Algebraic Stacks

Home , Algebraic space, Algebraic stack, Morphism of schemes, Prestack, Quotient stack, Stack (mathematics)

Algebraic Stacks

Contents

Introduction 2

1 Stack 3 1.1 Basic Grothendieck topologies ...... 3 1.1.1 Examples and definitions ...... 4 1.1.2 Representable presheaves ...... 4 1.2 Fibered categories ...... 5 1.2.1 Definition of fibered categories and categories fibered in groupoids . . . .6 1.2.2 2-Yoneda and PSh to CFG ...... 6 1.2.3 Fiber products ...... 8 1.3 Towards stacks ...... 10 1.3.1 Effective descent data ...... 11 1.3.2 Definition of stacks ...... 12 1.3.3 Representability ...... 14

2 Algebraic spaces and algebraic stacks 15 2.1 More background materials ...... 16 2.1.1 Different topologies ...... 16 2.1.2 Properties of sheaves and morphisms ...... 17 2.2 Algebraic spaces ...... 18 2.2.1 Deﬁnition of algebraic spaces ...... 18 2.2.2 Sheaf quotient ...... 19 2.2.3 Geometry on algebraic spaces ...... 22 2.3 Algebraic stacks ...... 23 2.3.1 Deﬁnition of algebraic stacks ...... 23 2.3.2 Geometry on algebraic stacks ...... 24 2.3.3 Deligne-Mumford stacks ...... 26

3 Various examples of algebraic stacks 27 3.1 Quotient stacks [X/G] ...... 27 3.1.1 Groupoids in algebraic spaces ...... 27 3.1.2 Torsor and principal bundles ...... 29 3.1.3 Brief touch on Keel-Mori ...... 31

3.2 Mg and Mg ...... 31

3.2.1 The compactiﬁcation Mg ...... 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 few things. First, my intention is not to give a comprehensive review, but only a concise introduction on the subject of stacks. In particular, I am only assuming some prior knowledge on sheaves and schemes, at the level of Part III algebraic geometry course, and some elementary category theory. Second, when learning something new, I love to keep intuitions at mind which would often provide a second perspective into how things work, and I organized this essay with this hope 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 categorical construction of stacks. In the second chapter, we look at how we can put geometry on this category and motivate the definition of algebraic stacks. Lastly in the third chapter, we look at two very important classes of algebraic stacks. The motivation for stacks, personally, comes from vector bundles. We know schemes generalize varieties. As a result, we can do geometry on objects like grassmannians, which is, in some sense, a classifying object (functor) that sends each scheme S to the set of n − k dimensional subspaces. It is a scheme in the sense that this functor is representable by a scheme. Another similar functor is vector bundles, which sends a scheme S to all rank r vector bundles on the scheme. However, this functor is not a scheme anymore. And we would want to generalize schemes to stacks to include this object. Eventually we will see that stacks are just sheaves taking values in categories. In chapter 2, we solve the problem that the construction of stacks is too categorical to do geometry. To solve this problem, we look back to some familiar cases, for example, on complex manifolds. We will have defined presheaves and sheaves on a category and there would be a picture C ⊂ Sh(C) ⊂ PSh(C). We ask the question: of all the sheaves on the category C of open sets in Cn with holomorphic maps, which are complex manifolds? The answer is the locally representable ones. Similarly on the category of affine schemes, schemes are the locally representable 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 is known as Artin stacks or algebraic stacks. In particular, it is still covered by schemes, and by saying a morphism of algebraic stacks can be represented by schemes or algebraic spaces, we could 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 are not 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 written by Deligne and Mumford, who studied closely a certain type of objects within algebraic stacks called Deligne-Mumford stacks.

2 Chapter 1

Stack

In this chapter, we will discuss the definition and construction of stacks. In particular, we will need 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 very beginning. This will serve as a natural motivation and map for what we are discussing and what is to come. Throughout this chapter, we will be dealing with one particular example many times: the functor Vr taking all rank r vector bundles on a scheme. We could see how these new definitions agree 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 the following two conditions hold:

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

2. For any covering {Xi → X} of an object X ∈ C, any descent data with respect to this covering 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 a topology on a category, and in particular, how to deﬁne a sheaf on a category.

3 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 on X:

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

Traditionally, when we talk about the topology of X, we are essentially talking about all open covers of X, denoted by J(X) = {{Ui → X}i∈I : Ui cover X}. In particular, if we require each S Ui → 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. an element from each X(Ui). If we treat X as a (contravariant) functor that sends a scheme U to all U-points X(U), an open cover will be a subfunctor. We call a subfunctor of the Yoneda functor

X a sieve. So a traditional open cover is a sieve. Specifying which Ui form a cover is roughly speaking specifying which sieves are “covering”. We will be mainly dealing with the category of S-schemes, in which ﬁber products exist. In this 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 of

C, a set Cov(X) of collections of morphisms {Xi → X}i∈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 category C, 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 X i∈I is a cover for X, and for each i, Vij Xi j∈Ji is a cover for Xi, then all composites

{ → → } Vij Xi X i∈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 treat them 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.

4 A sheaf is a presheaf that satisfies the gluing and identity properties (since we already have a 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 (where Yo(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 the presheaf 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 taking our base category to be reasonable enough (i.e. maps glue, and the category of X-schemes works ﬁne), what we said in the above example is that representable presheaves are sheaves. So we have 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 that sends 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 vector bundles 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 information on 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, which we discuss next.

1.2 Fibered categories

In this section, we will fix a base category C. There will no topology involved. A category over C is a category F equipped with a functor p : F → C. We also write F(U) the fiber category whose objects are u ∈ F with p(u) = U (object of F over U), and morphisms are f : u → u0 such that p( f ) = id. By definition, a morphism φ : u → v is a commutative diagram

φ u v

p(φ) p(u) p(v).

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

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

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

φ w λ u v

p(φ) p(w) h p(u) p(v).

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

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 every morphism f : U → V in C and v ∈ F(V), there exists a cartesian morphism φ : u → v such that p(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 → C such 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 we can 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 a pullback (exists by definition of fibered category) φ0 : u0 → v. Then by definition of pullbacks, we get a map α : u → u0, which will necessarily be an isomorphism as F(U) is a groupoid. By universal property, φ is cartesian.

Just for completion, a map between two categories over C ﬁbered in groupoids will be a functor 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) to include operations like “taking all rank r vector bundles”. Let’s call the category of all categories ﬁbered in groupoids over C as CFG(C). To see the relationship between PSh and CFG, essentially we need the 2-Yoneda lemma, which concerns 2-categories.

6 We will only be making some informal comments about 2-categories without going too deep into it. Roughly speaking, a 0-category consists of some objects without any relations. For example, 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. For example, the category of categories has 0-objects all categories, 1-objects all functors between categories (which is again a category), and furthermore some 2-objects natural transformations between 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 functors α : F → F0 such that p = p0 ◦ α; and the 2-objects are natural transformations between two functors α, β which project to identity on C. These 2-objects can also be defined on any fibered categories, and we want to give it a name.

Deﬁnition 1.11. If α, β : F → F0 are morphisms of ﬁbered categories, then a base preserving natural transformation τ : α → β is a natural transformation of functors such that for every u ∈ F, the 0 morphism τu : α(u) → β(u) in F projects to the identity morphism. We denote by

0 HOMC(F, F ) the category whose objects are morphisms of ﬁbered categories F → F0, and whose morphisms are 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 and whose morphisms are commutative triangles. This is a ﬁbered category over C with functor C/X → C sending Y → X to Y.

Lemma 1.12. The functor

ξ : 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 of categories ﬁbered in sets.

Definition 1.13. A category fibered in sets is a fibered category p : F → C such that the only morphisms 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 CFG is denoted by SF. The objects of SF are

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

7 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 C ﬁbered 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 the set (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 defined ∗ by the maps g . The natural map SF → G is an equivalence since it induces an equivalence in the fiber over any U.

By definition, since any identity morphism is necessarily invertible, any category fibered in sets 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 usual universal property does not capture enough information. We need what is called a 2-categorical fiber product, and the corresponding 2-universal property. Let’s start by considering fiber products of groupoids. Let

G1 f g G2 G 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

(x0, y0, σ0) → (x, y, σ) is a pair of isomorphisms a : x0 → x and b : y0 → y such that the diagram commutes:

0 f (x0) σ g(y0)

f (a) g(b) f (x) σ g(y).

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

p1 G1 ×G G2 G1

p2 f g G2 G.

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 (λ1) f ◦ α f ◦ p1 ◦ h

γ Σ◦h g(λ2) g ◦ β g ◦ p2 ◦ h commutes. The data

(h, λ1, λ2) is unique up to a unique isomorphism. Now let C be our base category, and let

F1 c d F2 F3 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 that γ : 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

(G F ) × (G F ) HOMC , 1 HOMC(G,F3) HOMC , 2 .

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

Lemma 1.16. Let F, F0 be two categories ﬁbered in groupoids over C. Then the category

0 HOMC(F, F ) of morphisms of CFG F → F0 is a groupoid.

9 Proof. Suppose there is a morphism ξ : f → g between two objects (morphisms of CFG f , g : F → F0). It sufﬁces to show for each u ∈ F, the map

ξu : f (u) → g(u)

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

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

(H G) → (H F ) × (H F ) HOMC , HOMC , 1 HOMC(H,F3) HOMC , 2 (†) (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 language of groupoids.

Proposition 1.17. There exists a collection of data (G, α, β, γ) as above, such that for every CFG H over C, the map (†) is an isomorphism. It satisﬁes the universal property that if (G0, α0, β0, γ0) is another collection with property above, then there exists a triple (F, u, v), where F : G → G0 is an equivalence of ﬁbered categories, u : α → α0 ◦ F and v : β → β0 ◦ F are isomorphisms of base-preserving natural transformations, such that the following diagram commutes:

◦ c ◦ α c u c ◦ α0 ◦ F γ γ0 ◦ d ◦ β d v d ◦ β0 ◦ F.

Moreover, it is unique in the sense that if (F0, u0, v0) is a second such triple, then there exists a unique isomorphism σ : F0 → F such that the diagrams:

0 α u α0 ◦ F u σ α0 ◦ F and

0 β v β0 ◦ F v σ β0 ◦ F 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).

10 Then we enlarged PSh to CFG, where CFG are, roughly speaking, presheaves taking values in categories. 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 X should be entirely determined by the fiber categories over a cover of X, and the isomorphisms need to be a sheaf. We begin by making sense of the first statement. Just as we need the cocycle conditions on overlaps 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, and Uijk = Ui ×U Uj ×U Uk.

Deﬁnition 1.18. Let {σi : Ui → U} be a covering in C. An object with descent data ({Ei}, {φij}) on ∗ ∗ this cover is a collection of objects Ei ∈ F(Ui), together with isomorphisms φij : pr2 Ej → pr1 Ei in F(Uij) (here pr1 is the projection to Ui, and pr2 to Uj; similar afterwards), satisfying the cocycle condition: ∗ ∗ ∗ ∗ ∗ pr13φik = pr12φij ◦ pr23φjk : pr3 Ek → pr1 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

0 {αi} : ({Ei}, {φij}) → ({Ei }, {ψij})

0 is a collection of arrows αi : Ei → Ei in F(Ui) with the property that for each i, j, the diagram commutes:

pr∗α ∗ 2 j ∗ 0 pr2 Ej pr2 Ej

φij ψij ∗ ∗ pr1 αi ∗ 0 pr1 Ei pr1 Ei.

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 overlaps”. We still need another that says every such collection will glue, i.e. there exists an object E in F(U) and pullbacks maps σ˜i : Ei → E lying over σi. For this, we deﬁne a natural functor

e : F(U) → F({σi : Ui → U}) as follows. ∗ ∗ ∗ For each object E of F(U), take the objects Ei = σi E; and the isomorphisms φij : pr2 σj E → ∗ ∗ pr1 σ1 E are the isomorphisms that come from the fact that both are pullbacks of E to Uij, and so they could be identiﬁed. 0 ∗ ∗ ∗ 0 Given an arrow α : E → E in F(U), we get arrows σi α : σ E → σ E , yielding an arrow from

11 the object with descent data associated with E to that with E0. So this deﬁnes a functor e as required.

Deﬁnition 1.19. An object with descent data ({Ei}, {φij}) in F({Ui → U}) is effective if it is isomorphic 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 morphisms Y → X, and morphisms are commutative triangles1. The fibered structure is by sending Y → X to Y. Now let X ∈ C be an object and let x, x0 ∈ F(X) be two objects in the fiber over X. We can define a presheaf on C/X: Isom(x, x0) : (C/X)op → Set as follows. For any object f : Y → X in C/X, choose pullbacks f ∗x, f ∗x0 and set

0 ∗ ∗ 0 Isom(x, x )( f : Y → X) = IsomF(Y)( f x, f x ).

g f For a composition Z −→ Y −→ X, the pullback ( f g)∗x is a pullback along g of f ∗x and ( f g)∗x0 is a pullback along g of f ∗x0. Therefore there is a canonical map

g∗ : Isom(x, x0)( f : Y → X) → Isom(x, x0)( f : Z → X) compatible with composition2.

0 Example 1.20. In the context of Vr of all rank r vector bundles, let X be a scheme, and E, E be two vector bundles over X. Then isomorphisms are a sheaf if, for any open covering {Xi → X} | → 0| of X, and for every collection of isomorphisms αi : E Xi E Xi in the ﬁber over Xi such that | = | → 0 = αi 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 the following 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 this covering is effective.

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

1 Here we assume C/X to have what is called the comma topology, i.e. {Ui → X} is a covering for U → X iff {Ui} is a covering for U in C. 2The presheaf Isom is independent of the choice of pullbacks up to canonical isomorphisms.

12 Proposition 1.21. A category ﬁbered in groupoids p : F → C is a stack if for every object X ∈ C and covering {Xi → X}i∈I, the functor

e : F(X) → F({Xi → X}i∈I ) is an equivalence of categories.

Proof. Condition 2 is equivalent to saying the functor e : F(X) → F({Xi → X}) deﬁned in section 1.4.1 is essentially surjective. It remains to prove condition 1 is equivalent to saying this functor e is fully faithful.

We ﬁrst assume condition 1 holds. Take an object X of C, a covering {Xi → X}, and 0 x, y ∈ F(X). The map e gives two descent data ({Ei}, {αij}) and ({Ei }, {βij}) associated with 0 x and y. Now a morphism between these two descent data is a collection of arrow φi : Ei → Ei 0 such that φi, φj agree on pullbacks of E, E to Uij. If Isom is a sheaf, then this ensures that any such morphism comes from a unique map x → y in F(X), i.e. e is fully faithful. The converse is similar.

Definition 1.22. A category fibered in groupoids F → C is a prestack if F(X) → F({Xi → X}) is fully 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 c d F2 F3 be a diagram of stacks fibered in groupoids over C. Then the fiber product (as categories fibered in groupoids) F1 ×F3 F2 is also a stack fibered in groupoids.

Proof. For any covering {Xi → X}, the maps

(F × F )(X) → F (X) × F (X) 1 F3 2 1 F3(X) 2 and (F × F )({X → X}) → F ({X → X}) × F ({X → X}) 1 F3 2 i 1 i F3({Xi→X}) 2 i 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 C and a morphism of ﬁbered categories q : F → Fa such that for any stack G over C, the induced functor

a HOMC(F , 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 so far. Recall we deﬁned:

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

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

h F

hX G.

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

Deﬁnition 1.26. A subfunctor U ⊂ X is a representable map of functors which is pointwise a monomorphism 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 which are representable.

Likewise, if we apply it op the category of afﬁne schemes, we would recover schemes. 0 0 Consider a map M → M where M is an object in C (or hM0 ) and M is a presheaf on C. In the common case where the open embedding of C are monomorphisms, M0 is an open subfunctor of M iff for any X ∈ C, s ∈ M(X), there is an open subset U ⊂ X and a map b : U → M such that ∗ ∗ ∗ s|U = b , and for any f : Y → X, the image of f is contained in U iff f (s) = g (t) for some map g : Y → M0. So:

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

Deﬁnition 1.29. Let F : M0 → M be a morphism of CFG. We say F is representable if for every object X of C, and any morphism from (the representable CFG associated to) X to M, the ﬁber 0 product M ×M X is representable.

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

14 Chapter 2

Algebraic spaces and algebraic stacks

A careful study of the ﬁrst chapter seems to suggest that stacks are nothing more than a categorical construction, bearing very little geometry for us to play with. Indeed this is the case. To talk about various geometric properties, we must impose some other conditions on stacks. Some special types of stacks with rich geometry are algebraic stacks, Deligne-Mumford stacks, and algebraic spaces. This chapter aims to give a nice intuition and precise deﬁnitions of the conditions imposed on stacks to become algebraic and/or Deligne-Mumford. In particular, we would eventually see the hierarchy of some abstract notions involved in the world of stacks. Figure below is taken from [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 would work 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 sense of this.

Definition 2.2. If A → B is a ring homomorphism, then we way that B is of finite presentation over 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 every affine open Spec(A) ⊂ Y and affine open Spec(B) ⊂ f −1(Spec(A)), the A-algebra B is of finite presentation over A.

Definition 2.4. A morphism of schemes f : X → Y is called formally smooth (resp. formally unramified, formally étale) if for every affine Y scheme Y0 → Y and every closed immersion 0 Y0 ,→ Y defined by a nilpotent ideal, the map

0 0 HomY(Y , X) → HomY(Y0, X) is surjective (resp. injective, bijective). If f is also of finite presentation, then f is called smooth (resp. unramified, étale).

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) be the category of X-schemes. We put an fppf (resp. smooth, big etale´ , big Zariski) topology on it by declaring that Cov(U) is the set of collections {Ui → U}i∈I of X-morphisms for which each morphism Ui → U is ﬂat and locally of ﬁnite presentation (resp. smooth, etale´ , an oepn embedding), and the map G Ui → U i∈I 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 properties of schemes and morphisms in terms of the corresponding sheaves on sites. We will take our base category 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 is in S if and only if each Ui is in S. We call a property P of objects of C stable if the class of objects satisfying 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

X0 X f 0 f Y0 Y for which the morphism f is in D, we have f 0 ∈ 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 coverings xi {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 are morphisms satisfying P. Then P is stable (resp. local on domain) if the subcategory DP ⊂ C is stable (resp. local on domain).

In subsection 1.3.3 Representability, we introduced the notion when a morphism of sheaves/stacks is representable. What we said is that the base change against a representable object is another representable 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 (as

sheaves), 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, we

say f has property P if for every S-scheme T, the morphism of schemes pr2 : F ×G T → T has property P.

17 2.2 Algebraic spaces

To motivate our next topic, we ask the question: what conditions we should put on stacks to make 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 to impose 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 smooth manifolds without the Hausdorff condition. It turns out that a manifold is Hausdorff if and only if 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 our objects.

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 ´etale topology.

We saw that this statement is true for Zariski topology. This proposition follows from the result 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 representable in the Zariski topology.

• An algebraic space is an fppf sheaf (on the category of afﬁne schemes) which is locally representable 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)

∼ C0 LocRep(C0) Sh(C0) PSh(C0) where C is the category of afﬁne schemes, and C0 is locally representable sheaves on afﬁne schemes, or equivalently schemes. We could then identify Sh(C) =∼ Sh(C0) and LocRep(C) =∼ LocRep(C0) (not true for presheaves, as they don’t care about topology at all). So the category of algebraic 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 → Set such 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 new category.

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 is representable by schemes. Then any morphism f : T → F for an S-scheme T is representable by schemes.

Proof. For schemes T, T0, the ﬁber products of these two diagrams are the same:

0 T T ×S T

f f ×g g T0 F F ∆ F × F.

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 of algebraic spaces translates to this: for any S-scheme T and morphism T → X, the morphism

U ×X T → T is etale´ and surjective. So, in the way schemes are covered by afﬁne schemes in the Zariski topology, algebraic spaces are 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 sheaf quotients. 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 a sheaf X/R which will be an algebraic space. In fact, every algebraic space can be described in this way. We begin by deﬁning such an equivalence relation and the quotient X/R.

Deﬁnition 2.13. An ´etaleequivalence 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 maps s, 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 inclusion

R ,→ X ×S X

is an ´etaleequivalence 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 some descent theory we have not established near the end of the proof, and results on that can be found 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 test scheme T0 into Y would be representable by schemes, the projection X → Y is representable by schemes. This is the etale´ surjection we need. Too see this, note there are cartesian squares in the diagram below where s is etale´ surjective, and the map T ×Y X → T will be etale´ surjective:

t T ×Y X R X s T X Y.

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 U over X ×S X. Then RU is an etale´ equivalence relation on U, and we therefore have an induced map

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 assume they 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 W0 → W such that the composite map 0 W → W → Y ×S Y

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

R X ×S X

∆ Y Y ×S Y.

0 0 So if F := F ×W W , then 0 ∼ × 0 ∼ × 0 F = Y Y×SY W = R X×S X W .

In particular, F0 is a scheme and F0 → W0 is a monomorphism, which implies that F0 is a separated S-scheme since W0 is a separated S-scheme. Since the square

F0 W0

F W is cartesian, the morphism F0 → F is representable and etale´ surjective. Therefore F is the sheaf quotient of F0 by the equivalence relation

0 0 0 0 0 R := F ×F F ⊂ F ×S F .

0 0 0 Since F → W is a monomorphism, the two projections pi : W ×W W → W satisfy that the squares

0 0 0 0 F ×F F W ×W W 0 p pi i F0 W0.

0 0 are cartesian. So R is a scheme and that the two projections pi are quasi-compact and etale.´

21 0 0 0 0 Lemma 2.17. Let U ⊂ F be a quasi-compact open subset, and let FU0 denote the quotient U /RU0 . Let 0 0 j : FU0 → F be the natural representable open embedding, and let FU0 ⊂ F denote the ﬁber product of the diagram

j FU0 F.

0 0 0 Then FU0 is a quasi-compact open subset of F containing U . 0 −1 0 0 Proof. Indeed, FU0 = p1(p2 (U )) and pi are quasi-compact and U was quasi-compact. 0 0 Since we can also write FU0 = FU0 ×W W , it suffices to check FU0 is a scheme for every 0 0 0 0 quasi-compact open subset U of F . This follows from the fact that FU0 → W is quasi-affine by Zariski’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 have various 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 an algebraic space X has property P if there exists an etale´ surjection U → X from a scheme U with property 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 → Y be a morphism of algebraic spaces that is representable (by schemes) and P is a property of morphisms of schemes which is stable in the etale´ topology. In the sense of definition 2.8, we say f has property P if there is an etale´ cover V → Y with V ×Y X → V has property P. Examples of such properties are proper, dominant, quasi-compact, (open/closed) embedding, etc. A few other examples are being flat, etale´ , smooth, surjective, locally of finite type, etc. These properties are, 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 etale´ topology, and let f : X → Y be a morphism of algebraic spaces. Then f has property P if there exist 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:

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. locally separated, 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 these properties 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-Mumford stacks. 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 called representable 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 two definitions of algebraic stacks. The first one helps us think about algebraic stacks, and we need the second one to do any concrete computations. Recall from 2.2.1 that an algebraic space is an etale´ -locally representable sheaf on the category of schemes with the etale´ topology, an algebraic stack is a smooth-locally representable stack on the 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 a scheme 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 an algebraic stack is representable (by schemes).

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

Y y Z z X with Y, Z algebraic spaces, the ﬁber product Y ×X Z is an algebraic space. In particular, any morphism f : 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 isomorphic ∗ ∗ to the sheaf Isom(pr1y, pr2z) over Y ×S Z, is an algebraic space. This follows from the obvious lemma below.

Lemma 2.24. Let X/S be a stack. The diagonal ∆ : X → X ×S X is representable if and only if for every S-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

u1×u2

X X ×S X.

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 in this 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 c Y d Z, 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 the set 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 morphism 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 etale´ surjective morphism from a scheme with property P.

The reason we need this terminology is that we can transfer some geometric properties of schemes to algebraic stacks. For example, locally noetherian, regular, and locally of ﬁnite type over 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 commutative diagram:

g 0 f G1 X G2 q p p f X Y 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 on domain in smooth topology. Then a morphism f : X → Y of algebraic stacks has property P if there 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 geometric properties are discussed in section 2.2.3. So we could use those languages to describe some more properties of algebraic stacks.

Deﬁnition 2.29. Let P be a morphism of algebraic spaces which is stable in smooth topology on the category of algebraic spaces. We say a representable morphism of algebraic stacks f : X → Y has property P if for every morphism Y0 → Y from an algebraic space, the morphism of algebraic spaces 0 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 to deﬁne a few more properties, we can do the same thing here, assume that ﬁber products exist in the 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 X be 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 structure morphism 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 a scheme.

We mention without proof here a few useful criteria for algebraic stacks to be DM or algebraic spaces, 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, meaning that 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 automorphism 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 are Deligne-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, namely quotient 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 a group 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))

F is a monomorphism. Here by R = G × X, we mean the scheme R = g∈G X. Now j is a monomorphism means we have an equivalence relation in the sense of Deﬁnition 2.13, since R is a disjoint union of X and therefore R → X is etale´ . As a result of Proposition 2.15, we could take such 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 a stack. 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 of algebraic 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 the target, and c the composition). This gives a functor

( )op → Sch/S fppf Groupoids S0 7→ (U(S0), R(S0), s, t, c).

Similar to Proposition 1.14 where we discussed we can view presheaves of sets as categories fibered in sets, here a presheaf of groupoid can be seen as a category fibered in groupoid. By Theorem 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 spaces over 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 an algebraic space over B and m : G ×B G → G is a morphism of algebraic spaces over B, such that for 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 action a : 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).

0 0 0 0 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 of G 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).

28 3.1.2 Torsor and principal bundles

This section explains what the stack [X/G] looks like, and provide some practical tools to do computations 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 a sheaf 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 map G × 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 of G(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) → (F0, ρ0) is a sheaf morphism h : F → F0 such that the diagram commutes:

id ×h G × F G G × F0

ρ ρ0 F h F0.

The idea of torsor is closely related to another idea called principal bundles. Here we fix a base scheme X and work over Sch/X with fppf topology. We also assume G above is representable by a 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, locally finitely presented, surjective morphism of schemes, and

ρ : G0 ×X P → P is a morphism such that:

1. The diagram commutes:

id ×ρ G0 G0 ×X G0 ×X P G0 ×X P

m×idP ρ ρ G0 ×X P P,

where m is the group law on G.

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

(eπ,idP) ρ P −−−−→ G0 ×X P −→ P

is the identity map on P.

3. The map

(ρ, pr2) : G0 ×X P → P ×X P is an isomorphism.

0 0 0 A morphism of principal G0-bundles (P, ρ) → (P , ρ ) is a morphism of X-schemes f : P → P such that the diagram commutes:

id × f G0 G0 ×X P G0 ×X P ρ ρ0 f P P0.

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

(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

φ P X

U b B.

2. A morphism is a pair ( f , g) : (U, b, P, φ) → (U0, b0, P0, φ0) where f : U → U0 is a morphism of schemes, and g : P → P0 is a G-equivariant morphism over f which induces an ∼ 0 0 isomorphism P = U × f ,U0 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 fppf GU-torsors P with a G-equivariant morphism to X. So the ﬁber categories are groupoids.

30 But it is actually a stack. The proof uses some result from descent theory, which we did not cover 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 exists a 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 where k 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, we introduce the notion of a coarse moduli space, which is, in some sense, the closest approximation to 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 with X 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 an algebraic 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 noetherian base 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 base change, meaning if Y → X is a flat morphism of algebraic spaces, then M ×X Y → Y is a coarse moduli space.

3.2 Mg and Mg

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

Deﬁnition 3.11. Let g be a non-negative integer at least 2. Let Mg be the category whose objects are 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 are equivalent.

31 that for every point s ∈ S, the ﬁber Cs is a geometrically connected proper smooth curve of genus g. A morphism (S0, C0 → S0) → (S, C → S) in Mg is a cartesian square

C0 C

S0 S.

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

Lemma 3.12. Mg is a stack for the ´etaletopology.

The proof uses descent theory for polarized schemes. We deﬁne the category Pol to consist of pairs ( f : X → Y, L) where f is a ﬂat morphism of schemes and L a relatively ample invertible sheaf on X. Descent theory for polarized schemes implies that any fppf covering S0 → S is an effective 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)

0 0 Mg(S → S) Pol(S → S).

1 0 where C → S is sent to (C → S, ΩC/S). So the functor Mg(S) → Mg(S → S) is fully faithful, and Mg 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 [Mfg/G], and then using Lemma 3.8 to show it is in fact Deligne-Mumford. We begin with a lemma.

1 ⊗3 Lemma 3.14. Let (S, f : C → S) ∈ Mg, and let LC/S be the invertible sheaf (ΩC/S) . 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 : S0 → S, the natural map

∗ 0 g f∗LC/S → f∗LC0/S0

is an isomorphism, where f 0 : C0 → S0 is the base change of f to S0.

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

0 1 h (C, LC/S) = 3 deg(ΩC/S) + 1 − g = 3(2g − 2) + 1 − g = 5g − 5.

Assuming these results, we can deﬁne Mfg as follows.

Deﬁnition 3.15. Let Mfg denote the functor on the category of schemes which to any scheme S associates the isomorphism classes of pairs

5g−5 ∼ ( f : C → S, σ : OS = f∗LC/S), where (S, f : C → S) ∈ Mg(S). An isomorphism

0 0 0 0 5g−5 ∼ 0 5g−5 ∼ ( f : C → S , σ : OS = f∗LC0/S) → ( f : C → S, σ : OS = f∗LC/S) is given by an isomorphism of curves α : C0 → C such that σ = α ◦ σ0.

The amazing thing about Mfg is that it is representable by a quasi-projective scheme, and we can deﬁne an action of G := GL5g−5 on Mfg. 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 : Mfg → Mg, where (C/S, σ) 7→ (S, C). For any scheme S and morphism f : S → Mg corresponding to a curve C/S, the fiber product Mfg ×Mg S is the Gs- 5g−5 ∼ torsor of isomorphisms σ : OS → f∗LC/S. Thus we have an isomorphism MG = [Mfg/G], and in particular, Mg is algebraic. To show it is Deligne-Mumford, it suffices to show that for any algebraically closed field k and smooth genus g curve C/k, the group scheme Autk(C) is reduced. For this, we can show that if A0 → A is a surjective morphism of k-algebras with square-zero kernel I, then the map

0 Autk(C)(A ) → Autk(C)(A) is injective. This is true because for any α : CA → CA an automorphism, the set of liftings to CA0 are given by the set of dotted arrows in the diagram

α CA CA CA0

0 CA0 Spec(A ).

By the universal property of differentials, the set of such dotted arrows form a torsor under the isomorphism Hom(α∗Ω1 , I ⊗ O ) =∼ H0(C , α∗T , I ⊗ O ), CA/A A CA A CA/A A C which is zero since it is zero in every ﬁber.

33 3.2.1 The compactiﬁcation Mg

In our construction of Mg, we required f : C → S to be a smooth map. The idea of compactiﬁca- tion, or adding new points to Mg, 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 construction below. First we formalize the notions of special points, and use this idea to deﬁne what stable curves are.

Definition 3.16. Let k be an algebraically closed field and X/k be a finite type k-scheme of dimension 1. A closed point x ∈ X(k) is called a node (or ordinary double point) if the complete local ring Oˆ X,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 that for 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 the deﬁnition here.

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

i g U −→ P −→ Y where i is a regular embedding of codimension e and g is smooth of relative dimension d + e for some e. A morphism f : X → Y is a local complete intersection if there exists an integer d such that f 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 suppose there exists a factorization i g X −→ P −→ Y of f with i a regular embedding of codimension e and g smooth of relative dimension d + e, and let ωP/Y 1 denote the (d + e)-th exterior power of ΩP/Y. Then

∗ e i E xt (i∗O , ω ) OP X P/Y is a locally free sheaf of rank 1 on X, which is independent of the choice of the factorization of f . This invertible 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 : Ce → C, we say a point q ∈ Ce(k) is special if π(q) is a node. It can be shown that, for any irreducible component Ci ⊂ C with normalization Cei, the degree of ωC/k restricted to Cei is

2gi − 2 + #{special points on Cei}.

34 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 Cei} > 0 for every irreducible component. If f : C → S is a prestable curve over a scheme S, we say C is stable if for every geometric point x → S the ﬁber Cx is a stable curve over k(x).

Deﬁnition 3.21. Fix g ≥ 2. Let Mg denote the ﬁbered category over Spec(Z) whose objects are pairs (S, f : C → S) where S is a scheme and f : C → S is a stable curve of genus g. A morphism (S0, f 0 : C0 → S0) → (S, f : C → S) is a cartesian square

C0 C f 0 f S0 S.

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 relative ⊗n dualizing sheaf ωC/S is relatively very ample. Then by descent theory on polarized schemes, we would get Mg is a stack.

The proof that Mg is algebraic is very similar to showing that Mg is algebraic. In particular, let N g denote the functor that associates to a scheme S the set of isomorphism classes of pairs ( → O5g−5 → ⊗3 ) N N → M π : C S, ı : S π∗ωC/S . Then g is a scheme, and the map g g is a smooth surjection.

The last thing to check is that the diagonal is unramified; or equivalently, if T0 ,→ T is a closed embedding of affine schemes defined by a square-zero ideal, and if α : C → C is an automorphism of a stable curve over T such that the reduction α0 : C0 → C0 of α to T0 is the identity, then α is the identity. For this it suffices to consider when T = Spec(A) of an artin local ring 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(Ω1 , O ) = 0. Ck/k Ck

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

35 Conclusion and future perspectives

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

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

2. For algebraic stacks, we ﬁrst introduced algebraic spaces as fppf sheaves that locally look like schemes. Then an algebraic stack is a stack over S-schemes with fppf topology, whose diagonal 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, or equivalently no inﬁnitesimal automorphisms.

It does seem like an overkill to deﬁne algebraic stacks using so much other ideas. Of course, we could have directly deﬁne algebraic stacks and maybe skip some of the prerequisites above; but then 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 might also 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 this could 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 stacks for example).

The reason these topics are not included is that they usually require more prerequisite than a ﬁrst course in algebraic geometry, and also they mostly stand alone from the facts that we already discussed. However, these materials could be easily grasped should anyone want to go further from 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 marked points. They should be a direct extension of the theory discussed in our last section of the essay.

2. Coarse moduli spaces. We only introduced the idea of a coarse moduli space in section 3.1.3. 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.

36 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 given genus. Publications math´ematiquesde 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 Colloquium Publications. 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.