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Algebraic Stacks 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 gen- eralize 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 2 C and objects x, y 2 F(X), the presheaf Isom(x, y) on C/X (which will be defined later) is a sheaf; and 2. For any covering fXi ! Xg of an object X 2 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) = ffUi ! Xgi2I : Ui cover Xg. 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 fXi ! Xgi2I, satisfying the three axioms: 1. (Isomorphism is a cover) If V ! X is an isomorphism, then fV ! Xg 2 Cov(X). 2. (Pullback cover exists) If fXi ! Xg is a covering, and Y ! X any morphism in the category C, then there is a fibered covering fXi ×Y Y ! Yg which is in Cov(Y). f ! g f ! g 3. (Cover of a cover is a cover) If Xi X i2I is a cover for X, and for each i, Vij Xi j2Ji is a cover for Xi, then all composites f ! ! g Vij Xi X i2I,j2Ji 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.
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