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Contents 1 Introduction (Martin Olsson) STAX Contents 1 Introduction (Martin Olsson) 1 2 Background: schemes as functors (Andrew Niles) 2 2.1 Category theory . .2 2.2 Schemes . .3 2.3 Sheaves . .3 3 Sites and sheaves (Slater Stich) 4 4 Fibered categories (Aaron Mazel-Gee) 6 4.1 Definitions and basic facts . .6 4.2 The 2-Yoneda lemma . .7 4.3 Categories fibered in groupoids . .8 4.3.1 ... coming from co/groupoid objects . .8 4.3.2 ... and 2-categorical fiber product thereof . .9 5 Faithfully flat descent (Andrew Dudzik) 10 6 Algebraic spaces (Yuhao Huang) 12 7 Coarse moduli spaces, part 1 (Shelly Manber) 13 8 Quasicoherent sheaves of modules on algebraic spaces (Jason Ferguson) 15 9 Algebraic stacks (Doosung Park) 17 10 Quasicoherent sheaves on algebraic stacks (Katrina Honigs) 19 10.1 Stacks and algebraic stacks . 19 10.2 Quasicoherent sheaves on stacks . 20 11 Mg is a Deligne-Mumford stack (Martin Olsson) 20 12 Coarse moduli spaces and the Keel-Mori theorem (Piotr Achinger) 22 12.1 Moduli problems . 22 12.2 The Keel-Mori theorem . 23 12.3 Tameness and quasicoherent sheaves . 24 13 Gerbes (Peter Mannisto) 24 13.1 Torsors . 24 13.2 Gerbes . 25 1 Introduction (Martin Olsson) Motivation for stacks comes from: • algebraic topology; 1 • moduli theory; • not working with one arm tied behind your back. In algebraic topology, we might like to talk about classifying spaces. But one quickly learns that algebraic varieties have such rigid structure that we can't make the same constructions. In topology, cohomology is a representable functor: for any abelian group A and nonnegative number n, there is a space K(A; n) such that Hn(X; A) ' [X; K(A; n)] (where the square brackets denote homotopy classes of maps). In algebraic geometry, H1(X; A) is defined in terms of torsors and H2(X; A) is defined in terms of gerbes, which are certain kinds of stacks. It is an extremely profitable business to think about cohomology classes in this way. However, the story of stacks really begins with moduli theory: the classification of algebraic varieties. For example, curves are classified by their genus (so at g = 0 we have only P1). But what about fX2+Y 2+Z2 = 0g ⊂ P2? If we're over the real numbers, we probably don't want to call this P1! To give the correct answer takes a lot of machinery involving Grothendieck topologies etc., but the final answer is that the moduli stack of P1 should be thought of as BP GL2. 1 Without going into details, we have the following picture. If S is a space, we can make a fiber bundle P ,! X S { each fiber is a copy of P1, but X =6∼ S × P1. For example, if S = S1 is the circle, we can cover S1 with two open intervals U1 and U2 over which the bundle is trivial, and then to specify such a bundle we must specify how the bundles are supposed to glue on U1 \ U2. These gluings can be nontrivial { indeed, P GL2 gives the automorphisms of P1 { and so we can get nontrivial bundles over S. To do this properly in algebraic geometry requires faithfully flat descent, ´etale topology, and other things. At g = 1, there are two interesting things that happen. First of all, we can look at pointed genus-1 curves (E; e) (i.e. elliptic curves) or unpointed genus-1 curves. As it turns out, one can make families of E ! S where all fibers 0 0 are genus-1 curves and there exists an ´etalesurjection S ! S so that E ×S S is a scheme but E is an algebraic space which is not a scheme. An ´etalemorphism should just be thought of as a local homeomorphism, so S0 ! S 0 0 0 ∗ 0 ∼ ∗ 0 should be thought of as something like a covering space. Then in S ×S S ⇒ S ! S, choices p1E ! p2E plus cocycle conditions should glue to an element E0 over S0. It is impossible to do this in the world of schemes. 2 3 2 3 Example 1. Consider the two (pointed) elliptic curves E1 : y = x −Dx and E2 : y = x +Dx over Q. These are isomorphic once we have a cube-root of D, but they are non-isomorphic over Q itself. This illustrates the fact that the statement \elliptic curves are classified by their j-invariant" really only holds over algebraically closed fields. Thus, the functor M1;1 which takes a scheme S to an elliptic curve E ! S over S (ie. each fiber is an elliptic curve, 1 with basepoints given by a section e : S ! E) has a map M1;1 ! Aj , which we call a coarse moduli space. Example 2. When g ≥ 2, we have a map Mg ! Mg; the first is a stack and the second is a singular space. If for example we'd like to study intersection theory, the correct place for defining cycles is Mg. 2 Background: schemes as functors (Andrew Niles) 2.1 Category theory op Let C be a (locally small) category. For any X 2 C, we have a functor hX = HomC(−;X): C ! Sets. op Lemma 1 (Yoneda). For any functor F : C ! Sets, the function HomF un(hX ;F ) ! F (X) defined by f 7! fx(idX ) is a bijection. op Corollary 1. The mapping X 7! hX defines a fully faithful functor C! F un(C ; catSets). Thus we will use X and hX interchangeably. We say that a functor F : Cop ! Sets is representable by an object X 2 C if there is an isomorphism of functors ∼ hX = F . We make the enormous abuse of notation and say that F \is" an object of C (since it's in the essential image of the Yoneda embedding). Let F; G : Cop ! Sets and let f : F ! G be a morphism of functors. We say that f is relatively representable if for every X 2 C and every morphism of functors hx ! G, the fiber product F ×G hX is representable. The fiber product is by definition computed pointwise, i.e. (F ×G hX )(Y ) = F (Y ) ×G(Y ) hX (Y ). 2 2.2 Schemes Let us fix C = AffS, the category whose objects are affine schemes Spec(R) with a fixed morphism Spec(R) ! S op and whose morphisms are just morphisms of S-schemes. (If S = Spec(A) is affine, then this is just (AlgA) .) Note op that for any S-scheme X, we still have a functor hX : AffS ! Sets. op Let F; G : AffS ! Sets be functors and let f : F ! G be a morphism of functors. We say that f is an affine open (resp. closed) immersion if • f is relatively representable, and • for all hX ! G, the morphism F ×G hX ! hX in the diagram - F ×G hX hX ? ? F - G \is" an open (resp. closed) immersion. ∼ That is, we choose an isomorphism hY = F ×G hX , and then the morphism hY ! hX corresponds to an open (resp. closed) immersion Y ! X of schemes. 2.3 Sheaves Recall the notion of a sheaf F (of sets) on a scheme X. This is a functor F : Op(X)op ! Sets with the property that for any open cover fUig of an open subset U ⊂ X, Y Y F(U) ! F(Ui) ⇒ F(Ui \ Uj) i i;j is an equalizer diagram (in Sets). op op Now, let F : AffS ! Sets be a functor. We say that F is a big Zariski sheaf if for all U 2 AffS and for all open affine covers fUig of U, Y Y F(U) ! F(Ui) ⇒ F(Ui \ Uj) i i;j is an equalizer diagram (in Sets). (Note that we actually take the intersection to be a fiber product in the category op AffS , so there's no worry that this won't be affine.) Of course this is very similar to the usual sheaf condition; the only difference is that U is no longer sitting inside a fixed scheme X. We will soon learn that this is nothing but a sheaf on a \Grothendieck topology". A morphism f : F ! G of big Zariski sheaves (which is by definition just a morphism of functors) is called op surjective if for all U 2 AffS and u 2 G(U), there is a covering fUig of U in AffS such that ujUi 2 Im(fUi : F (Ui) ! G(Ui)) for all i. op Proposition 1. Let F : AffS ! Sets be a functor. Then F is representable by a separated (not necessarily affine) S-scheme if and only if the following three conditions hold: 1. F is a big Zariski sheaf. 2. ∆ : F ! F × F is an affine closed immersion. ` 3. There is a surjection i hXi ! F of big Zariski sheaves, where each hXi ! F is an affine open immersion. op Also, the functor Sch=S ! F un(AffS ; Sets) given by X ! hX is fully faithful, and the above precisely describes this functor. This is nice because it can be a lot easier to describe a functor than to describe a scheme that represents it. Proof of ((). Let us begin with the Xi. Since each hXi ! F is an affine open immersion, then the fiber products ∼ hXi ×F hXj are affine, say hXi ×F hXj = hVij with open immersions Xi -Vij ,! Xj. Then we define X to be the Xi glued along the Vij. 3 If we want to talk about more general, not necessarily separated schemes, we need to make the following modification.
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