 10.6.3], as the directed system defining $!Z is not, ingeneral, a sequence of nilimmersions.

Fix y ∈ |Y| = |YK|. By [EGA, IV, 18.12.1], there is an etale neighborhood

(YK,y, yK) → (YK, y) such that (($K)!ZK) ×YKYK,y has an open and closed sub-

scheme UK,y containing the fiber over y such that UK,y → YK,y is finite. Using[EGA, IV, 18.1.2], we see that for any ideal of definition I of Y, one produces

a unique etale morphism (YI,y, yI) → (YI, y) lifting the morphism (YK,y, yK) →(YK, y). In particular, one obtains a unique, open and closed subscheme UI,y ↪→(($I)!ZI)×YI

YI,y lifting the finite YK,y-scheme UK,y. Note, that for any inclusion

of ideals of definition I ⊂ J, there is a canonical isomorphism YJ ×Y YI,y ∼= YJ,yand we let Yy be the locally noetherian formal scheme lim−→I

YI,y. For a fixed ideal

of definition I, consider the induced commutative diagram:

U ′I,y//

φI,y

��

ZI,y

��

vv

// XI,y

{{$I,y

��ZI

//

��

XI

$I

��

UI,y// ($I,y)!ZI,y

//

vv

YI,y

{{($I)!ZI

// YI,

where every square is cartesian, except for the squares at the front and back right.Note that the bottom square is cartesian by Corollary 2.6. We deduce from herethat the map φI,y : U ′I,y → UI,y is proper and Stein, so that OUI,y

= (φI,y)∗OU ′I,y

,

and that U ′I,y → XI,y is finite. In particular, since U′y = lim−→IU ′I,y is a locally

noetherian formal finite Xy = lim−→IXI,y-scheme by [EGA, I, 10.6.3], we deduce that

uy : U′y → Yy is an adic, proper morphism of locally noetherian formal schemes. By[EGA, III, 3.4.2], (uy)∗OU′y

is a finite OYy-algebra and an application of [EGA, III,

3.4.4] shows that there is a topological isomorphism (uy)∗OU′y∼= lim←−I

(uI,y)∗OU ′I,y

.

Thus, we define the finite Yy-scheme Uy = SpfYy(uy)∗OU′y

, and we conclude that

it is the formal weak geometric colimit of the directed system {UI,y}I.Let Y =

∐y∈|Y| Yy and set Y2 = Y ×Y Y. Note that Y → Y is adic etale

and surjective. Now take X = Y ×Y X, X2 = Y2 ×Y X and let $ : X → Y and

$2 : X2 → Y2 be the induced maps. Further, set Z = X ×X Z, Z2 = X2 ×X Z

and we obtain an adic etale equivalence relation [Z2 ⇒ Z]. By Lemma 2.17, it isclear that Z is a uniform formal geometric quotient of this equivalence relation.


We also have that ($I)!ZI is the uniform geometric quotient of the equivalence

relation [($2I)!Z

2I

sI //tI// ($I)!ZI] and so one concludes that $!Z is the quotient of

the equivalence relation [$2! Z

2 s //t// $!Z] in the category of topologically ringed

spaces. Note that since we have not yet shown that $!Z and $2! Z

2 are locallynoetherian formal schemes which are adic, locally quasi-finite, and separated overY, we are not in the situation to apply Lemma 2.17; we will, however, slice thisequivalence relation, so that we can.

For any ideal of definition I of Y, we set UI =∐y∈|Y| UI,y, and observe that

UI → ($I)!ZI is an etale surjection. If we set RI = s−1I (UI) ∩ t−1

I (UI), then

[RI

sI //tI// UI] is an etale equivalence relation, and the scheme ($I)!ZI is a uniform

geometric quotient of this equivalence relation. Next, we let R′I ⊂ Z2I and U ′I ⊂ ZI

denote the preimages. In particular, the etale equivalence relation [R′IsI //tI// U ′I]

has geometric quotient the scheme ZI. By Corollary 2.5, we have ($I)!U′I = UI

and ($2I)!R

′I = RI. Hence, ($I)!ZI is the uniform geometric quotient of the etale

equivalence relation [($2I)!R

′I

sI //tI// ($I)!U

′I] .

Define U := lim−→IUI =

∐y∈|Y| Uy ⊂ $!Z, which is a clopen subspace and a locally

noetherian formal scheme, that is adic, locally quasi-finite, and separated over Y.

Next, take R := lim−→IRI = s−1(U) ∩ t−1(U) ⊂ $2

! Z2, then as topologically ringed

spaces, we have:

R =∐

ys,yt∈|Y|

s−1(Uys) ∩ t−1(Uyt).

Repeating the arguments showing that Uy is a locally noetherian formal scheme,

which is finite over Y, one concludes that s−1(Uy) (resp. t−1(Uy)) is a locally

noetherian formal scheme for all y ∈ |Y|, which is finite over Y2. In particular,we conclude immediately that R is a locally noetherian formal scheme that is adic,locally quasi-finite, and separated over Y. Moreover, the maps s, t : R → U areadic etale, surjective and we obtain an adic etale equivalence relation [R ⇒ U]. ByLemma 2.17, this equivalence relation has a uniform formal geometric quotient, V,which is a locally noetherian formal scheme, adic, locally quasi-finite, and separatedover Y. Note that the topological space |V| = |($K)!ZK| is quasi-compact and soV ∈ QF(Y). Further, by Lemma 1.14 we have isomorphisms of topologically ringedspaces:

$!Z = lim−→I

($I)!ZI∼= lim−→

I

lim−→[

($2I)!R

′I

sI //tI// ($I)!U

′I

]∼= lim−→

[lim−→I

($2I)!R

′I

lim−→IsI//

lim−→I

tI

// lim−→I($I)!U

′I

]∼= lim−→

[R

s //t// U]∼= V.

Since the colimits appearing above were all weak formal geometric colimits, we haveshown that $!Z is a locally noetherian formal scheme, which is the weak formalgeometric colimit of the directed system {($I)!ZI}I. By Lemma 1.14, we concludethat $!Z is the categorical colimit of its defining directed system in QF(Y). Also,there is an adic, proper, surjective, and Stein map $Z : Z→ $!Z.

We now check that Z 7→ $!Z defines a functor. So, given a morphism Zf−→ Z0,

there is a canonical morphism fI : ZI → Z0I for any ideal of definition I of Y. Since

($I)! is a functor, we obtain a morphism ($I)!fI : ($I)!ZI → ($I)!Z0I . Taking the

categorical colimit of this family of morphisms in QF(Y) produces a morphism offormal schemes $!f : $!Z→ $!Z

0. That this construction respects composition isimmediate from the functoriality of ($I)!.


To show that ($!, $∗) is an adjoint pair, let Z ∈ QFp.f.(X/Y) and Z1 ∈ QF(Y),

then

HomQF(Y)($!Z,Z1) = HomQF(Y)(lim−→

I

($I)!ZI,Z1) = lim←−

I

HomQF(Y)(($I)!ZI,Z1)

= lim←−I

HomQF(YI)(($I)!ZI, Z1I ) = lim←−

I

HomQFp.f.(XI/YI)(ZI, ($I)∗Z1

I )

= lim←−I

HomQFp.f.(X/Y)(ZI, $∗Z1) = HomQFp.f.(X/Y)(Z, $

∗Z1).

The remainder of the claims are easily verified, so they are omitted. �

Before we prove Corollary 2.15, we require a basic result on adic, proper, andStein morphisms of locally noetherian formal schemes.

Lemma 2.18. Let $ : X → Y be a proper, adic, morphism of locally noetherianformal schemes. Given an adic flat map p : Y′ → Y of locally noetherian formalschemes, let $′ : X′ → Y′ be the pullback of $ by p. Let F be a coherent OX-module.Then for any q ≥ 0, the base change homomorphism p∗Rq$∗F → Rq$′∗p

′∗F is atopological isomorphism. In particular, proper, adic, Stein morphisms are preservedunder adic flat base change.

Proof. Note that this result is well-known for the case of a morphism of schemes.The Lemma can be extracted from a general result [AJL99, Prop. I.7.2], but weinstead include a direct proof (suggested by B. Conrad). Fix q ≥ 0, and by [EGA,III, 3.4.5.1], the statement is Zariski local on Y and Y′, so we may assume thatY = SpfI R for some I-adic noetherian ring R. Set Y′ = SpfIR′ R

′ so that R→ R′

is an adic flat morphism of adic noetherian rings. In particular, we observe thatR→ R′ is a flat morphism of abstract rings. We need to show that the morphismHq(X,F)⊗RR′ → Hq(X′, p′∗F) is a topological isomorphism. By [EGA, III, 3.4.2],the R-module Hq(X,F) and the R′-module Hq(X′, p′∗F) are coherent, so it sufficesto show that the base change morphism Hq(X,F)⊗RR′ → Hq(X′, p′∗F) is bijective.

For each k ≥ 0, set Rk = R/Ik+1, R′k = R′⊗R Rk, Xk = X⊗R Rk, X ′k = X′⊗R′R′k, Fk = F|Xk , and F ′k = (p′∗F)|X′k . By [EGA, III, 3.4.4.2] there is a topological

isomorphism of R-modules Hq(X,F) ∼= lim←−kHq(Xk, Fk) and the projective system

of R-modules {Hq(Xk, Fk)}k≥0 is cofinal with the projective system of R-modules{Hq(X,F) ⊗R Rk}k≥0. Tensoring both of these projective systems of R-moduleswith R′, we see that the projective system of R′-modules {Hq(Xk, Fk)⊗RR′}k≥0 iscofinal with the projective system of R′-modules {Hq(X,F)⊗RR′k}k≥0, thus admitthe same limit. The latter system of R′-modules clearly has limit Hq(X,F)⊗R R′.

By flat base change for schemes, there are natural isomorphisms Hq(Xk, Fk)⊗RR′ ∼= Hq(X ′k, F

′k). By [EGA, III, 3.4.4.2] we see that the limit of the projective

system {Hq(Xk, Fk)⊗R R′}k≥0 is the R′-module Hq(X′, p′∗F). �

Lemma 2.19. Consider a commutative diagram of locally noetherian formal schemes:

Xf

��g

��Y

h// Z

where f and g are adic, proper, surjective, and have geometrically connected fibers.If h is quasi-finite, then it is a universal homeomorphism. If, in addition, f and gare Stein, then h is an isomorphism of formal schemes.

Proof. By restricting to the underlying reduced closed subschemes, the first claimfollows from Lemma 2.8. The latter claim is obvious. �


Proof of Corollary 2.15. We note that (3) is clear, and concentrate on (1) and (2).

As in the proof of Corollary 2.6, for (Zσ−→ X) ∈ QFp.f.(X/Y) we obtain a commu-

tative diagram:

p′∗Z

''

//

yy ��

Z

��

σ

&&X′ p′ //

��

X

��$′!p

′∗Z //

,,

p∗$!Z //

&&

$!Z

%%Y′ p // Y.

We note that the base change map $′!p′∗Z → p∗$!Z is quasi-finite. Moreover, by

Theorem 2.12(1b), the map p′∗Z → $′!p′∗Z is adic, proper, Stein, surjective, and

has geometrically connected fibers. Similarly, we conclude that the map p′∗Z →p∗$!Z is adic, proper, surjective and has geometrically connected fibers, and if pis flat, then the morphism is also Stein by Lemma 2.18. By Lemma 2.19 the basechange map ∆ : $′!p

′∗Z→ p∗$!Z is a finite, universal homeomorphism and it is anisomorphism if p is flat. �

2.3. The Theorem on Formal Functions. Let π : X → Y be a proper morphismof locally noetherian schemes, and let |V | ⊂ |Y | be a closed subset. Then there isa 2-commutative diagram of categories:

QF(Y )c/|V | //

π∗

��

QF(Y/|V |)

π∗

��QF(X)

c/π−1|V | // QF(X/π−1|V |).

From this, we obtain a natural transformation of functors from QFp.f.(X/Y ) to

QF(Y ):

Φπ,|V | : π! ◦ c/π−1|V | =⇒ c/|V | ◦ π!.

Theorem 2.20 (Formal Functions). Suppose that π : X → Y is a proper morphismof locally noetherian schemes and let |V | ⊂ |Y | be a closed subset, then Φπ,|V | isan isomorphism of functors. That is, for any Z in QFp.f.(X/Y ), there is a naturalisomorphism

π!Z −→ (π!Z)∧.

Proof. Let K / OY denote the unique radical ideal with support |V |; this is the

largest ideal of definition for the locally noetherian formal scheme Y/|V | and any

other ideal of definition for Y/|V | is given by an ideal I ⊂ K with support |V |. Forany ideal of definition I, we set YI = V (I), XI = X ×Y YI, ZI = Z ×X XI anddenote the induced map as πI : XI → YI. By Corollary 2.6(1), there is a natural,finite, universal homeomorphism I : (πK)!ZK → (πI)!ZI.

By the construction in Theorem 2.12, we see that π!Z is the topologically locallyringed space (|(πK)!ZK|, lim←−I

−1I O(πI)!ZI

). Also, (π!Z)∧ is defined as the topolog-

ically locally ringed space (|(π!Z)K|, Oπ!Z). The map Φ : π!Z → (π!Z)∧ is on thelevel of topological spaces |Φ| : |(πK)!ZK| → |(π!Z)K|, which is a finite, universalhomeomorphism by Corollary 2.6(1) . On the level of sheaves of topological rings,

the map is Φ] : Oπ!Z → Φ∗ lim←−I−1I O(πI)!ZI

. By Theorem 2.3(1b), we have natural,

proper and Stein morphisms πZ : Z → π!Z and πZI

I : ZI → (πI)!ZI so there areisomorphisms of sheaves Oπ!Z = (πZ)∗OZ and O(πI)!ZI

= (πZI)∗OZI. Thus, the


map on sheaves of topological rings is seen to be

(πZ∗ OZ)∧ −→ Φ∗ lim←−I

−1I (πZI)∗OZI

,

which is a topological isomorphism by [EGA, III, 3.4.4]. �

3. The Existence Theorem

Definition 3.1. For a morphism of algebraic stacks π : X → Y , let QFp(X/Y )denote the full subcategory of QFp.f.(X/Y ) consisting of those (Z → X) such thatthe composition Z → X → Y is proper.

Let R be an I-adic noetherian ring, set S = SpecR and Sn = SpecR/In+1.Further, suppose that π : X → S is a morphism of locally noetherian algebraicstacks. We set Xn = X×S Sn and let πn : Xn → Sn denote the induced morphism.Consider the I-adic completion functor

Ψπ,I : QFp(X/S) −→ lim←−n

QFp(Xn/Sn) : (Z → X) 7→ (Z ×S Sn → Xn)n≥0,

where the category lim←−n QFp(Xn/Sn) is the 2-categorical limit of the inverse system

of categories {QFp(Xn/Sn)}n≥0. The purpose of an existence theorem is to providesufficient conditions on the morphism π : X → S for the functor Ψπ,I to be anequivalence of categories.

In the case that the morphism X → S is locally of finite type and separated,the category QFp(X/S) is equivalent to the category of finite algebras over X withS-proper support. A consequence of [Ols05, Thm. 1.4] is that the functor Ψπ,I isan equivalence. In full generality, we have the following

Proposition 3.2. Let S be the spectrum of an I-adic noetherian ring R. Considera morphism of algebraic stacks π : X → S that is locally of finite type. Then thefunctor Ψπ,I is fully faithful.

Proof. Consider objects Z, Z ′ ∈ QFp(X/S) and an adic system of morphisms(fn)n≥0 : (Zn)n≥0 → (Z ′n)n≥0. In particular, (fn)n≥0 is an adic system of finitemorphisms to a proper S-stack, so by [Ols05, Thm. 1.4], there is a finite S-morphismf : Z → Z ′ such that for every n ≥ 0, there is a 2-isomorphism fSn

∼= fn. We mustnow show that f is automatically an X-morphism, which will follow if we can provethat given two S-morphisms s, t : Z → X such that for every n ≥ 0 we have a2-isomorphism sXn

∼= tXn , then s ∼= t. Working smooth-locally and considering theinduced morphisms of formal schemes, we can apply [EGA, I, 10.9.4] and concludethat s = t on an open subset of the stack Z, containing the closed fiber. Since thestack Z is S-proper and S is I-adic, we conclude immediately that s = t on thestack Z. Thus, we have shown that the functor Ψπ,I is full.

To obtain the faithfulness, we suppose that we have two finite X-morphisms f ,g : Z → Z ′ such that we have an equality of morphisms (Ψπ,I)∗(f) = (Ψπ,I)∗(g) inthe category lim←−n QFp(Xn/Sn), then we have to show that f = g in QFp(X/S).

Once again, we apply [EGA, I, 10.9.4] and conclude as before. Thus, the functorΨπ,I is also faithful. �

Definition 3.3. For a morphism of algebraic stacks π : X → Y , define the categoryQFp,fin,∆(X/Y ) to be the full subcategory of QFp(X/Y ) consisting of those objects(Z → X) such that the composition of morphisms Z → X → Y has finite diagonal.

Remark 3.4. In the case that the morphism of algebraic stacks π : X → Y has affinestabilizers, then we have an equality of categories QFp,fin,∆(X/Y ) = QFp(X/Y )—this holds in particular for schemes, algebraic spaces and stacks with quasi-finitediagonal.


Retaining the notation used above, there is an induced completion functor:

Ψπ,I : QFp,fin,∆(X/S) −→ lim←−n

QFp,fin,∆(Xn/Sn).

What follows is the main result of this section.

Theorem 3.5. Let R be an I-adic noetherian ring, set S = SpecR. If the mor-phism of algebraic stacks π : X → S is locally of finite type with quasi-compact and

separated diagonal, then the functor Ψπ,I is an equivalence.

The new content is the removal of the separation hypothesis. In particular,this recovers the consequences of [Ols05, Thm. 1.4] for all morphisms of algebraicstacks π : X → S that are locally of finite type and have affine stabilizers. Someinteresting special cases of this are:

(1) global quotient stacks;(2) stacks with quasi-finite diagonal;(3) algebraic spaces;(4) schemes.

We will prove Theorem 3.5 using a method of devissage—once to handle the caseof schemes, and twice to handle algebraic stacks. These methods will require anumber of lemmata to set up. We start with a variant of the Artin–Rees lemma.

Lemma 3.6. Let Y be a noetherian formal scheme. Consider an inclusion ofcoherent OY-modules F′ ⊂ F such that for a coherent ideal J /OY we have JF′ = 0.Then there is a k ≥ 0 such that (JkF) ∩ F′ = 0.

Proof. Since Y is quasi-compact, this is Zariski local on Y. Then it is simply amatter of applying the Artin–Rees Lemma [AM69, Cor. 10.10]. �

Let f : Z′ → Z be a finite morphism of locally noetherian formal schemes. Wedenote the induced map on sheaves of rings by f ] : OZ → f∗OZ′ . As is customary,we will define the conductor ideal of f to be Cf := AnnOZ

(f∗OZ′/OZ).

Lemma 3.7. Suppose that X is a noetherian scheme and that |V | ⊂ |X| is a closedsubset. Consider a commutative diagram of noetherian formal schemes:

Z′f //

s′ !!

Z

s~~X/|V |

where s, s′ are adic, quasi-finite, and separated; f is finite, surjective, and Z′ iseffectivizable. If

(1) there is a coherent ideal J/OZ such that J∩ker f ] = 0 and V (J), V (JZ′) are

effectivizable, then there is a unique factorization f : Z′ → Z′′β−→ Z such that

β is finite, surjective, β] : OZ → β∗OZ′′ is injective, Z′′ is effectivizable, the

formation of Z′′β−→ Z commutes with flat base change on X, and Cf ⊂ Cβ;

or(2) there is a coherent OX-ideal I such that ker f ] and coker f ] are annihilated

by IZ, and V (InZ) is effectivizable for all n ≥ 1. Then Z is effectivizable.

Proof. For (1), we define Z′′ to be the colimit of the diagram [V (J)← V (JZ′) ↪→ Z′]

in QF(X/|V |), which exists and is effectivizable by Theorem 1.18 and Corollary

1.19. There is an induced factorization Z′ → Z′′β−→ Z which commutes with flat

base change on X. It remains to show that the map β] is injective. This is a local


problem, so we may assume that Z = SpfI A and Z′ = SpfIB B for some finite mapG : A → B of noetherian rings, where B is IB-adic. We write J = Γ(SpfI A, J),K = kerG, and Theorem 1.18 implies that Z′′ = SpfI′(A/J ×B/BJ B), where I ′ isthe ideal generated by I in A/J ×B/BJ B. Thus, we have to show that the finitemap of rings b : A→ A/J ×B/BJ B is injective, which is obvious, since J ∩K = 0.It is clear that coker b = BJ/J and so CG ⊂ Cb.

For (2), by Lemma 3.6, we may replace IZ by some power such that ker f ]∩IZ =

0. By (1), there is a factorization of f : Z′ → Z as Z′ → Z′′β−→ Z, where Z′′ is

effectivizable, the map of formal schemes β is finite and surjective, β] injective, andcokerβ] is annihilated by IZ. In particular, since IZ ⊂ Cβ , there is a cocartesian

diagram of formal X-schemes:

V (IZ′′)/|V |� � //

��

Z′′

β��

V (IZ) �� // Z.

By Corollary 1.19, we conclude that Z is effectivizable. �

We have a simple lemma that will be used shortly.

Lemma 3.8. Let R be an I-adically complete noetherian ring. Define V = SpecR

and V = SpfI R. Let Z be a noetherian formal scheme and let Z → V be an adicmorphism. Fix a coherent sheaf F on Z. Let p be an open prime of R, and let R′ be

the p-adic completion of Rp. Take V ′ = SpfIR′ R′. Suppose that FV ′ = 0 on ZV ′ ,

then there is an open neighborhood U of v = [p] ∈ V such that FU = 0 on ZU.

Proof. Let V0 = Spec(R/I), V ′0 = Spec(R′/IR′), Z0 = Z ×V V0, and F0 = F|Z0.

Next, observe that R′/IR′ is the completion of Rp/IRp with respect to the p-adic topology and thus Rp/IRp → R′/IR′ is faithfully flat. Hence, if W0 =Spec(Rp/IRp), then since F0|(Z0)V ′0

= 0, we have F0|(Z0)W0= 0. In particular,

it follows from the result for schemes, that there is an open neighborhood U0 of vin V0 such that F0|(Z0)U0

= 0. One now obtains from U0 a formal open subscheme

U of V such that the pullback of F|U to (Z0)U0is 0. Since ZU is a noetherian formal

scheme and F|U is coherent, it follows that F|U = 0 on ZU. �

The next result is the analogue of [EGA, III, 5.3.4] in our situation, and addressesthe existence of an effectivizable formal scheme Z′ which would be used in Lemma3.7.

Lemma 3.9. Let Y be a noetherian scheme, let t : Y ′ → Y be a proper morphism,and let U ⊂ Y be an open subset such that t−1U → U is an isomorphism. LetJ / OY be an ideal such that supp(OY /J)c = U . Further, suppose that |V | ⊂ |Y | is

a closed subset, and let t : Y ′ → Y denote the induced map on completions along

|V |. If (Eγ−→ Y ) ∈ QF(Y ), and η : t!t

∗E → E denotes the adjunction morphism,then there is a k > 0 such that the kernel and cokernel of the map OE → η∗Ot! t∗Eis annihilated by the OE-ideal JkE.

Proof. Since Y is quasi-compact, the result is local on Y for the Zariski topology.

So, we may assume that Y = SpecR, J = J , for J /R and let I be an ideal defining

|V | ⊂ |Y |. Let R denote the I-adic completion of R. We are free to replace R

by R and we consequently assume that R is I-adic. Since R is I-adic, any openneighbourhood of |V | = |Spec(R/I)| is all of SpecR. Thus, it suffices to exhibitfor every y ∈ |V |, an open neighbourhood Yy of y such that the result is true over


Yy. By Lemma 3.8 combined with Corollary 2.15, we may thus assume that R islocal and maximal-adically complete.

We now proceed to prove the result by induction on dimR = d. In the case thatd = 0, then R is local artinian, and the result is trivial. Otherwise, assume that d >0, and we have proven the result for all complete local rings S with dimS < d. Note,that by what we have proven in the previous paragraph, the inductive hypothesisensures that we have proven the result for all noetherian schemes Y such that all thelocal rings of Y have dim < d. Now, since R is maximal-adically complete, R/In+1

is maximal-adically complete for any n ≥ 0. Thus, if En = E×SpfI RSpec(R/In+1),

then by [EGA, IV, 18.12.3], there is a decomposition En = V 1nqV 2

n , where V 1n → En

is finite and contains the special fiber, and V 2n is disjoint from the special fiber.

These decompositions are compatible, in the sense that V in ×En En−1 = V in−1 fori = 1, 2 and all n ≥ 1 and so we find that there is a decomposition of the formalscheme E = V1 qV2 such that V1 is finite over SpfI R and V2 misses the specialfiber.

Combining Corollary 2.16 with [EGA, III, 5.3.4] applied to V1 → SpecR we see

that there is a k such that JkV1 annhilates the kernel and cokernel of the map OV1 →η∗Ot! t∗V1 . Now note that V2 → W := (SpecR) − {m}, where m is the maximalideal of R, and since W is a noetherian scheme with all local rings of dimension< d, applying Corollaries 2.14 and 2.15 to V2 → W we may use the inductive

hypothesis to obtain that JkV2annihilates the kernel and cokernel of this map (by

possibly increasing k). Thus, since OE = OV1 ×OV2 , we find that there is a k such

that JkE annihilates the kernel and cokernel of the morphism OE → η∗Ot! t∗E. �

Let π : X → S be a morphism, locally of finite type, between locally noetherian

schemes, and let πI : XI → S denote the I-adic completion. In particular, we define

QFp(XI/S) as the full subcategory of QF(XI) consisting of those (Zσ−→ XI) such

that πI ◦ σ is adic and proper. There is an equivalence of categories:

QFp(XI/S)→ lim←−n

QFp(Xn/Sn).

By abuse of notation, we will consider Ψπ,I to be a functor with target category

QFp(XI/S). It will sometimes be convenient to further abuse notation, and sup-

press the subscript I in the definition of XI .

Proposition 3.10 (Birational devissage). Let π : X → S be a morphism of noe-therian schemes, such that for any closed subscheme V ↪→ X, there is a proper,birational (on V ) morphism τV : V ′ → V for which Ψπ◦τV ,I is an equivalence.Then Ψπ,I is an equivalence.

Proof. First, let ıV : V ↪→ X be a closed subscheme. Then there is a commutativediagram of categories:

QFp(V/S)

(ıV )!

��

Ψπ◦ıV ,I // QFp(V /S)

(ıV )!��

QFp(X/S)Ψπ,I

// QFp(X/S)

.

The vertical maps are fully faithful inclusions of subcategories. We observe that

the essential image of the functor (ıV )! is the set of objects (Zσ−→ X) ∈ QFp(X/S)

such that the natural map (ıV )!(ıV )∗Z → Z is an isomorphism. Similarly for the

essential image of ıV .


By Proposition 3.2, it suffices to prove that the functor Ψπ,I is essentially surjec-tive. We will show this by noetherian induction on the closed subsets of X. Moreprecisely, for a closed subset |V | ↪→ |X| we will consider the statement P (|V |):for any closed subscheme ıW : W ↪→ X with |W | ⊂ |V |, the functor Ψπ◦ıW ,I isessentially surjective. Since P (∅) is trivially true, by the principle of noetherianinduction, the result will follow if we can show that if the statement P (|V ′|) istrue for all proper closed subsets |V ′| ( |V |, then the statement P (|V |) true. Thestatement of the proposition allows us to equivalently prove the following: if thefunctor Ψπ◦ıV ,I is essentially surjective for any closed subscheme ıV : V → X suchthat |V | ( |X|, then the functor Ψπ,I is an equivalence.

We let Z := (Zσ−→ X) be an object of QFp(X/S) such that for any closed

subscheme ıV : V ↪→ X, with |V | ( |X|, the map (ıV )!(ıV )∗Z → Z is not an

isomorphism, and by noetherian induction, it remains to show that there is a Z in

QFp(X/S) such that Z ∼= Z.By assumption, there is a proper and birational morphism of schemes τ : X ′ → X

which is an isomorphism over an open subscheme : U ↪→ X, and the functorΨπ◦τ,I is an equivalence. Now define the closed subset |D| = |U |c ⊂ |X|, andtake ı : D → X to denote the inclusion of a subscheme D, supported preciselyon |D|, with coherent ideal sheaf J. By Lemma 3.9, there is an n > 0 such that

the OZ-ideal JnZ annihilates the kernel and cokernel of the map OZ → η∗Oτ!τ∗Z,where η : τ!τ

∗Z→ Z is the adjunction map. Next, observe that the formal scheme

τ∗Z belongs to QFp(X ′/S) and by hypothesis, there is an E in QFp(X ′/S) such

that E ∼= τ∗Z. Furthermore, Theorem 2.20, implies that we have isomorphisms

τ!τ∗Z ∼= τ!E ∼= (τ!E). Setting Z ′ = τ!E, we obtain a finite and surjective map

η : Z ′ → Z such that the kernel and cokernel are annihilated by the OZ-ideal JnZ.Since X is a noetherian scheme, we may now apply Lemma 3.7(2) to the map

η : Z ′ → Z and conclude that Z is effectivizable. �

With the devissage technique for schemes fully developed, we now proceed todevelop a devissage technique for algebraic stacks.

Lemma 3.11. Let X be a noetherian scheme, and let π : X ′ → X be a finitesurjective morphism that is flat over an open subset U ⊂ X. Let π2 : X ′ ×X X ′ →X denote the induced morphism. Furthermore, let J be a coherent OX-ideal with

support |X \ U |. Let I be another coherent OX-ideal, and let X (resp. X ′) denotethe formal scheme obtained by completing X along the closed subset V (I) (resp.

V (IX′)). Let (Z → X) ∈ QF(X) and consider the diagram [π2!(π2)∗Z ⇒ π!π

∗Z].

Assume that this diagram has a uniform formal geometric colimit Z (this is true in

the case that π∗Z and (π2)∗Z are effectivizable, by Corollary 1.19), so that there is

an induced finite map γπ,Z : Z → Z. Then, there is a k such that the kernel and

cokernel of the map γ]π,Z is annihilated by JkZ.

Proof. Repeating verbatim the reductions in Lemma 3.9, it suffices to prove the re-sult in the case that X = SpecR, where R is a local and maximal-adically completenoetherian ring. Further, let R′ be the finite R-algebra such that X ′ = SpecR′. SetI = Γ(X, I) and J = Γ(X, J). We will prove the result by induction on dimR = d.The case where d = 0 is trivial (since R is local and artinian). Otherwise, as-sume that d > 0, and we have proven the result for all complete local rings S withdimS < d. As in Lemma 3.9, this ensures that the result has been proven for allnoetherian schemes X such that all the local rings of X have dim < d.

Note that Z = Z1qZ2, where Z1 → X is finite, contains the special fiber, and Z2

misses the special fiber. In particular, Z2 → W := (SpecR)− {m}, where m is the


maximal ideal of R. Since R is local, all of the local rings of W have dim < d and bythe inductive hypothesis we can find a k2 such that Jk2Z2 annihilates the kernel and

cokernel of γ]π,Z2 . Now, since the map Z1 → X is finite, then Z1 = SpfIT T , where

T is a finite R-algebra. By definition, if we set T = ker(T ⊗RR′h−→ T ⊗RR′⊗RR′),

where h(t⊗ r) = t⊗1⊗ r− t⊗ r⊗1, then Z1 = SpfIT T . Let f ∈ J , then Rf → R′fis flat. By flat descent for modules, we conclude that Tf → Tf is an isomorphism.

Thus, the kernel and cokernel of γ]π,Z1 : T → T is annihilated by some fixed power

of f . Since J is coherent, we conclude that there is a k1 such that Jk1 annihilates

the kernel and cokernel of γ]π,Z1 . It is now clear that taking k = max{k1, k2} has

the property that JkZ annihilates the kernel and cokernel of γ]π,Z. �

Proposition 3.12 (Generically finite flat devissage). Let π : X → S be a noe-therian algebraic stack of finite type with schematic diagonal. For any S-morphismp : Y → X, define p2 : Y ×X Y → X. Suppose that for any closed substackıV : V ↪→ X there is a finite and generically flat (on V ) morphism τV : V ′ → Vsuch that Ψπ◦τV ,I and Ψπ◦τ2

V ,Iare equivalences. Then Ψπ,I is an equivalence.

Proof. The principle of noetherian induction implies that we may assume thatΨπ◦ıV ,I is an equivalence for any closed substack ıV : V ↪→ X with |V | ( |X|. ByProposition 3.2, it suffices to show that Ψπ,I is essentially surjective.

By assumption, there is a finite map τ : X1 → X such that over a dense openU ⊂ X the induced map τ : τ−1U → U is flat with the property that Ψπ◦τ,I andΨπ◦τ2,I are equivalences. It will be convenient to set τ1 := τ and X2 := X1×X X1

so that we have τ2 : X2 → X. Let (Zn)n≥0 ∈ lim←−n QFp(Xn/Sn) and for each

n ≥ 0 we set Zin = Zn ×Xn Xin, so that (Zin)n≥0 ∈ lim←−n QFp(Xi

n/Sn) for i = 1, 2.

By assumption, for i = 1, 2, there is a unique (Zi → Xi) ∈ QFp(Xi/S) such that

Ψπ◦τ i,I(Zi → Xi) = (Zin)n≥0. Note that from the two finite projection maps s,

t : X2 → X1 we obtain a coequalizer diagram [Z2 ⇒ Z1] in QF(X). By Theorem1.10, this diagram has a uniform categorical and uniform geometric coequalizer(C → X) in QF(X) and Z1 → C is finite and surjective. As Z1 is S-proper, itfollows that C is S-proper so that (C → X) ∈ QFp(X/S).

Let p1 : V 1 → X be a smooth surjection from a scheme V 1. Consider an integerj ≥ 1 and let V j be the j-fold fiber product of V 1 over X. Since X has schematicdiagonal, V j is an S-scheme. The V j-scheme CV j is the categorical and geometriccoequalizer of the diagram [Z2

V j ⇒ Z1V j ] in QF(V j). For each n ≥ 0, the morphism

(CV 1)n → V 1n comes with descent data for the smooth covering V 1

n → Xn. Wealso note here that for each n, the morphism (Zn)V 1 → V 1

n comes equipped withdescent data for the smooth covering p1

n : V 1n → Xn.

Define the formal V j-scheme ZV j = lim←−n(Zn ×X V j) ∈ QF(V j) and let σj :

ZV j → V j denote the structure map. By Theorem 1.18, CV j is the uniform cat-

egorical and uniform formal geometric coequalizer of the diagram [Z2V j ⇒ Z1

V j ]

in QF(V j), and whence there is a unique finite morphism of formal V j-schemes

φj : CV j → ZV j .Consider the closed subset |D| = |U |c ⊂ |X|, and fix a coherent sheaf of ideals

J / OX with support |D|. By Lemma 3.11 there is a k > 0 for j = 1, 2, such that(JV j )

kZV j

annihilates the kernel and cokernel of (φj)]. By Lemma 3.7(2), we thus

conclude that ZV j is effectivizable for j = 1, 2. Thus, we may find (W j → V j) ∈QF(V j) such that W j ∼= ZV j . By smooth descent of quasi-finite, separated andrepresentable morphisms, we may conclude that there is a quasi-finite, separatedand representable morphism (Z → X) ∈ QF(X) such that ZV j ∼= W j for j = 1, 2.


Further, since smooth descent commutes with arbitrary fiber products, we concludethat Z×X Xn

∼= Zn. Finally, to see that (Z → X) ∈ QFp(X), we simply note thatthe X-map φ : C → Z is finite and surjective (this is verified smooth locally on X).Thus, we have proved that (Zn)n≥0 ∈ lim−→n

QFp(Xn/Sn) lies in the essential image

of Ψπ,I . �

We may finally prove Theorem 3.5.

Proof of Theorem 3.5. By Proposition 3.2, it remains to show that the functor Ψπ,I

is essentially surjective, which we divide into a number of cases.Basic case. Here, we assume that we have a morphism of schemes π : X → S

of finite type, which factors as Xp−→ Y

q−→ S where p is etale and q is projective.

Now, suppose that (Zσ−→ X) ∈ QFp(X/S). Then we observe that the composition

of morphisms Z→ X → Y is quasi-finite and proper, since Y is separated and Z is

S-proper. Hence, the morphism Z→ Y is finite, and as q : Y → S is projective, by

[EGA, III, 5.4.4], there is a finite Y -scheme Z such that (Z → Y ) ∼= (Z → Y ) in

QFp(Y /S).It remains to construct an S-morphism s : Z → X such that s = σ. To do this,

it suffices to construct a section to the morphism X ×Y Z → Z which restricts tothe section given by the morphism σn : Zn → Xn for all n. The hypotheses onX → Y are all preserved by base change, so we conclude that X×Y Z → Z is quasi-compact and etale. Hence, the etale sheaf on Z given by U 7→ HomZ(U,X ×Y Z)is constructible. Since we have a section Z0 → X0×Y0

Z0, by [Mil80, Thm. VI.2.1],we obtain a unique section Z → X ×Y Z which restricts to the one we started with(hence to all other σn’s).Quasi-compact and schematic case. Here, we assume that we have any mor-phism of schemes π : X → S of finite type. Note that every closed subscheme V ofX is of finite type. In particular, by [RG71, Cor. 5.7.13], there is always a diagram:

V ′

��V

��

P

��S,

where V ′ → V is proper and birational on V , P → S is projective, and V ′ → P isetale. By the basic case considered above, we know that ΨV ′→S,I is an equivalence.Thus, we have met the criteria to apply Proposition 3.10 and we conclude that Ψπ,I

is an equivalence for any morphism of schemes π : X → S of finite type.Quasi-compact with quasi-finite and separated diagonal case. Now weassume that we have a morphism of algebraic stacks π : X → S of finite type withquasi-finite and separated diagonal. By Proposition 3.12 and the case for schemesalready proved, it suffices to show that for any closed substack V ↪→ X, there is afinite surjection V ′ → V from a scheme V ′, which is flat over a dense open of V .This is [Ryd09, Thm. B].General case. Suppose that π : X → S is locally of finite type with quasi-compact

and separated diagonal. It remains to show that the functor Ψπ,I is essentiallysurjective. Let Y ↪→ X be the open substack of X with quasi-finite and separateddiagonal. If (Zn)n≥0 ∈ lim←−n QFp,fin,∆(Xn/Sn), then for each n ≥ 0, the structure

map Zn → X factors uniquely through Y . Hence, it suffices to prove that the

functor Ψπ,I is essentially surjective in the case that π : X → S has quasi-finiteand separated diagonal.


Let OX denote the set of quasi-compact open substacks X. It is ordered byinclusion, and we note that {|W ×X Z0|}W∈OX is an open cover of the quasi-compact topological space |Z0|. Thus, there is a W ∈ OX such that the canonicalX-morphism W ×X Z0 → Z0 is an isomorphism. In particular, we conclude thatthe map Z0 → X factors uniquely as Z0 → W ↪→ X. Since the open immersionW ↪→ X is etale, it follows that the maps Zn → X also factor compatibly throughW and thus (Zn)n≥0 ∈ lim←−QFp(Wn/Sn). We can thus replace X with W andconclude by the previous case. �

4. Proof of Theorem 2

Before we get to the proof Theorem 2, we will require an analysis of some spacesof sections. Fix a scheme T , and let s : V ′ → V be a representable morphismof algebraic T -stacks. Define the T -sheaf SecT (V ′/V ) to be the sections of themorphism s. We require an improvement of [Ols06, Prop. 5.10] and [Lie06, Lem.2.10], which we prove using [Hal12a, Thm. B].

Proposition 4.1. Fix a scheme T and a proper, flat, and finitely presented mor-phism of algebraic stacks p : Z → T . Consider a quasi-finite, separated, finitelypresented, and representable morphism s : Q→ Z. Then the T -sheaf SecT (Q/Z) isrepresented by a quasi-affine T -scheme.

Proof. By a standard limit argument [Ryd09, Prop. B.3], we can assume that Tis noetherian. By Zariski’s Main Theorem [LMB, Thm. 16.5(ii)], there is a finiteZ-morphism Q → Z and an open immersion Q ↪→ Q. In particular, we see thatthere is a natural transformation of T -sheaves SecT (Q/Z) → SecT (Q/Z) which isrepresented by quasi-compact open immersions. Hence, we may assume for theremainder that the morphism s : Q → Z is finite. Next, observe that SecT (Q/Z)is a closed subfunctor of the T -presheaf HomOZ/T

(s∗OQ,OZ). By [Hal12a, Thm.

B], the sheaf HomOZ/T(s∗OQ,OZ) is represented by a scheme that is affine over T ,

and the result follows. �

Corollary 4.2. Fix a scheme T , and a pair of proper, flat, and finitely presentedmorphisms of algebraic stacks pi : Zi → T for i = 1, 2, with finite diagonals.Suppose that X is an algebraic T -stack, that is locally of finite presentation andhas quasi-compact and separated diagonal, and that we have quasi-finite and rep-resentable morphisms si : Zi → X for i = 1, 2. Then the functor on Sch/T ,given by T ′ 7→ HomXT ′ ((Z1)T ′ , (Z2)T ′), is represented by a scheme that is quasi-affine over T . In particular, the open subfunctor IsomX(Z1, Z2) ⊂ HomX(Z1, Z2)parameterizing isomorphisms is represented by a scheme that is quasi-affine over T .

Proof. Note that HomX(Z1, Z2) = SecT ((Z1 ×X Z2)/Z1) and since X → T hasseparated diagonal, it follows that for each i = 1, 2, the morphisms si are separated.Thus, by Proposition 4.1 the functor HomX(Z1, Z2) has the asserted properties. �

Proof of Theorem 2. The results of [Hal12b, §9.2], together with Theorem 3.5, showthat the stack HSX/S is algebraic and locally of finite presentation over S. We nowapply Corollary 4.2 to obtain the asserted separation property. �

References

[AJL99] L. Alonso Tarrıo, A. Jeremıas Lopez, and J. Lipman, Studies in duality on Noetherianformal schemes and non-Noetherian ordinary schemes, Contemporary Mathematics,vol. 244, American Mathematical Society, Providence, RI, 1999.

[AK10] V. Alexeev and A. Knutson, Complete moduli spaces of branchvarieties, J. ReineAngew. Math. 639 (2010), 39–71.

[AM69] M. F. Atiyah and I. G. Macdonald, Introduction to commutative algebra, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1969.


[Aok06] M. Aoki, Homs stacks: erratum, Manuscripta Math. 121 (2006), no. 1, 135.

[Art69] M. Artin, Algebraization of formal moduli. I, Global Analysis (Papers in Honor of K.

Kodaira), Univ. Tokyo Press, Tokyo, 1969, pp. 21–71.[Art74] , Versal deformations and algebraic stacks, Invent. Math. 27 (1974), 165–189.

[Con] B. Conrad, Formal GAGA for Artin stacks, Available on homepage.[DM69] P. Deligne and D. Mumford, The irreducibility of the space of curves of given genus,

Inst. Hautes Etudes Sci. Publ. Math. (1969), no. 36, 75–109.

[EGA] A. Grothendieck, Elements de geometrie algebrique, I.H.E.S. Publ. Math. 4, 8, 11, 17,

20, 24, 28, 32 (1960, 1961, 1961, 1963, 1964, 1965, 1966, 1967).[Fal03] G. Faltings, Finiteness of coherent cohomology for proper fppf stacks, J. Algebraic

Geom. 12 (2003), no. 2, 357–366.

[FGA] A. Grothendieck, Fondements de la geometrie algebrique. Extraits du Seminaire Bour-baki, 1957–1962, Secretariat mathematique, Paris, 1962.

[FGI+05] B. Fantechi, L. Gottsche, L. Illusie, S. L. Kleiman, N. Nitsure, and A. Vistoli, Funda-mental algebraic geometry, Mathematical Surveys and Monographs, vol. 123, American

Mathematical Society, Providence, RI, 2005, Grothendieck’s FGA explained.

[Hal12a] J. Hall, Coherence results for algebraic stacks, Preprint, Jun 2012, arXiv:1206.4179.[Hal12b] , Openness of versality via coherent functors, Preprint, Jun 2012,

arXiv:1206.4182.

[Has03] B. Hassett, Moduli spaces of weighted pointed stable curves, Adv. Math. 173 (2003),no. 2, 316–352.

[Høn04] M. Hønsen, A compact moduli space parameterizing Cohen-Macaulay curves in projec-

tive space, Ph.D. thesis, MIT, 2004.[Kol11] Janos Kollar, Quotients by finite equivalence relations, Current Developments in Alge-

braic Geometry, Math. Sci. Res. Inst. Publ., vol. 59, Cambridge Univ. Press, Cambridge,

2011, pp. 227–256.[Lie06] M. Lieblich, Remarks on the stack of coherent algebras, Int. Math. Res. Not. (2006),

Art. ID 75273, 12.[LMB] G. Laumon and L. Moret-Bailly, Champs algebriques, Ergebnisse der Mathematik und

ihrer Grenzgebiete. 3. Folge., vol. 39, Springer-Verlag, Berlin, 2000.

[LS08] C. Lundkvist and R. Skjelnes, Non-effective deformations of Grothendieck’s Hilbertfunctor, Math. Z. 258 (2008), no. 3, 513–519.

[Mil80] J. S. Milne, Etale cohomology, Princeton Mathematical Series, vol. 33, Princeton Uni-versity Press, Princeton, N.J., 1980.

[Ols05] M. Olsson, On proper coverings of Artin stacks, Adv. Math. 198 (2005), no. 1, 93–106.

[Ols06] , Hom-stacks and restriction of scalars, Duke Math. J. 134 (2006), no. 1, 139–

164.

[OS03] M. Olsson and J. M. Starr, Quot functors for Deligne-Mumford stacks, Comm. Algebra31 (2003), no. 8, 4069–4096, Special issue in honor of Steven L. Kleiman.

[RG71] M. Raynaud and L. Gruson, Criteres de platitude et de projectivite. Techniques de

“platification” d’un module, Invent. Math. 13 (1971), 1–89.[Ryd07] David Rydh, Existence and properties of geometric quotients, Preprint, Aug 2007,

arXiv:0708.3333v2, To appear in J. Algebraic Geom.

[Ryd09] D. Rydh, Noetherian approximation of algebraic spaces and stacks, Preprint, 2009,arXiv:0904.0227v2.

[Ryd11] , Representability of Hilbert schemes and Hilbert stacks of points, Comm. Alge-

bra 39 (2011), no. 7, 2632–2646.[Sch91] D. Schubert, A new compactification of the moduli space of curves, Compositio Math.

78 (1991), no. 3, 297–313.[Smy09] D. I. Smyth, Towards a classification of modular compactifications of the moduli space

of curves, Preprint, 2009, arXiv:0902.3690.

[Vis91] A. Vistoli, The Hilbert stack and the theory of moduli of families, Geometry Semi-nars, 1988–1991 (Italian) (Bologna, 1988–1991), Univ. Stud. Bologna, Bologna, 1991,pp. 175–181.

Department of Mathematics, KTH Royal Institute of Technology, SE-100 44 Stock-

holm, SwedenE-mail address: [email protected]

Department of Mathematics, KTH Royal Institute of Technology, SE-100 44 Stock-holm, Sweden

E-mail address: [email protected]

http://arXiv.org/abs/1206.4179


http://arXiv.org/abs/0708.3333v2

http://arXiv.org/abs/0904.0227v2


introduction - kth › ~dary › hilbertstacksnonsep20120626.pdf · unlike schemes, the norm for...

Documents