MODULI SPACES AND DEFORMATION THEORY, CLASS 7

RAVI VAKIL

Contents

1. Left-over remarks 1 2. Stacks 1 2.1. Random remarks 3 2.2. Summary of Deligne-Mumford stacks 3 2.3. Sheaves on Deligne-Mumford stacks 3

1. Left-over remarks

Reason for mentioning adding a final object M to moduli topology: This is a straightforward way of making sense of etale cohomology of moduli space. All we need are (i) tensor products (done via isom), and (ii) definition of etale coverings. Etale, and all curve over algebraically closed fields exist.

Jason has pointed out another good reference for stacks: Andrew Kresch’s homepage at penn. http://www.math.upenn.edu/˜ kresch/teaching/stacks.html

Other remarks: The Hilbert functor Hilb(X/S) is the functor which associates to a morphism f : T → S, the set of all closed subschemes of T ×_{S} X which are flat and proper over T. Without the “proper” condition, Hilb(X/S) isn’t necessarily representable after all. Assuming this, it is true that Hilb(X/S) is representable by a separated S-scheme which is a disjoint union of quasi-projective S-schemes, if X is also finitely presented over S. (Note that quasiprojective doesn’t imply finitely presented: let S be some hideous pathological scheme, and X a closed subscheme given by a non-finitely generated ideal.)

2. Stacks

Topologies on X: classical (if complex), Zariski, etale, smooth, flat.

Important exercise. If X is a scheme, then consider the following functor: Y → Mor(Y, X). Of course you’ve seen this functor before. Show that this is a sheaf in the Zariski topology. State precisely what results would be necessary for it to be true in the etale topology.

Date: Thursday, September 28, 2000.

We want to generalize this notion to “groupoids over {Sch}”. We know longer have a functor to sheaves, or groupoids. So here’s the right stand-in.

Definition. A groupoid F over {Sch} is a stack if

(i) (Isomorphisms are a sheaf) For any X ∈ {Sch} and any two objects a and b in the groupoid F (X), the functor Isom_{X}(a, b) : {Sch} → (Sets) (which associates to morphism f : Y → X the set of isomorpihsms in F (Y) between f^{∗}a and f^{∗}b, is a sheaf in the etale topology.

Exercise. State precisely what you need for this to be true for a moduli groupoid.

(ii) Let {X_{i} → X} be a covering of X ∈ {Sch} in the etale topology. Let a_{i} ∈ F (X_{i}), and let

ϕ_{ij} : a_{j}|_{X_{i}×_{X}X_{j}} → a_{i}|_{X_{i}×X_{j}}

be isomorphisms in F (X_{i} ×_{X} X_{j}) satisfy the cocycle condition. Then there is a ∈ F (X) with isomorphisms ψ_{i} : a|_{X_{i}} ˜→a_{i}, such that

ψ_{ij} = (ψ_{i}|_{X_{i}×_{X}X_{j}}) ∘ (ψ_{j}|_{X_{i}×_{X}X_{j}})^{−1}.

Translation: gluability for objects; Identify for objects follows from the first one. Hence objects “form a sheaf”. (Some funkiness here too — explain why we can’t hope for anything more.)

Exercise. Check precisely what you need for this to be true for a moduli groupoid. Show that these conditions are immediate if we were working in the Zariski topology.

In general, you can have stacks in various topologies.

Definition. A stack F is of Deligne-Mumford type (i.e. is a DM-stack) if

(i) The diagonal Δ_{F} is representable, quasicompact, and separated. (ii) There is a scheme U and an etale surjective morphism U → F. Such a

morphism U → F is called an atlas.

(Remark: I’d prefer if you didn’t call it an algebraic stack.)

Checking this isn’t so bad for a moduli functor. We already have the representability of the diagonal in “good situations”. More generally, this comes down to checking that the Isom sheaf is representable, separated, and quasicompact, and then finding an atlas.

Exercise: Say this precisely.

Remark: forces finite automorphism group of objects. Precisely: ...

You can make sense of the dimension of a DM-stack as well. (Explain.)

2.1. Random remarks. Jason’s example of elliptic curves:

Even in the case when S = Spec(k) and X/S is projective, it can happen that Aut(X/S) is not quasi-compact. An example of a projective X with Aut(X/S) not quasi-compact is X = E × E with E an elliptic curve over S. For any matrix

(\begin{matrix}a & b \\ c & d \end{matrix})

in SL(2, Z), we can form the automorphism f : E × E → E × E by f(x, y) = (ax + by, cx + dy). Thus Aut(X/S) contains a copy of SL(2, Z), so it isn’t quasicompact.

Define Artin stack. Additional condition: lfpr; can define dimensions, so dim BG is −3.

2.2. Summary of Deligne-Mumford stacks. How we got there: contravariant functor from sheaves to sets. Then to groupoids. Then a moduli groupoid (or a groupoid over {Sch}).

Stack: isomorphisms form a sheaf; objects “form a sheaf”.

Think of it as a scheme, with stacky points.

The great thing is that you can work on your atlas U, as everything you’re interested in descends.

2.3. Sheaves on Deligne-Mumford stacks. Define Quasicoherent sheaves, coherent sheaves, locally free sheaves (vector bundles), homomorphisms of locally free sheaves.

A quasicoherent sheaf S on a Deligne-Mumford stack F is the following data (i) to each atlas U → F a quasicoherent sheaf S_{U} on U. (Yes, a Zariski-sheaf!) (ii) For each commutative diagram U →^{ϕ} V over F with U and V atlases, an isomorphism a_{ϕ} : S_{U} → ϕ^{∗}S_{V}. These isomorphisms satisfy the cocycle condition, i.e. for any

three atlases U →^{ϕ} V →^{ψ} W over F, we have

a_{ψ∘ϕ} = a_{ϕ} ∘ ϕ^{∗}a_{ψ} : S_{U} →^{∼} (ψ ∘ ϕ)^{∗}S_{W} = ϕ^{∗}(ψ^{∗}S_{W}).

A coherent sheaf and locally free sheaf are defined similarly. A homomorphism is these things is what you think it is. You can define inverse and direct images.

Exercise: This is sensible because all 4 concepts descend under etale covers.

Example: Hodge bundle. L_{i}, tautological line bundles. Structure sheaf.