Algebraic Stacks

Contents

Introduction 2

1 Stack 31.1 Basic Grothendieck topologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.1.1 Examples and definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.1.2 Representable presheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2 Fibered categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.2.1 Definition of fibered categories and categories fibered in groupoids . . . . 61.2.2 2-Yoneda and PSh to CFG . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.2.3 Fiber products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.3 Towards stacks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.3.1 Effective descent data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.3.2 Definition of stacks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.3.3 Representability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2 Algebraic spaces and algebraic stacks 152.1 More background materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.1.1 Different topologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.1.2 Properties of sheaves and morphisms . . . . . . . . . . . . . . . . . . . . . 17

2.2 Algebraic spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.2.1 Definition of algebraic spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 182.2.2 Sheaf quotient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.2.3 Geometry on algebraic spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.3 Algebraic stacks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.3.1 Definition of algebraic stacks . . . . . . . . . . . . . . . . . . . . . . . . . . 232.3.2 Geometry on algebraic stacks . . . . . . . . . . . . . . . . . . . . . . . . . . 242.3.3 Deligne-Mumford stacks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3 Various examples of algebraic stacks 273.1 Quotient stacks [X/G] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.1.1 Groupoids in algebraic spaces . . . . . . . . . . . . . . . . . . . . . . . . . 273.1.2 Torsor and principal bundles . . . . . . . . . . . . . . . . . . . . . . . . . . 293.1.3 Brief touch on Keel-Mori . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.2 Mg andMg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.2.1 The compactificationMg . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

Conclusion and future perspectives 36

References 37

1

Introduction

This is my Cambridge mathematics Part III essay on algebraic stacks, and that means a fewthings. First, my intention is not to give a comprehensive review, but only a concise introductionon the subject of stacks. In particular, I am only assuming some prior knowledge on sheavesand schemes, at the level of Part III algebraic geometry course, and some elementary categorytheory. Second, when learning something new, I love to keep intuitions at mind which wouldoften provide a second perspective into how things work, and I organized this essay with thishope in mind. Many statements, therefore, are stated in two ways: one with precise definitions,and another that helps to think.

Structure and motivations

The whole structure of this essay is rather simple. In the first chapter, we introduce the categoricalconstruction of stacks. In the second chapter, we look at how we can put geometry on thiscategory and motivate the definition of algebraic stacks. Lastly in the third chapter, we look attwo very important classes of algebraic stacks.

The motivation for stacks, personally, comes from vector bundles. We know schemes gen-eralize varieties. As a result, we can do geometry on objects like grassmannians, which is, insome sense, a classifying object (functor) that sends each scheme S to the set of n− k dimensionalsubspaces. It is a scheme in the sense that this functor is representable by a scheme. Anothersimilar functor is vector bundles, which sends a scheme S to all rank r vector bundles on thescheme. However, this functor is not a scheme anymore. And we would want to generalizeschemes to stacks to include this object. Eventually we will see that stacks are just sheaves takingvalues in categories.

In chapter 2, we solve the problem that the construction of stacks is too categorical to dogeometry. To solve this problem, we look back to some familiar cases, for example, on complexmanifolds. We will have defined presheaves and sheaves on a category and there would bea picture C ⊂ Sh(C) ⊂ PSh(C). We ask the question: of all the sheaves on the category C ofopen sets in Cn with holomorphic maps, which are complex manifolds? The answer is thelocally representable ones. Similarly on the category of affine schemes, schemes are the locallyrepresentable sheaves; but we will be mostly dealing with a different topology on the category of(affine) schemes, where locally representable (affine) schemes are the notion of algebraic spaces,which slightly generalizes schemes and bear many scheme-level geometry.

Once we pass this stage, we can ask a generalization to locally representable stacks. This isknown as Artin stacks or algebraic stacks. In particular, it is still covered by schemes, and bysaying a morphism of algebraic stacks can be represented by schemes or algebraic spaces, wecould make similar definitions of geometric properties of algebraic stacks and their maps.

A very special and wide-occurring class of algebraic stacks is the quotient stacks. They arenot so easy to construct, but they do take a very important place in the theory of algebraic stacks.We will look at them in our last chapter. We will also discuss the main ideas from a paper writtenby Deligne and Mumford, who studied closely a certain type of objects within algebraic stackscalled Deligne-Mumford stacks.

2

Chapter 1

Stack

In this chapter, we will discuss the definition and construction of stacks. In particular, we willneed lots of new definitions that are given and explored only when required.

The structure of this chapter will be roughly the following. Contrary to most other authors’style that tons of definitions culminate at stacks, we give a precise definition of stacks at the verybeginning. This will serve as a natural motivation and map for what we are discussing and whatis to come.

Throughout this chapter, we will be dealing with one particular example many times: thefunctor Vr taking all rank r vector bundles on a scheme. We could see how these new definitionsagree with the old familiar case. This idea is further explored in [Fan].

Definition 1.1. A category over a base category C fibered in groupoids p : F → C is a stack if thefollowing two conditions hold:

1. For any X ∈ C and objects x, y ∈ F(X), the presheaf Isom(x, y) on C/X (which will bedefined later) is a sheaf; and

2. For any covering Xi → X of an object X ∈ C, any descent data with respect to thiscovering is effective.

So naturally the rest of this chapter serves to explain the meanings of:

1. a category over C fibered in groupoids,

2. topology and sheaves on categories,

3. the presheaf Isom(x, y) on C/X, and

4. an effective descent datum.

1.1 Basic Grothendieck topologies

To discuss why we need a category fibered in groupoids, we need to talk about how to give atopology on a category, and in particular, how to define a sheaf on a category.

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1.1.1 Examples and definitions

We want to explore the idea that “knowing an object is the same as knowing the maps to it”.Let’s say we have an S-scheme X. Given any other scheme U, we can talk about the U-points onX:

X(U) = Homscheme(U, X).

Traditionally, when we talk about the topology of X, we are essentially talking about all opencovers of X, denoted by J(X) = Ui → Xi∈I : Ui cover X. In particular, if we require eachUi → X to be an open embedding and X =

⋃Ui, then this is known as the big Zariski topology on

the category of S-schemes.

But we could do more here. An open cover, in the sense above, is a family of Ui-points, i.e. anelement from each X(Ui). If we treat X as a (contravariant) functor that sends a scheme U to allU-points X(U), an open cover will be a subfunctor. We call a subfunctor of the Yoneda functorX a sieve. So a traditional open cover is a sieve. Specifying which Ui form a cover is roughlyspeaking specifying which sieves are “covering”.

We will be mainly dealing with the category of S-schemes, in which fiber products exist. Inthis case, we could use a simpler set of axioms to define a topology1 as follows:

Definition 1.2. Let C be a category. A Grothendieck topology on C consists of, for each object X ofC, a set Cov(X) of collections of morphisms Xi → Xi∈I , satisfying the three axioms:

1. (Isomorphism is a cover) If V → X is an isomorphism, then V → X ∈ Cov(X).

2. (Pullback cover exists) If Xi → X is a covering, and Y → X any morphism in the categoryC, then there is a fibered covering Xi ×Y Y → Y which is in Cov(Y).

3. (Cover of a cover is a cover) If Xi → Xi∈I is a cover for X, and for each i, Vij → Xij∈Ji

is a cover for Xi, then all composites

Vij → Xi → Xi∈I,j∈Ji

is a cover for X.

A category with a Grothendieck topology is called a site.

1.1.2 Representable presheaves

Definition 1.3. A presheaf on a category C is a contravariant functor

F : Cop → Set.

A presheaf map will be a natural transformation between functors.

In what is to come, presheaves will be the things that we work with. So it is better to treatthem as objects instead of functors. In particular, we can build a “functor category” PSh(C),whose objects are presheaves, and morphisms are natural transformations.

1Otherwise we will be defining what is known as a pretopology.

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A sheaf is a presheaf that satisfies the gluing and identity properties (since we already havea topology here). We then define a map of sheaves to be a map of presheaves. By definition,Sh(C) will be a full subcategory of PSh(C). We have our usual categorical Yoneda lemma (whereYo(X) := Mor(−, X)):

Lemma 1.4. Suppose X ∈ C, and F ∈ PSh(C). There is a bijection

Mor(Yo(X), F)↔ F(X)

that is functorial in both X and F.

Definition 1.5. We say a presheaf F is representable if it is isomorphic to Yo(X) for some X ∈ C.

Example 1.6. Take C to be the category of X schemes with big Zariski topology1, and take thepresheaf F to be the Yoneda embedding Mor(−, Z) for any Z. Then F is a sheaf.

Now taking F = Yo(Y) in the Yoneda lemma, we get that the Yoneda embedding C → PSh(C)sending Z 7→ Mor(−, Z) is a fully faithful functor, and thus embeds C → PSh(C). When takingour base category to be reasonable enough (i.e. maps glue, and the category of X-schemes worksfine), what we said in the above example is that representable presheaves are sheaves. So wehave a picture:

C → Sh(C) → PSh(C).

Now that we can talk about sheaves on X-schemes, we can look at the “to-be” stack Vr thatsends an X-scheme Y to the “collection” of all rank r vector bundles over Y. This is not a sheaf,but is “like a sheaf”.

So what do we mean by collection? First, isomorphism classes don’t work, because all vectorbundles are locally trivial, and therefore Vr will not be separated. Then we might want non-isomorphic classes. But we can only glue the maps if we are given some additional informationon the overlaps. In particular, it won’t even be a functor from X-schemes to Sets.

To solve this problem, we need the notion of categories fibered in sets and groupoids, whichwe discuss next.

1.2 Fibered categories

In this section, we will fix a base category C. There will no topology involved. A category over Cis a category F equipped with a functor p : F → C. We also write F(U) the fiber category whoseobjects are u ∈ F with p(u) = U (object of F over U), and morphisms are f : u → u′ such thatp( f ) = id.

By definition, a morphism φ : u→ v is a commutative diagram

u v

p(u) p(v).

φ

p(φ)

1A more highbrow version of this example is to say the big Zariski topology is subcanonical.

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We say a morphism φ : u→ v is cartesian if it satisfies the universal property in the sense that ifw ∈ F is another object, ψ : w→ v a morphism, and p(ψ) factors as

p(w)h−→ p(u)

p(φ)−−→ p(v),

then there exists a unique morphism λ : w→ u such that φ λ = ψ and p(λ) = h. In a picture,

w u v

p(w) p(u) p(v).

ψ

λ φ

h p(φ)

We call u the pullback of v along p(φ). By an easy argument, pullbacks are unique up to a uniqueisomorphism.

1.2.1 Definition of fibered categories and categories fibered in groupoids

Definition 1.7. A fibered category over C is a category p : F → C over C such that for everymorphism f : U → V in C and v ∈ F(V), there exists a cartesian morphism φ : u→ v such thatp(φ) = f , i.e. we can always do pullbacks.

Definition 1.8. A category fibered in groupoids over a category C is a fibered category p : F → Csuch that each fiber F(U) is a groupoid.

Sometimes it is easier to work with the following criteria.

Proposition 1.9. A category F over C is fibered in groupoids over C iff the two conditions hold:

1. Pullback exists with its universal property, and

2. All morphisms in F are pullbacks.

Proof. If F satisfies two conditions above, and φ : u→ v a morphism in F(U), by condition 1 wecan find an inverse over the inverse of the identity map p(φ), and therefore φ is invertible.

Conversely, given any morphism φ : u→ v, which maps to p(φ) : U → V, we first choose apullback (exists by definition of fibered category) φ′ : u′ → v. Then by definition of pullbacks,we get a map α : u → u′, which will necessarily be an isomorphism as F(U) is a groupoid. Byuniversal property, φ is cartesian.

Just for completion, a map between two categories over C fibered in groupoids will be afunctor that commutes with projection to C and sends pullbacks to pullbacks.

1.2.2 2-Yoneda and PSh to CFG

As we discussed at the end of subsection 1.1.2, we are trying to enlarge the class of PSh(C) toinclude operations like “taking all rank r vector bundles”. Let’s call the category of all categoriesfibered in groupoids over C as CFG(C). To see the relationship between PSh and CFG, essentiallywe need the 2-Yoneda lemma, which concerns 2-categories.

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We will only be making some informal comments about 2-categories without going too deepinto it. Roughly speaking, a 0-category consists of some objects without any relations. Forexample, a set is a 0-category.

A 1-category has objects and some morphisms, and is what we used to think as a category.The category of sets is a 1-category.

A 2-category has some objects, and morphisms between these objects form a category. Forexample, the category of categories has 0-objects all categories, 1-objects all functors betweencategories (which is again a category), and furthermore some 2-objects natural transformationsbetween the 1-objects.

Example 1.10. CFG(C) is a 2-category.

The 0-objects are all the categories over C fibered in groupoids; the 1-objects are functorsα : F → F′ such that p = p′ α; and the 2-objects are natural transformations between twofunctors α, β which project to identity on C. These 2-objects can also be defined on any fiberedcategories, and we want to give it a name.

Definition 1.11. If α, β : F → F′ are morphisms of fibered categories, then a base preserving naturaltransformation τ : α → β is a natural transformation of functors such that for every u ∈ F, themorphism τu : α(u)→ β(u) in F′ projects to the identity morphism. We denote by

HOMC(F, F′)

the category whose objects are morphisms of fibered categories F → F′, and whose morphismsare base preserving natural transformations.

Now we can state the 2-Yoneda. Just as the normal Yoneda deals with the Yoneda embedding,here we have a category C/X for some object X ∈ C, whose objects are morphisms Y → X andwhose morphisms are commutative triangles. This is a fibered category over C with functorC/X → C sending Y → X to Y.

Lemma 1.12. The functorξ : HOMC((C/X), F)→ F(X)

sending a morphism of fibered categories g : C/X → F to g(X → X) is an equivalence of categories.

We omit the proof here as it is just a categorical construction. It is in [Ols16, Vis08].

We need one more definition to see the relationship between PSh and CFG: the notion ofcategories fibered in sets.

Definition 1.13. A category fibered in sets is a fibered category p : F → C such that the onlymorphisms in the fiber category F(U) are the identity morphisms (that is, F(U) is a set).

Let’s construct a map PSh(C) to CFG(C). For a presheaf F : Cop → Set, say the image in CFGis denoted by SF. The objects of SF are

Ob(SF) = (X, x) : X ∈ Ob(C), x ∈ Ob(F(X)).

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Morphisms between (X, x) and (Y, y) are

MorSF ((X, x), (Y, y)) = f ∈ MorC(X, Y) : F( f )(y) = x.

The fibered category structure is p(X, x) = X. It’s straightforward to see SF is a category over Cfibered in sets.

Proposition 1.14. The functor

S− : (presheaves on C)→ (categories fibered in sets over C)

sending F to SF defined above is an equivalence of categories.

Proof. Given a category fibered in sets p : G → C, define F : Cop → Set by sending U ∈ C to theset (by 2-Yoneda)

G(U) ∼= HOMC((C/U), G)

and a morphism g : V → U to g∗ : G(U)→ G(V).

Then SF is a fibered category whose fiber over U is G(U) and whose morphisms are definedby the maps g∗. The natural map SF → G is an equivalence since it induces an equivalence inthe fiber over any U.

By definition, since any identity morphism is necessarily invertible, any category fibered insets is a category fibered in groupoids. So we embedded presheaves on C into CFG.

In particular, it makes sense now to define a representable CFG:

Definition 1.15. A representable CFG is a CFG that is isomorphic to some SF.

1.2.3 Fiber products

The main obstacle in defining fiber products for CFG is that it is a 2-category, and the usualuniversal property does not capture enough information. We need what is called a 2-categoricalfiber product, and the corresponding 2-universal property.

Let’s start by considering fiber products of groupoids. Let

G1

G2 G

fg

be a diagram of groupoids. The fiber product G1×G G2 has objects triples (x, y, σ), where x ∈ G1,y ∈ G2 are objects, and σ : f (x)→ g(y) is an isomorphism in G. A morphism

(x′, y′, σ′)→ (x, y, σ)

is a pair of isomorphisms a : x′ → x and b : y′ → y such that the diagram commutes:

f (x′) g(y′)

f (x) g(y).

σ′

f (a) g(b)

σ

8

There are functors pj : G1 ×G G2 → Gj for j = 1, 2, and a natural isomorphism of functorsΣ : f p1 → g p2:

G1 ×G G2 G1

G2 G.

p1

p2 fg

It has the universal property that, if H is another groupoid and that

α : H → G1, β : H → G2, γ : f α→ g β

are two functors and an isomorphism of functors, then there exists a collection of data

(h : H → G1 ×G G2, λ1, λ2)

where h is a functor,λ1 : α→ p1 h, λ2 : β→ p2 h

are isomorphisms of functors, and the diagram

f α f p1 h

g β g p2 h

f (λ1)

γ Σhg(λ2)

commutes. The data(h, λ1, λ2)

is unique up to a unique isomorphism.

Now let C be our base category, and let

F1

F2 F3

c

d

be a diagram of CFG over C. What would be the correct notion of a universal property here?

Suppose we have a CFG G over C, and morphisms α : G → F1 and β : G → F2 such thatγ : c α→ d β is an isomorphism of morphisms between CFG G → F3. Giving the data (α, β, γ)

is equivalent to giving an object of

HOMC(G, F1)×HOMC(G,F3)HOMC(G, F2).

The next lemma shows HOMC(G, F1) is actually a groupoid, and therefore we can use theuniversal property of groupoids we just developed.

Lemma 1.16. Let F, F′ be two categories fibered in groupoids over C. Then the category

HOMC(F, F′)

of morphisms of CFG F → F′ is a groupoid.

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Proof. Suppose there is a morphism ξ : f → g between two objects (morphisms of CFG f , g : F →F′). It suffices to show for each u ∈ F, the map

ξu : f (u)→ g(u)

is an isomorphism. By definition, ξu is a morphism in F′(pF(u)), which is a groupoid, andtherefore it is, indeed, an isomorphism.

Using this lemma, the data (α, β, γ) defines, for any other CFG H, a morphism of groupoid:

HOMC(H, G)→ HOMC(H, F1)×HOMC(H,F3)HOMC(H, F2) (†)

(h : H → G) 7→ (α h, β h, γ h)

And now we can phrase the universal property of fibered products for CFG using the languageof groupoids.

Proposition 1.17. There exists a collection of data (G, α, β, γ) as above, such that for every CFG H overC, the map (†) is an isomorphism. It satisfies the universal property that if (G′, α′, β′, γ′) is anothercollection with property above, then there exists a triple (F, u, v), where F : G → G′ is an equivalenceof fibered categories, u : α → α′ F and v : β → β′ F are isomorphisms of base-preserving naturaltransformations, such that the following diagram commutes:

c α c α′ F

d β d β′ F.

cu

γ γ′

dv

Moreover, it is unique in the sense that if (F′, u′, v′) is a second such triple, then there exists a uniqueisomorphism σ : F′ → F such that the diagrams:

α α′ F

α′ F

u′

uσ

and

β β′ F

β′ F

v′

vσ

commute.

The proof is, again, very categorical in flavor, and can be found in [Ols16].

1.3 Towards stacks

As we discussed in the previous section, we first had this picture

C → Sh(C) → PSh(C).

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Then we enlarged PSh to CFG, where CFG are, roughly speaking, presheaves taking values incategories. What would the corresponding notion for “sheaves taking values in categories”?

As before, for a presheaf to be a sheaf, we would want it to glue, i.e. the fiber category over Xshould be entirely determined by the fiber categories over a cover of X, and the isomorphismsneed to be a sheaf.

We begin by making sense of the first statement. Just as we need the cocycle conditions onoverlaps for vector bundles to glue, a similar notion is required here, called descent.

The main reference for this section is [Vis08].

1.3.1 Effective descent data

Let F be a category fibered over C. Given a covering Ui → U, we write Uij = Ui ×U Uj, andUijk = Ui ×U Uj ×U Uk.

Definition 1.18. Let σi : Ui → U be a covering in C. An object with descent data (Ei, φij) onthis cover is a collection of objects Ei ∈ F(Ui), together with isomorphisms φij : pr∗2 Ej → pr∗1 Ei inF(Uij) (here pr1 is the projection to Ui, and pr2 to Uj; similar afterwards), satisfying the cocyclecondition:

pr∗13φik = pr∗12φij pr∗23φjk : pr∗3 Ek → pr∗1 Ei

for any triple indices i, j, k.

The isomorphisms φij are called transition isomorphisms of the object with descent data.

An arrow between objects with descent data

αi : (Ei, φij)→ (E′i, ψij)

is a collection of arrows αi : Ei → E′i in F(Ui) with the property that for each i, j, the diagramcommutes:

pr∗2 Ej pr∗2 E′j

pr∗1 Ei pr∗1 E′i .

pr∗2 αj

φij ψij

pr∗1 αi

Together these make objects with descent data a category, denoted by F(Ui → U).

For vector bundles, the definition above captures the idea that “information agree on over-laps”. We still need another that says every such collection will glue, i.e. there exists an object Ein F(U) and pullbacks maps σi : Ei → E lying over σi. For this, we define a natural functor

ε : F(U)→ F(σi : Ui → U)

as follows.

For each object E of F(U), take the objects Ei = σ∗i E; and the isomorphisms φij : pr∗2 σ∗j E→pr∗1 σ∗1 E are the isomorphisms that come from the fact that both are pullbacks of E to Uij, and sothey could be identified.

Given an arrow α : E→ E′ in F(U), we get arrows σ∗i α : σ∗E→ σ∗E′, yielding an arrow from

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the object with descent data associated with E to that with E′. So this defines a functor ε asrequired.

Definition 1.19. An object with descent data (Ei, φij) in F(Ui → U) is effective if it isisomorphic to the image of an object of F(U).

1.3.2 Definition of stacks

The last bit we need to define a stack is the Isom presheaf. We give the construction here.

Recall that in discussing 2-Yoneda, we defined a category C/X, where objects are morphismsY → X, and morphisms are commutative triangles1. The fibered structure is by sending Y → Xto Y.

Now let X ∈ C be an object and let x, x′ ∈ F(X) be two objects in the fiber over X. We candefine a presheaf on C/X:

Isom(x, x′) : (C/X)op → Set

as follows.

For any object f : Y → X in C/X, choose pullbacks f ∗x, f ∗x′ and set

Isom(x, x′)( f : Y → X) = IsomF(Y)( f ∗x, f ∗x′).

For a composition Zg−→ Y

f−→ X, the pullback ( f g)∗x is a pullback along g of f ∗x and ( f g)∗x′ is apullback along g of f ∗x′. Therefore there is a canonical map

g∗ : Isom(x, x′)( f : Y → X)→ Isom(x, x′)( f : Z → X)

compatible with composition2.

Example 1.20. In the context of Vr of all rank r vector bundles, let X be a scheme, and E, E′ betwo vector bundles over X. Then isomorphisms are a sheaf if, for any open covering Xi → Xof X, and for every collection of isomorphisms αi : E|Xi → E′|Xi in the fiber over Xi such thatαi|Xij = αj|Xij , there is a unique isomorphism α : E→ E′ such that αXi = αi.

Now we can say what a stack is.

Definition 1.1. A category over a base category C fibered in groupoids p : F → C is a stack if thefollowing two conditions hold:

1. For any X ∈ C and objects x, y ∈ F(X), the presheaf Isom(x, y) on C/X is a sheaf; and

2. For any covering Xi → X of an object X ∈ C, any descent data with respect to thiscovering is effective.

To relate to some other definitions found in some references, we have an equivalent way to definestacks.

1Here we assume C/X to have what is called the comma topology, i.e. Ui → X is a covering for U → X iff Ui isa covering for U in C.

2The presheaf Isom is independent of the choice of pullbacks up to canonical isomorphisms.

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Proposition 1.21. A category fibered in groupoids p : F → C is a stack if for every object X ∈ C andcovering Xi → Xi∈I , the functor

ε : F(X)→ F(Xi → Xi∈I)

is an equivalence of categories.

Proof. Condition 2 is equivalent to saying the functor ε : F(X)→ F(Xi → X) defined in section1.4.1 is essentially surjective. It remains to prove condition 1 is equivalent to saying this functor ε

is fully faithful.

We first assume condition 1 holds. Take an object X of C, a covering Xi → X, andx, y ∈ F(X). The map ε gives two descent data (Ei, αij) and (E′i, βij) associated withx and y. Now a morphism between these two descent data is a collection of arrow φi : Ei → E′isuch that φi, φj agree on pullbacks of E, E′ to Uij. If Isom is a sheaf, then this ensures that anysuch morphism comes from a unique map x → y in F(X), i.e. ε is fully faithful. The converse issimilar.

Definition 1.22. A category fibered in groupoids F → C is a prestack if F(X)→ F(Xi → X) isfully faithful, i.e. satisfying condition 1 of stack definition.

Using Proposition 1.21 above, we can get a slick proof that stacks have fiber products.

Lemma 1.23. Let

F1

F2 F3

c

d

be a diagram of stacks fibered in groupoids over C. Then the fiber product (as categories fibered ingroupoids) F1 ×F3 F2 is also a stack fibered in groupoids.

Proof. For any covering Xi → X, the maps

(F1 ×F3 F2)(X)→ F1(X)×F3(X) F2(X)

and(F1 ×F3 F2)(Xi → X)→ F1(Xi → X)×F3(Xi→X) F2(Xi → X)

are equivalences of groupoids.

Just as one can associate a sheaf to any presheaf, we could associate a stack to any CFG:

Theorem 1.24. Let p : F → C be a category fibered in groupoids. Then there exists a stack Fa over Cand a morphism of fibered categories q : F → Fa such that for any stack G over C, the induced functor

HOMC(Fa, G)→ HOMC(F, G)

is an equivalence of categories.

A proof can be found in [Aut].

13

1.3.3 Representability

We collect here some notions of being representable in different settings that we introduced sofar.

Recall we defined:

Definition 1.5. A presheaf F ∈ PSh(C) is representable if it is isomorphic to hY := Mor(−, Y) forsome Y ∈ C.

Definition 1.25. A morphism F → G in PSh(C) is representable if the base change h against amap hX → G is representable:

h F

hX G.

Next we discuss when a presheaf is locally representable. For this, the best example in mindwould be to take our base category as open balls in Cn, and the locally representable sheaves onC should be the category of complex manifolds.

Definition 1.26. A subfunctor U ⊂ X is a representable map of functors which is pointwise amonomorphism in C. It is open if, on representable objects, it base changes to an open embedding.

Definition 1.27. A presheaf is locally representable if it admits a cover by open subfunctors whichare representable.

Likewise, if we apply it op the category of affine schemes, we would recover schemes.

Consider a map M′ → M where M′ is an object in C (or hM′ ) and M is a presheaf on C. In thecommon case where the open embedding of C are monomorphisms, M′ is an open subfunctor ofM iff for any X ∈ C, s ∈ M(X), there is an open subset U ⊂ X and a map b : U → M such thats|U = b∗, and for any f : Y → X, the image of f is contained in U iff f ∗(s) = g∗(t) for some mapg : Y → M′. So:

Proposition 1.28. In the above setting, M is locally representable if there is some set Mi consistingof objects of C and elements t ∈ M(Mi) such that for any X and s ∈ M(X), there is an open coverX =

⋃i Ui such that s|Ui is uniquely the pullback of t by some map bi : Ui → Mi.

We also defined when a CFG is representable.

Definition 1.15. A CFG is representable if it is isomorphic to some SF, where S− is the functorthat embeds presheaves as categories fibered in sets in CFG.

Having constructed fiber products for CFG, we can now say when a map of CFG is repre-sentable.

Definition 1.29. Let F : M′ → M be a morphism of CFG. We say F is representable if for everyobject X of C, and any morphism from (the representable CFG associated to) X to M, the fiberproduct M′ ×M X is representable.

Just as we can pass these notions from presheaves to sheaves, we could do the same thingsfrom CFG to stacks. In particular, we would like our “geometric object” to be locally representablestacks, admitting an “open” cover.

14

Chapter 2

Algebraic spaces and algebraicstacks

A careful study of the first chapter seems to suggest that stacks are nothing more than a categoricalconstruction, bearing very little geometry for us to play with. Indeed this is the case. To talkabout various geometric properties, we must impose some other conditions on stacks. Somespecial types of stacks with rich geometry are algebraic stacks, Deligne-Mumford stacks, andalgebraic spaces.

This chapter aims to give a nice intuition and precise definitions of the conditions imposedon stacks to become algebraic and/or Deligne-Mumford. In particular, we would eventually seethe hierarchy of some abstract notions involved in the world of stacks. Figure below is takenfrom [Vak]:

category fibered in groupoids

prestack presheaf

stack sheaf separated presheaf

algebraic (Artin) stack locally representable sheaf

DM stack “geometric space”

algebraic space

scheme

Just like the previous chapter, this should be our map and guide for this chapter.

15

2.1 More background materials

In this section, we list a few definitions that in our later discussion we would take for granted.

2.1.1 Different topologies

In this subsection, we give a few more topologies on the category of schemes that we wouldwork with. These include fppf, etale, and smooth sites.

Definition 2.1. A morphism of schemes f : X → Y is flat if for every x ∈ X the map

OY, f (x) → OX,x

is flat.

Etale maps are like local homeomorphisms in topology. The following definitions make senseof this.

Definition 2.2. If A → B is a ring homomorphism, then we way that B is of finite presentationover A (or B is a finitely presented A-algebra) if there exists a surjection

π : A[X1, . . . , Xs]→ B

with ker(π) a finitely generated ideal in A[X1, . . . , Xs].

Definition 2.3. A morphism of schemes f : X → Y is called locally of finite presentation if for everyaffine open Spec(A) ⊂ Y and affine open Spec(B) ⊂ f−1(Spec(A)), the A-algebra B is of finitepresentation over A.

Definition 2.4. A morphism of schemes f : X → Y is called formally smooth (resp. formallyunramified, formally etale) if for every affine Y scheme Y′ → Y and every closed immersionY′0 → Y defined by a nilpotent ideal, the map

HomY(Y′, X)→ HomY(Y′0, X)

is surjective (resp. injective, bijective). If f is also of finite presentation, then f is called smooth(resp. unramified, etale).

Using these descriptions for morphisms of schemes, we can build a few more sites:

Example 2.5. (fppf/smooth/big etale/big Zariski site). Let X be a scheme and let (Sch/X) bethe category of X-schemes. We put an fppf (resp. smooth, big etale, big Zariski) topology onit by declaring that Cov(U) is the set of collections Ui → Ui∈I of X-morphisms for whicheach morphism Ui → U is flat and locally of finite presentation (resp. smooth, etale, an oepnembedding), and the map ⊔

i∈IUi → U

is surjective.

16

2.1.2 Properties of sheaves and morphisms

In this subsection, we develop the language we need to speak about many geometric propertiesof schemes and morphisms in terms of the corresponding sheaves on sites. We will take our basecategory C to be a subcanonical, i.e. representable presheaves are sheaves.

Definition 2.6. A class of objects S ⊂ C is stable if for every covering Ui → U, the object U isin S if and only if each Ui is in S. We call a property P of objects of C stable if the class of objectssatisfying P is stable.

Definition 2.7. Let C be a site.

(i) A subcategory D ⊂ C is closed if

(a) D contains all isomorphisms, and

(b) for all cartesian diagrams in C

X′ X

Y′ Y

f ′ f

for which the morphism f is in D, we have f ′ ∈ D.

(ii) A closed subcategory D ⊂ C is stable if for all f : X → Y in C and all coverings Yi → Y,the morphism f is in D if and only if all maps fi : X×Y Yi → Yi are in D.

(iii) A stable subcategory D ⊂ C is local on domain if for all f : X → Y in C and all coveringsXi

xi−→ X, the morphism f is in D if and only the composites f xi are in D.

(iv) If P is a property of morphisms in C satisfied by isomorphisms and closed under composi-tion, let DP be the subcategory of C with the same objects as C, but whose morphisms aremorphisms satisfying P. Then P is stable (resp. local on domain) if the subcategory DP ⊂ Cis stable (resp. local on domain).

In subsection 1.3.3 Representability, we introduced the notion when a morphism of sheaves/stacksis representable. What we said is that the base change against a representable object is anotherrepresentable sheaf/stack.

Here we generalize the idea to say when a morphism is representable by some other categories.The most important one is when a morphism of sheaves on Sch/S is representable by schemes.

Definition 2.8. Let f : F → G be a morphism of sheaves on Sch/S with etale topology.

1. We say f is representable by schemes if for every S-scheme T and morphism T → G (assheaves), the fiber product F×G T is a scheme.

2. Let P be a stable property of morphisms of schemes. If f is representable by schemes, wesay f has property P if for every S-scheme T, the morphism of schemes pr2 : F×G T → Thas property P.

17

2.2 Algebraic spaces

To motivate our next topic, we ask the question: what conditions we should put on stacks tomake them have some geometry?

First we saw the notion of being locally representable in section 1.3.3. It basically says a stack,if it is locally representable, admits a cover by schemes. This is one condition we would like toimpose on stacks.

Another one is to consider the case when we take our base category to be open balls in Rn

with smooth maps, and then locally representable sheaves on this category would be smoothmanifolds without the Hausdorff condition. It turns out that a manifold is Hausdorff if and onlyif the diagonal map M→ M×M is a closed embedding.

By the universal property of fiber products, the diagonal map is always defined for stacks,and we can ask for various properties of it, and ask how it would affect the “geometry” of ourobjects.

2.2.1 Definition of algebraic spaces

Following the motivation above, we first consider various notions of being locally representable.

We quote a result without proof here.

Proposition 2.9. Affine schemes are sheaves with the fppf topology, and in particular with the etaletopology.

We saw that this statement is true for Zariski topology. This proposition follows from theresult on faithfully flat descent for quasi-coherent sheaves. See [Ols16] for a proof.

As before, we have the following picture:

Affine schemes → Shfppf(Affine schemes) → PShfppf(Affine schemes).

Between affine schemes and fppf sheaves, there are various notions of being locally representable:

• A scheme is an fppf sheaf (on the category of affine schemes) which is locally representablein the Zariski topology.

• An algebraic space is an fppf sheaf (on the category of affine schemes) which is locallyrepresentable in the etale topology.

Still we would like to work with schemes instead of affine schemes. This is done via the following:

C LocRep(C) Sh(C) PSh(C)

C′ LocRep(C′) Sh(C′) PSh(C′)∼

where C is the category of affine schemes, and C′ is locally representable sheaves on affineschemes, or equivalently schemes. We could then identify Sh(C) ∼= Sh(C′) and LocRep(C) ∼=LocRep(C′) (not true for presheaves, as they don’t care about topology at all). So the category ofalgebraic spaces is the category of etale-locally representable sheaves on the category of schemes.

18

It is probably surprising that this condition also guarantees it has a representable diagonal(by schemes), which uses Zariski’s main theorem to prove and is in [Ols16].

Put these words differently, we have the following definition:

Definition 2.10. Let S be a scheme. An algebraic space over S is a functor X : (Sch/S)op → Setsuch that the following hold:

1. X is a sheaf with respect to the big etale topology.

2. ∆ : X → X×S X is representable by schemes.

3. There exists an S-scheme U → S and a surjective etale morphism U → X.

Morphisms of algebraic spaces over S are morphisms of functors. Together they form a newcategory.

For condition 3 to make sense, we need the following lemma:

Lemma 2.11. Let F be a sheaf on Sch/S with etale topology, and suppose that ∆ : F → F × F isrepresentable by schemes. Then any morphism f : T → F for an S-scheme T is representable by schemes.

Proof. For schemes T, T′, the fiber products of these two diagrams are the same:

T T ×S T′

T′ F F F× F.

f f×gg ∆

Since ∆ is representable by schemes, any f is also representable by schemes.

Now being etale and surjective are stable properties, and this means that condition 3 ofalgebraic spaces translates to this: for any S-scheme T and morphism T → X, the morphismU ×X T → T is etale and surjective.

So, in the way schemes are covered by affine schemes in the Zariski topology, algebraic spacesare covered by schemes in the etale topology, precisely what we meant to be locally representable.

Example 2.12. In particular, schemes (via Yoneda embedding functors) are algebraic spaces.

While not obvious, algebraic spaces occur in algebraic geometry quite natually as sheafquotients. We discuss this in the next subsection.

2.2.2 Sheaf quotient

Roughly speaking, by defining an equivalence relation on a scheme X over S, we can form asheaf X/R which will be an algebraic space. In fact, every algebraic space can be described inthis way.

We begin by defining such an equivalence relation and the quotient X/R.

Definition 2.13. An etale equivalence relation on an S-scheme X is a monomorphism of schemes

R → X×S X

such that the following hold:

19

1. For every S-scheme T, the subset of T-valued points

R(T) ⊂ X(T)× X(T)

is an equivalence relation on X(T).

2. The two mapss, t : R→ X

induced by the two projections from X×S X are etale.

Definition 2.14. In the context above, we write X/R for the sheafification of the presheaf

(Sch/S)op → Set, T 7→ X(T)/R(T)

with respect to the etale topology on (Sch/S)op.

The main task is to prove the following proposition.

Proposition 2.15. Every algebraic stack arises as a quotient of a scheme by an etale equivalence relation.In other words,

1. X/R is an algebraic space.

2. If Y is an algebraic space over S, and X → Y is an etale surjective morphism with X a scheme, then

R := X×Y X

is a scheme and the inclusionR → X×S X

is an etale equivalence relation. Moreover, the natural map

X/R→ Y

is an isomorphism.

The proof to this statement takes up the rest of this section. In particular, we will need somedescent theory we have not established near the end of the proof, and results on that can befound in [Vis08, Ols16].

To prove 1, it is enough to show the diagonal of Y := X/R

∆ : Y → Y×S Y

is representable. Once this is shown, by lemma 2.11, which says any morphism from a testscheme T0 into Y would be representable by schemes, the projection X → Y is representable byschemes. This is the etale surjection we need. Too see this, note there are cartesian squares in thediagram below where s is etale surjective, and the map T ×Y X → T will be etale surjective:

T ×Y X R X

T X Y.

t

s

20

Thus by remark after lemma 2.11, X → Y is the map we want.

Let j : U → X be an open subscheme, and let RU denote the fiber product of R and U ×S Uover X×S X. Then RU is an etale equivalence relation on U, and we therefore have an inducedmap

j : U/RU → Y.

A fact that we will use is the following, and we will assume this is true.

Lemma 2.16. The morphism j is representable by open embeddings.

Now to check ∆Y is representable, let f : W → Y×S Y be a morphism, and F := Y×Y×SY W.We need to check F is a scheme. Notice we can work Zariski locally on S and W, so we assumethey are both affine schemes. In this case we prove F is a scheme as follows.

Since X → Y is a surjective morphism of sheaves, there exists an etale covering W ′ → Wsuch that the composite map

W ′ →W → Y×S Y

factors through X×S X, and we may even assume W ′ is affine since W is quasi-compact. Nowwe have a cartesian square

R X×S X

Y Y×S Y.∆

So if F′ := F×W W ′, thenF′ ∼= Y×Y×SY W ′ ∼= R×X×SX W ′.

In particular, F′ is a scheme and F′ → W ′ is a monomorphism, which implies that F′ is aseparated S-scheme since W ′ is a separated S-scheme. Since the square

F′ W ′

F W

is cartesian, the morphism F′ → F is representable and etale surjective. Therefore F is the sheafquotient of F′ by the equivalence relation

R′ := F′ ×F F′ ⊂ F′ ×S F′.

Since F → W is a monomorphism, the two projections pi : W ′ ×W W ′ → W ′ satisfy that thesquares

F′ ×F F′ W ′ ×W W ′

F′ W ′.

p′i pi

are cartesian. So R′ is a scheme and that the two projections p′i are quasi-compact and etale.

21

Lemma 2.17. Let U′ ⊂ F′ be a quasi-compact open subset, and let FU′ denote the quotient U′/R′U′ . Letj : FU′ → F be the natural representable open embedding, and let F′U′ ⊂ F′ denote the fiber product of thediagram

F′

FU′ F.j

Then F′U′ is a quasi-compact open subset of F′ containing U′.

Proof. Indeed, F′U′ = p1(p−12 (U′)) and pi are quasi-compact and U′ was quasi-compact.

Since we can also write F′U′ = FU′ ×W W ′, it suffices to check FU′ is a scheme for everyquasi-compact open subset U′ of F′. This follows from the fact that F′U′ → W ′ is quasi-affine byZariski’s main theorem, and descent theory on quasi-affine morphisms. So F is indeed a scheme,and this concludes the proof to statement 1.

For statement 2, note there is a commutative square

R X×S X

Y Y×S Y,∆

and R is a scheme. So the rest are immediate.

2.2.3 Geometry on algebraic spaces

Since we defined an algebraic space by saying it is covered by schemes, and we already havevarious geometry to talk about on schemes, we can transfer them to algebraic spaces.

Definition 2.18. Let P be a property of schemes which is stable in the etale topology. Then analgebraic space X has property P if there exists an etale surjection U → X from a scheme U withproperty P.

For example, we can now say an algebraic space is locally noetherian, reduced, regular,normal, etc.

Similarly we can do the same thing for morphisms of algebraic spaces. Let f : X → Ybe a morphism of algebraic spaces that is representable (by schemes) and P is a property ofmorphisms of schemes which is stable in the etale topology. In the sense of definition 2.8, we sayf has property P if there is an etale cover V → Y with V ×Y X → V has property P. Examplesof such properties are proper, dominant, quasi-compact, (open/closed) embedding, etc. A fewother examples are being flat, etale, smooth, surjective, locally of finite type, etc. These propertiesare, in addition to being stable, local on domain.

Definition 2.19. Let P be a property of schemes which is stable and local on domain in the etaletopology, and let f : X → Y be a morphism of algebraic spaces. Then f has property P if thereexist etale covers v : V → Y and u : U → X such that the projection

U ×Y V → V

22

has property P.

One crucial property of algebraic spaces is that fiber products exist. A proof can be found in[Ols16].

Proposition 2.20. For any diagram of algebraic spaces:

X2

X1 X3,

the fiber product in the category of sheaves is an algebraic space.

As a consequence, any morphism of algebraic space f : X → Y has a diagonal map ∆X/Y : X →X×Y X, which can be used to describe some more properties.

Definition 2.21. A morphism f : X → Y of algebraic spaces is quasi-separated (resp. locallyseparated, separated) if the diagonal map

∆X/Y : X → X×Y X

is quasi-compact (resp. an embedding, a closed embedding). An algebraic space X has theseproperties if the morphism X → S satisfies the conditions above.

2.3 Algebraic stacks

In this section, we discuss some special stacks, namely algebraic stacks and Deligne-Mumfordstacks.

Throughout this section, we work over the category of S-schemes with the etale topology.And by stack, we mean a stack over S-schemes with etale-topology. A stack morphism is calledrepresentable if it is representable by algebraic spaces.

2.3.1 Definition of algebraic stacks

Similar to how we motivated the definition of algebraic spaces in subsection 2.2.1, we give twodefinitions of algebraic stacks. The first one helps us think about algebraic stacks, and we needthe second one to do any concrete computations.

Recall from 2.2.1 that an algebraic space is an etale-locally representable sheaf on the categoryof schemes with the etale topology, an algebraic stack is a smooth-locally representable stack onthe category of schemes with the smooth topology.

Similar to the way we described algebraic spaces, the characterization above translate to:

Definition 2.22. A stack X/S is an algebraic stack (aka. Artin stack) if the following hold:

1. The diagonal∆ : X → X×S X

is representable by algebraic spaces.

23

2. There exists a smooth surjective morphism π : Y → X with Y a scheme.

A morphism of algebraic stacks is a morphism of stacks.

Similar to Lemma 2.11, condition 1 here implies that every morphism t : T → X with T ascheme is again representable, and therefore condition 2 makes sense.

Also, condition 2 here can be replaced by: there exists a smooth surjective morphism f : Y →X with Y an algebraic space. This is because any morphism from an algebraic space to analgebraic stack is representable (by schemes).

Proposition 2.23. Let X/S be an algebraic stack. Then for any diagram

Y

Z X

y

z

with Y, Z algebraic spaces, the fiber product Y×X Z is an algebraic space. In particular, any morphismf : Y → X from an algebraic space Y is representable.

To prove this, we use the fact that the fiber product in the above setting, which is isomorphicto the sheaf Isom(pr∗1y, pr∗2z) over Y×S Z, is an algebraic space. This follows from the obviouslemma below.

Lemma 2.24. Let X/S be a stack. The diagonal ∆ : X → X×S X is representable if and only if for everyS-scheme U and two objects u1, u2 ∈ X(U), the sheaf Isom(u1, u2) is an algebraic space.

Proof. We have a cartesian square here

Isom(u1, u2) U

X X×S X.

u1×u2

And the assertion follows from our construction of Isom.

Since we want to do geometry on algebraic stacks, we would expect fiber products to exist inthis category. Again we won’t prove it here; a proof can be found in [Ols16].

Proposition 2.25. For a diagram of algebraic S-stacks

X

Y Z,

c

d

let W be the stack fiber product. Then W is algebraic.

2.3.2 Geometry on algebraic stacks

Recall we defined that a property P of S-schemes is called stable in the smooth topology if theset of every S-schemes satisfying this property is stable.

24

Definition 2.26. An algebraic stack X/S has property P if there exists a smooth surjective mor-phism π : Y → X with Y a scheme having property P.

Remark. Similarly we could say the same thing for algebraic spaces that there exists an etalesurjective morphism from a scheme with property P.

The reason we need this terminology is that we can transfer some geometric properties ofschemes to algebraic stacks. For example, locally noetherian, regular, and locally of finite typeover S.

We could do similar things for a property P of morphisms of schemes.

Definition 2.27. Let f : X → Y be a morphism of algebraic stacks. A chart for f is the commuta-tive diagram:

G1 X′ G2

X Y

g

q

h

f

p p

f

where G1, G2 are algebraic spaces, the square is cartesian, and g and p are smooth and surjective.If G1, G2 are schemes, we call this a chart for f by schemes.

Definition 2.28. Let P be a property of morphisms of schemes which is stable and local ondomain in smooth topology. Then a morphism f : X → Y of algebraic stacks has property P ifthere exists a chart for f by schemes such that the morphism h (defined above) has property P.

For example, P could be being smooth, locally of finite presentation, surjective.

But a map of algebraic stacks can also be represented by algebraic spaces, whose geometricproperties are discussed in section 2.2.3. So we could use those languages to describe some moreproperties of algebraic stacks.

Definition 2.29. Let P be a morphism of algebraic spaces which is stable in smooth topology onthe category of algebraic spaces. We say a representable morphism of algebraic stacks f : X → Yhas property P if for every morphism Y′ → Y from an algebraic space, the morphism of algebraicspaces

X×Y Y′ → Y

has property P.

For example, we can say a representable morphism is etale, separated, proper, an embedding,etc.

Just as we did in section 2.2.3 where we used the diagonal of a map of algebraic spaces todefine a few more properties, we can do the same thing here, assume that fiber products exist inthe category of algebraic stacks (Proposition 2.23) and that the diagonal map is representable.

Definition 2.30. Let f : X → Y be a morphism of algebraic stacks over S, and ∆X/Y : X → X×Y Xbe the diagonal map. We say f is quasi-separated (resp. separated) if the diagonal ∆X/Y is quasi-compact and quasi-separated (resp. proper). We say X has such properties if the structuremorphism X → S satisfies the conditions above.

25

2.3.3 Deligne-Mumford stacks

In this last section, we give the definition of a special type of algebraic stacks, called Deligne-Mumford stack.

Definition 2.31. A Deligne-Mumford (DM) stack is an etale-locally representable algebraic stack.Or equivalently, an algebraic stack X/S is DM if there exists an etale surjection Y → X with Y ascheme.

We mention without proof here a few useful criteria for algebraic stacks to be DM or algebraicspaces, formalizing the last missing part of the diagram at the beginning of this chapter.

Theorem 2.32. An algebraic stack is Deligne-Mumford if the diagonal ∆ is formally unramified, meaningthat Ω∆ = 0. Informally, this means the stack has no “infinitesimal automorphisms”.

Corollary 2.33. For an algebraic stack X/S, every S-scheme U, and object x ∈ X(U), if the automor-phism group of x is trivial, then X is an algebraic space.

In particular, we will use the results here to prove that the examples in our next chapter areDeligne-Mumford (under certain conditions).

26

Chapter 3

Various examples of algebraic stacks

In our third and last chapter, we give two very important classes of algebraic stacks, namelyquotient stacks and the stack parametrizing stable curves of genus g. Throughout the discussion,we will also introduce some more advanced results and techniques.

3.1 Quotient stacks [X/G]

In the following discussion, we present the formal construction of the quotient stack [X/G] of agroup acting on a scheme.

Our goal is to study the quotient of a scheme X by a group action G. We write

ρ : G× X → X

for the action. It is well known that in the category of schemes, quotients might not exist.However, if the action is free, we would always get an algebraic space.

The action of G is called free if the map

j : (R := G× X)→ X× X, (g, x) 7→ (x, ρ(g, x))

is a monomorphism. Here by R = G × X, we mean the scheme R =⊔

g∈G X. Now j is amonomorphism means we have an equivalence relation in the sense of Definition 2.13, since R isa disjoint union of X and therefore R→ X is etale. As a result of Proposition 2.15, we could takesuch quotient to get an algebraic space, denoted by X/G.

If the action of G on X is no longer free, we would only be able to define the quotient as astack. It is, in fact, algebraic. The main reference for this section will be [Aut].

3.1.1 Groupoids in algebraic spaces

We work over a base scheme S, and an algebraic space B over S.

27

Definition 3.1. A groupoid in algebraic spaces over B is a collection of data,

(U, R, s, t, c)

where U and R are algebraic spaces over B, s, t : R→ U and c : R×s,U,t R→ R are morphisms ofalgebraic spaces over B, such that for any scheme T over B the quintuple

(U(T), R(T), s, t, c)

is a groupoid category (U(T) is the set of objects, R(T) arrows, s known as the source, t thetarget, and c the composition). This gives a functor

(Sch/S)opfppf → Groupoids

S′ 7→ (U(S′), R(S′), s, t, c).

Similar to Proposition 1.14 where we discussed we can view presheaves of sets as categoriesfibered in sets, here a presheaf of groupoid can be seen as a category fibered in groupoid. ByTheorem 1.24, we can stackify this.

Definition 3.2. Let B → S be an algebraic space, (U, R, s, t, c) a groupoid in algebraic spacesover B as above. The quotient stack

p : [U/R]→ (Sch/S)fppf

is the stackification of the category fibered in groupoids mentioned above.

The stack [X/G] we are going to define is a special case of the definition above. But for that,we need one more notion, a group algebraic space.

Definition 3.3. Let B→ S as above. A group algebraic space over B is a pair (G, m) where G is analgebraic space over B and m : G×B G → G is a morphism of algebraic spaces over B, such thatfor every scheme T over B, the pair (G(T), m) is a group.

For any group algebraic space (G, m) over B, X an algebraic space over B, and an actiona : G×B X → X of G on X, we get a groupoid in algebraic spaces in the following manner:

1. We set U = X and R = G×B X.

2. We set s : R→ U equal to (g, x) 7→ x.

3. We set t : R→ U equal to (g, x) 7→ a(g, x).

4. We set c : R×s,U,t R→ R equal to ((g, x), (g′, x′)) 7→ (m(g, g′), x′).

Now we can define [X/G].

Definition 3.4. Let (G, m) be a group algebraic space over B. Let a : G×B X → X be an action ofG on an algebraic space X over B. The quotient stack

p : [X/G]→ (Sch/S)fppf

is the stackification of the category fibered in groupoid (X, G×B X, s, t, c).

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3.1.2 Torsor and principal bundles

This section explains what the stack [X/G] looks like, and provide some practical tools to docomputations with them.

Definition 3.5. Let G be a sheaf of groups on a category C with a topology. A G-torsor on C is asheaf of sets F on C with a left action ρ : G× F → F, such that the following hold:

1. For every U ∈ C, there exists a covering Ui → U such that F(Ui) 6= ∅ for all i.

2. The mapG× F → F× F, (g, f ) 7→ ( f , g f )

is an isomorphism.

Note that the second condition is equivalent to saying that if F(U) 6= ∅, then the action ofG(U) on F(U) is simply transitive. We say that a torsor (F, ρ) is trivial if F has a global section.In this case if we fix a global section f , then we have an isomorphism

G → F, g 7→ g f ,

which identifies F with G and the action ρ with left-translation on F.

A morphism of G-torsors (F, ρ) → (F′, ρ′) is a sheaf morphism h : F → F′ such that thediagram commutes:

G× F G× F′

F F′.

idG×h

ρ ρ′

h

The idea of torsor is closely related to another idea called principal bundles. Here we fix a basescheme X and work over Sch/X with fppf topology. We also assume G above is representable bya flat locally finitely presented X-group scheme G0.

Definition 3.6. A principal G0-bundle over X is a pair (π : P → X, ρ) where π is flat, locallyfinitely presented, surjective morphism of schemes, and

ρ : G0 ×X P→ P

is a morphism such that:

1. The diagram commutes:

G0 ×X G0 ×X P G0 ×X P

G0 ×X P P,

idG0×ρ

m×idP ρ

ρ

where m is the group law on G.

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2. If e : X → G0 is the identity section, then the composition

P(eπ,idP)−−−−→ G0 ×X P

ρ−→ P

is the identity map on P.

3. The map(ρ, pr2) : G0 ×X P→ P×X P

is an isomorphism.

A morphism of principal G0-bundles (P, ρ)→ (P′, ρ′) is a morphism of X-schemes f : P→ P′

such that the diagram commutes:

G0 ×X P G0 ×X P

P P′.

idG0× f

ρ ρ′

f

For a principal G0-bundle (P, ρ) over X, we get a G-torsor (F, ρ) by letting F be the sheaf onSch/X represented by P, with action induced by the action ρ. This in fact defines a fully faithfulfunctor

(principal G0-bundles on X)→ (G-torsors on X).

Now back to the stack [X/G]. Set-up is the same as in Definition 3.3. Temporarily, we denote[[X/G]] a new category as follows:

1. An object of [[X/G]] consists of a quadruple (U, b, P, φ : P→ X) where

(a) U is a scheme in (Sch/S)fppf.

(b) b : U → B is a morphism over S.

(c) P is an fppf GU-torsor over U, where GU = U ×b,G G.

(d) φ : P→ X is a G-equivariant morphism fitting into the commutative diagram

P X

U B.

φ

b

2. A morphism is a pair ( f , g) : (U, b, P, φ) → (U′, b′, P′, φ′) where f : U → U′ is a mor-phism of schemes, and g : P→ P′ is a G-equivariant morphism over f which induces anisomorphism P ∼= U × f ,U′ P′, and has the property that φ = φ′ g.

Thus [[X/G]] is a category and

p : [[X/G]]→ (Sch/S)fppf, (U, b, P, φ) 7→ U

is a functor. Note that the fiber category over U is the disjoint union over b ∈ MorS(U, B) of fppfGU-torsors P with a G-equivariant morphism to X. So the fiber categories are groupoids.

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But it is actually a stack. The proof uses some result from descent theory, which we did notcover in this essay. See, for example, [Ols16, Aut] for more details1.

Lemma 3.7. The functor above defines an algebraic stack in groupoids over (Sch/S)fppf. And there existsa canonical equivalence

[X/G]→ [[X/G]]

of stacks in groupoids over (Sch/S)fppf.

Finally, we want to see when this quotient stack is DM. Again we only state the result here.

Lemma 3.8. The stack [X/G] is Deligne-Mumford if and only if for every point s : Spec(k)→ S wherek is algebraically closed, and t ∈ [X/G](k), the stabilizer group scheme Gt ⊂ Gs is etale over s, where Gs

is the pullback of G along s.

3.1.3 Brief touch on Keel-Mori

We might wonder for what conditions X/G would exist as an algebraic space. For this, weintroduce the notion of a coarse moduli space, which is, in some sense, the closest approximationto an algebraic stack by an algebraic space:

Definition 3.9. If M is an algebraic stack, a coarse moduli space of M is a morphism M→ X withX an algebraic space, such that

1. Any morphism from M to an algebraic space factors through X.

2. The map M → X induces an isomorphism on geometric points (a geometric point of analgebraic space X is a morphism from Spec k to X for k algebraically closed).

We do have an amazing theorem on the criterion for coarse moduli spaces to exist:

Theorem 3.10. (Keel-Mori). If M is an algebraic stack of local finite presentation over a locally noetherianbase scheme S with finite diagonal, then there exists a coarse moduli space M→ X.

Furthermore, if M is separated or proper, so is X. This construction is also preserved by flat basechange, meaning if Y → X is a flat morphism of algebraic spaces, then M×X Y → Y is a coarse modulispace.

3.2 Mg andMg

In this section, we aim to give a short introduction to the construction and various propertiesof the stackMg and its compactificationMg. Certain proofs to, for example, the stack is well-defined and various properties, will involve topics outside the scope of this essay, and thereforewe would only mention the general ideas. The original idea comes from [DM69]. Detailed proofscan be found in [Ols16].

Definition 3.11. Let g be a non-negative integer at least 2. LetMg be the category whose objectsare pairs (S, f : C → S) where S is a scheme, and f : C → S is a proper smooth morphism such

1Olsson defined the functor in his book in Definition 8.1.12 slightly different than the one here, but they areequivalent.

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that for every point s ∈ S, the fiber Cs is a geometrically connected proper smooth curve of genusg. A morphism

(S′, C′ → S′)→ (S, C → S)

inMg is a cartesian square

C′ C

S′ S.

The functor sending (S, f : C → S) to S makes Mg a fibered category over the category ofschemes.

Lemma 3.12. Mg is a stack for the etale topology.

The proof uses descent theory for polarized schemes. We define the category Pol to consistof pairs ( f : X → Y, L) where f is a flat morphism of schemes and L a relatively ample invertiblesheaf on X. Descent theory for polarized schemes implies that any fppf covering S′ → S is aneffective descent morphism for Pol, see [Ols16] section 4.4.10 for a proof. But for such a covering,we could also construct a commutative diagram:

Mg(S) Pol(S)

Mg(S′ → S) Pol(S′ → S).

where C → S is sent to (C → S, Ω1C/S). So the functorMg(S) →Mg(S′ → S) is fully faithful,

andMg is a stack.

But we could say much more:

Theorem 3.13. Mg is a Deligne-Mumford stack.

The way we show this is to first show it is algebraic of the form [Mg/G], and then usingLemma 3.8 to show it is in fact Deligne-Mumford.

We begin with a lemma.

Lemma 3.14. Let (S, f : C → S) ∈ Mg, and let LC/S be the invertible sheaf (Ω1C/S)

⊗3. Then

1. The sheaf f∗LC/S is a locally free sheaf of rank 5g− 5 on S.

2. The map f ∗∗ LC/S → LC/S is surjective, and the resulting S-map

C → P( f∗LC/S)

is a closed embedding.

3. For any morphism g : S′ → S, the natural map

g∗ f∗LC/S → f ′∗LC′/S′

is an isomorphism, where f ′ : C′ → S′ is the base change of f to S′.

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For 1 and 3, it is in fact enough to check when S = Spec(k) for an algebraically closedfield and C/k a smooth proper curve of genus g. Then it is an application of Serre duality andRiemann-Roch for curves in this case. For example, since the degree of the canonical divisor onC is 2g− 2, we have

h0(C, LC/S) = 3 deg(Ω1C/S) + 1− g = 3(2g− 2) + 1− g = 5g− 5.

Assuming these results, we can define Mg as follows.

Definition 3.15. Let Mg denote the functor on the category of schemes which to any scheme Sassociates the isomorphism classes of pairs

( f : C → S, σ : O5g−5S

∼= f∗LC/S),

where (S, f : C → S) ∈ Mg(S). An isomorphism

( f ′ : C′ → S′, σ′ : O5g−5S

∼= f ′∗LC′/S)→ ( f : C → S, σ : O5g−5S

∼= f∗LC/S)

is given by an isomorphism of curves α : C′ → C such that σ = α σ′.

The amazing thing about Mg is that it is representable by a quasi-projective scheme, and wecan define an action of G := GL5g−5 on Mg. Given an S-point where S is a scheme, define

g ∗ (C/S, σ) 7→ (C/S, σ g), g ∈ G(S).

Then consider the map π : Mg → Mg, where (C/S, σ) 7→ (S, C). For any scheme S andmorphism f : S → Mg corresponding to a curve C/S, the fiber product Mg ×Mg S is the Gs-

torsor of isomorphisms σ : O5g−5S → f∗LC/S. Thus we have an isomorphismMG ∼= [Mg/G],

and in particular,Mg is algebraic.

To show it is Deligne-Mumford, it suffices to show that for any algebraically closed field kand smooth genus g curve C/k, the group scheme Autk(C) is reduced. For this, we can showthat if A′ → A is a surjective morphism of k-algebras with square-zero kernel I, then the map

Autk(C)(A′)→ Autk(C)(A)

is injective. This is true because for any α : CA → CA an automorphism, the set of liftings to CA′

are given by the set of dotted arrows in the diagram

CA CA CA′

CA′ Spec(A′).

α

By the universal property of differentials, the set of such dotted arrows form a torsor under theisomorphism

Hom(α∗Ω1CA/A, I ⊗A OCA)

∼= H0(CA, α∗TCA/A, I ⊗A OC),

which is zero since it is zero in every fiber.

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3.2.1 The compactificationMg

In our construction ofMg, we required f : C → S to be a smooth map. The idea of compactifica-tion, or adding new points toMg, is to include curves that have some special non-smooth points.In particular,Mg is called the moduli stack of stable curves of genus g. We give the constructionbelow.

First we formalize the notions of special points, and use this idea to define what stable curvesare.

Definition 3.16. Let k be an algebraically closed field and X/k be a finite type k-scheme ofdimension 1. A closed point x ∈ X(k) is called a node (or ordinary double point) if the completelocal ring OX,x is isomorphic to k[[x, y]]/(xy). The scheme X is called a nodal curve (or at-worst-nodal) if every closed point x ∈ X is either a smooth point or a nodal point.

Definition 3.17. A prestable curve over a scheme S is a proper flat morphism π : C → S such thatfor every geometric point s→ S, the fiber Cs is a connected nodal curve over k(s).

A key feature of prestable curves is that there exists a relative dualizing sheaf. We give thedefinition here.

Definition 3.18. A closed embedding i : X → Y of schemes is called a regular embedding ofcodimension d if for every point x ∈ X, there exists an affine open Spec(A) ⊂ Y of x such thatthe ideal I ⊂ A defining X ∩ Spec(A) is generated by a regular sequence in A of length d. Amorphism f : X → Y is called a local complete intersection morphism of codimension d if for everyx ∈ X, there exists a neighborhood U ⊂ X of x and a factorization

U i−→ Pg−→ Y

where i is a regular embedding of codimension e and g is smooth of relative dimension d + e forsome e. A morphism f : X → Y is a local complete intersection if there exists an integer d such thatf is a local complete intersection morphism of codimension d.

Lemma 3.19. If f : X → Y is a local complete intersection morphism of codimension d, and supposethere exists a factorization

X i−→ Pg−→ Y

of f with i a regular embedding of codimension e and g smooth of relative dimension d + e, and let ωP/Y

denote the (d + e)-th exterior power of Ω1P/Y. Then

i∗ExteOP

(i∗OX , ωP/Y)

is a locally free sheaf of rank 1 on X, which is independent of the choice of the factorization of f . Thisinvertible sheaf is denoted ωX/Y and is called the relative dualizing sheaf of f .

Now for a prestable curve C over an algebraically closed field k with normalization π : C → C,we say a point q ∈ C(k) is special if π(q) is a node. It can be shown that, for any irreduciblecomponent Ci ⊂ C with normalization Ci, the degree of ωC/k restricted to Ci is

2gi − 2 + #special points on Ci.

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Definition 3.20. Let k be an algebraically closed field. We say a prestable curve C/k is stable if

2gi − 2 + #special points on Ci > 0

for every irreducible component. If f : C → S is a prestable curve over a scheme S, we say C isstable if for every geometric point x → S the fiber Cx is a stable curve over k(x).

Definition 3.21. Fix g ≥ 2. LetMg denote the fibered category over Spec(Z) whose objects arepairs (S, f : C → S) where S is a scheme and f : C → S is a stable curve of genus g. A morphism(S′, f ′ : C′ → S′)→ (S, f : C → S) is a cartesian square

C′ C

S′ S.

f ′ f

Theorem 3.22. Mg is a Deligne-Mumford stack.

We have, for a stable curve π : C → S of arithmetic genus g, and for n ≥ 3, the relativedualizing sheaf ω⊗n

C/S is relatively very ample. Then by descent theory on polarized schemes, wewould getMg is a stack.

The proof thatMg is algebraic is very similar to showing thatMg is algebraic. In particular,let N g denote the functor that associates to a scheme S the set of isomorphism classes of pairs(π : C → S, ı : O5g−5

S → π∗ω⊗3C/S). Then N g is a scheme, and the map N g → Mg is a smooth

surjection.

The last thing to check is that the diagonal is unramified; or equivalently, if T0 → T is aclosed embedding of affine schemes defined by a square-zero ideal, and if α : C → C is anautomorphism of a stable curve over T such that the reduction α0 : C0 → C0 of α to T0 is theidentity, then α is the identity. For this it suffices to consider when T = Spec(A) of an artin localring with residue field k and I annihilated by the maximal ideal of A.

Let Ck denote the reduction of C to k. This further reduces to check that

Ext0(Ω1Ck/k,OCk ) = 0.

The general idea here is to consider π : C → Ck the normalization, and D ⊂ C the preimage ofthe nodes. Then the above expression can be identified with those sections δ ∈ H0(C, TC) of thetangent bundles of C which vanish at each point of D. For component Γ ∈ C of genus ≥ 2, this isclear since TC has negative degree. For genus 1 component, there is no nonzero global sectionvanishing at a point. For genus 0 component, there are no nonzero global sections vanishing atthree points. Together we conclude the proof thatMg is Deligne-Mumford.

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Conclusion and future perspectives

We introduced the idea of algebraic stacks pretty much according to the historical lines in theusual manner:

1. Chapter 1 aimed to define stacks. For this, we assumed some basic algebra and schemetheory facts. Then we developed the theory of sheaves on sites and some very basic descenttheory. Using these languages, we defined stacks fibered in groupoids over a site as acategory fibered in groupoids satisfying that Isom is a sheaf and every descent datum iseffective.

2. For algebraic stacks, we first introduced algebraic spaces as fppf sheaves that locally looklike schemes. Then an algebraic stack is a stack over S-schemes with fppf topology, whosediagonal is representable by algebraic spaces and which has a smooth covering by schemes.

3. Deligne-Mumdord stacks are algebraic stacks that have formally unramified diagonal, orequivalently no infinitesimal automorphisms.

It does seem like an overkill to define algebraic stacks using so much other ideas. Of course, wecould have directly define algebraic stacks and maybe skip some of the prerequisites above; butthen algebraic stacks would be standing alone, without any other geometric objects to hold onto.In particular, we won’t be able to have some of the intuitions explained in this essay. This mightalso justify the existence of the encyclopedic Stacks Project.

Below we list some topics that could have been integrated into this introductory essay:

1. A more thorough discussion of descent theory on schemes and morphisms. In fact thiscould be the motivation for stacks.

2. A more careful study of geometry on algebraic spaces: topological properties, quasi-coherent sheaves on algebraic spaces, etc.

3. Quasi-coherent sheaves on algebraic stacks, and more construction of stacks (root stacksfor example).

The reason these topics are not included is that they usually require more prerequisite than a firstcourse in algebraic geometry, and also they mostly stand alone from the facts that we alreadydiscussed. However, these materials could be easily grasped should anyone want to go furtherfrom here. Some other directions from this point are:

1. The moduli stacks Mg,n and Mg,n of genus g smooth or stable curves with n markedpoints. They should be a direct extension of the theory discussed in our last section of theessay.

2. Coarse moduli spaces. We only introduced the idea of a coarse moduli space in section3.1.3. A more careful study would lead to, for example, Chow’s lemma for DM stacks,criterion for properness, and finiteness of cohomology.

3. Deformation theory and Artin’s criterion for algebraicity of stacks.

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References

[Aut] The Stacks Project Authors. The Stacks Project. URL: https://stacks.math.columbia.edu.

[DM69] Pierre Deligne and David Mumford. The irreducibility of the space of curves of givengenus. Publications mathematiques de l’I.H.E.S., 36:75–109, 1969.

[Fan] Barbara Fantechi. Stacks for everybody. URL: http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.68.3914&rep=rep1&type=pdf.

[Ols16] Martin Olsson. Algebraic Spaces and Stacks. American Mathematical Society ColloquiumPublications. American Mathematical Society, 2016.

[Vak] Ravi Vakil. Lecture notes. URL: http://virtualmath1.stanford.edu/~vakil/

17-245.

[Vis08] Angelo Vistoli. Notes on Grothendieck topologies, fibered categories and descent theory. 2008.URL: http://homepage.sns.it/vistoli/descent.pdf.

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