I’ve been trying to learn a bit about stacks lately. This is actually going to tie in quite nicely into the homotopy-theoretic story I was talking about earlier: the moduli stack of formal groups turns out to be a close approximation to stable homotopy theory. But before that, there are other more fundamental examples of stacks.

Here is a toy example of an Artin stack I would like to understand. Let be a noetherian scheme; we’d like a stack which parametrizes coherent sheaves on . This is definitely a “stack” rather than a scheme, because coherent sheaves are likely to have plenty of automorphisms.

Let’s be a little more precise. Given a test scheme (say, affine) , a map

should be a “family of coherent sheaves on ” parametrized by . One way of saying this is that we just have a quasi-coherent sheaf of finite presentation on . In order to impose a “continuity” condition on this family, we require that be flat over .

To make this more precise, we define it relative to a base:

Definition 1Let be a base noetherian scheme, and let be an -scheme of finite type. We define a functor

sending an affine to the groupoid of finitely presented quasi-coherent sheaves on , flat over . The pull-back morphisms are given by pull-backs of sheaves.

In this post, we will see that this is an Artin stack under certain conditions.

**1. First observations
**

The first observation is that is a (plain) stack: that is, it satisfies descent for étale coverings. This is a consequence of Grothendieck’s faithfully flat descent theorem: we can glue quasi-coherent sheaves and morphisms of quasi-coherent sheaves along *fpqc* coverings. A fortiori, we can do the same for quasi-coherent sheaves of finite presentation, since that is an fpqc local condition.

In order to do geometry with something like , we would like to have a smooth cover

where is only a scheme. We would also like certain finiteness and separation conditions on the automorphism group objects of elements of .

This is encapsulated by the definition:

Definition 2Let be a scheme. A functor satisfying descent is anArtin stackif

- The diagonal map is representable, separated, and quasi-compact.
- There is a smooth, surjective morphism where is a scheme

Condition 1 in this definition states that the automorphism objects are reasonable. Namely, it states that for any affine scheme and any morphism (that is, any choice of elements ), the sheaf (which is equivalent to the sheaf of groupoids )

is representable by a scheme (or at least, by an algebraic space—something which étale locally looks like a scheme) which is separated and quasi-compact. (A *Deligne-Mumford stack* is one where these automorphism objects are very small: it is one where the diagonal is unramified.)

Condition 2 is that there exists a “smooth presentation:” in order to geometry with , we can use the cover .

So we want to find conditions under which is an **Artin stack** (relative to ).

**2. The isomorphism objects**

So, let’s start by showing that the diagonal map

is representable, separated, and quasi-compact. In other words, we have to show that if is an -scheme and are -flat finitely presented sheaves on , then the functor ,

is representable by a quasi-compact, separated scheme over .

**Reduction**: We can assume that is finitely presented over . In fact, and there is a given map . Now is a filtered limit of , finitely presented over . Moreover, “descend” to a finitely presented *flat* sheaves over affine where is of finite presentation over . This is a consequence of difficult results on “noetherian descent.” I’ve never read the proofs of noetherian descent for flatness, but I’ve heard that they are difficult. If we prove representability of the isomorphism objects over , then we’ll be done.

Anyway, the real point of all this is that we can assume is *noetherian.* In fact, once we’ve said all this, we can even assume that (by replacing by ). So let’s assume we have -flat f.p sheaves on itself. And now, we can appeal to the following fact.

Theorem 3Let be noetherian, and let be projective. Let be coherent sheaves on , with flat over . Then the functor

is representable by a scheme affine and finite type over (in fact, a group scheme).

This is 5.8 of “FGA Explained,” and it’s also somewhere in EGA III. It doesn’t quite do what we want, because we want to show that the functor

is representable. The idea is that sits isomorphically as a subfunctor

where consists of pairs such that . So, we just need to show that is a closed subfunctor (hence representable).

But that’s because of the following. Fix a noetherian scheme and a projective morphism , two -flat coherent sheaves over , and two morphisms .

Then the following lemma implies that is relatively representable:

Proposition 4There exists a maximal closed subscheme such that and is terminal among morphisms to with this property.

In fact, this is because the scheme of maps is a linear scheme, and we take the zero section—without loss of generality, take .

So we find that one half the definition of an Artin stack is satisfied under projectivity:

Proposition 5If is projective, then the map is representable, separated, and quasi-compact.

Of course, to see that is an Artin stack we need a smooth presentation; we will need another hypothesis for this.

**3. The scheme**

In order to get a smooth presentation of , we will use schemes.

As before, let’s assume the map is projective, and let be an ample line bundle on , and we denote twisting by the usual way (by ).

Let be a coherent sheaf on which is flat over . Then the sheaves

are coherent on , and are flat (i.e., locally free) on for . Moreover, the maps are surjections for : this means that is a quotient of a locally free sheaf for .

A key observation is that, given a sheaf on , the space of (flat) quotients of is parametrized by a *scheme*.

Definition 6Let and let be a coherent sheaf on . We define the functor

which sends a -scheme to the set of isomorphism classes of pairs , where is a quotient of flat over .

Observe that this is actually a functor taking values in sets, rather than groupoids: the act of requiring a specific quotient map “rigidifies” the moduli problem.

Theorem 7 (Grothendieck)If is projective, then is representable by a scheme.

When , this scheme parametrizes families of subschemes of , and is called the **Hilbert scheme**; hopefully I’ll have more to say about this in the future.

**4. A cover of **

Let be a projective morphism. Assume moreover that , and that this holds universally: that is, after any base change .

For each , there is a map

which sends a flat quotient of to the underlying . Together, this induces a map

which still isn’t necessarily what we want, because a coherent sheaf on might not be a quotient (even étale locally) of a trivial vector bundle. (This isn’t even true when : for instance, is not such a quotient.) In other words, this map is not a surjection.

Instead, the strategy is to define a map

which sends a flat quotient to . Zariski locally, is a quotient of , for by Serre’s theorem. This map is a surjection, for that reason.

Now we want more than a surjective cover by a scheme, though: we want a smooth cover. To check smoothness, we have to fix a scheme , a map classifying a coherent sheaf on , and analyze the -scheme . What functor does this represent? For a -scheme , we have that

consists of the set of surjective maps . In other words, the set of surjective maps

or equivalently the set of maps

such that the adjoint map is surjective. (Here we have implicitly used the fact that universally.)

**Goal**: The functor that sends to the set of maps such that the adjoint is surjective should be representable by a smooth -scheme.

In order to make this happen, we will replace the Quot schemes by suitable open subfunctors.

**5. The smooth cover of **

Let us define a subfunctor of as follows.

Definition 8sends a scheme to the set of flat quotients of such that for and such that the adjoint map

is an isomorphism.

Let us note that this is literally an open subfunctor of . The first condition (that for ) is an open condition by the semicontinuity theorem. It implies that commutes with base change, and in particular, that the second condition after the first is imposed is an open condition.

Proposition 9The map is a smooth morphism.

*Proof:* In fact, we need to show that if and is coherent and -flat on , then the associated fiber product is representable by a smooth -scheme. Let be the open subscheme defined by the conditions for and such that has rank and is surjective. The second is an open condition, since is locally free on —this is implied by the other conditions.

So then maps into . Over , this fibered product is in fact a -torsor: it is

In fact, it consists of all maps which are isomorphisms and such that the adjoint maps are surjections; the second condition is implied by the first, though. Consequently, this torsor has to be smooth over .

The way we get a smooth cover of is to take the map

which we have just seen is smooth. The last thing is to see that it is surjective. Let’s say we have an object , a coherent sheaf on flat over . (We would have to do this for any mapping to , but the argument is the same.) But this is Serre’s theorem: for , has no cohomology on fibers, is a vector bundle, and is a surjection. Restricting to an open neighborhood, we can assume is a trivial bundle of rank : in that case, is in the image of .

Finally, this proves:

**Theorem: **If holds universally and is projective, then is an Artin stack.

June 4, 2012 at 9:17 am

This is definitely true under weaker hypotheses than projectivity of $X$. Perhaps you are already aware of this, but one reference is Lieblich’s “Remarks on …” There is also a result in this direction in one of my papers with de Jong or with Roth (I can’t remember which).

June 4, 2012 at 12:06 pm

Interesting; thanks for the references! I was only aware of what was in ch. 4 of Laumon/Moret-Bailly.

June 6, 2012 at 10:56 pm

The parenthetical remark at the end of Theorem 3 doesn’t look right, as it would imply that the hom sheaf is always a sheaf of groups, which can’t be true. It might be true for the Isom sheaves though.

June 7, 2012 at 11:39 am

I meant that it is a group scheme under addition of homomorphisms (not composition).

June 8, 2012 at 8:52 pm

Ah, I see. Thanks.

September 19, 2012 at 12:08 am

Do you know if there is a moduli space for ideal sheaves on a scheme under some assumptions? I saw this in a paper but don’t know how to make sense of it.

September 19, 2012 at 7:59 am

That sounds like the Quot scheme, which exists under properness assumptions (see e.g. Nitsure’s article in FGA explained). (Rather, that parametrizes closed subschemes.)