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 ${X}$ be a noetherian scheme; we’d like a stack ${\mathrm{Coh}(X)}$ which parametrizes coherent sheaves on ${X}$. 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) ${T}$, a map

$\displaystyle T \rightarrow \mathrm{Coh}(X)$

should be a “family of coherent sheaves on ${X}$” parametrized by ${T}$. One way of saying this is that we just have a quasi-coherent sheaf ${\mathcal{F}}$ of finite presentation on ${X \times_{\mathbb{Z}} T}$. In order to impose a “continuity” condition on this family, we require that ${\mathcal{F}}$ be flat over ${T}$.

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

Definition 1 Let ${S}$ be a base noetherian scheme, and let ${X \rightarrow S}$ be an ${S}$-scheme of finite type. We define a functor

$\displaystyle \mathrm{Coh}_{X/S}: \mathrm{Aff}_{/S}^{op} \rightarrow \mathrm{Grpd}$

sending an affine ${T \rightarrow S}$ to the groupoid of finitely presented quasi-coherent sheaves on ${T \times_S X}$, flat over ${T}$. 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 ${\mathrm{Coh}_{X/S}}$ 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 ${\mathrm{Coh}_{X/S}}$, we would like to have a smooth cover

$\displaystyle Y \rightarrow \mathrm{Coh}_{X/S}$

where ${Y}$ is only a scheme. We would also like certain finiteness and separation conditions on the automorphism group objects of elements of ${\mathrm{Coh}_{X/S}}$.

This is encapsulated by the definition:

Definition 2 Let ${S}$ be a scheme. A functor ${F: \mathrm{Aff}_{/S}^{op} \rightarrow \mathrm{Grpd}}$ satisfying descent is an Artin stack if

1. The diagonal map ${F \rightarrow F \times_S F}$ is representable, separated, and quasi-compact.
2. There is a smooth, surjective morphism ${Y \rightarrow F}$ where ${Y}$ is a scheme

Condition 1 in this definition states that the automorphism objects are reasonable. Namely, it states that for any affine scheme ${T}$ and any morphism ${T \rightarrow F \times_S F}$ (that is, any choice of elements ${x_1, x_2 \in F(T)}$), the sheaf (which is equivalent to the sheaf of groupoids ${T \times_{F \times_S F} F}$)

$\displaystyle \mathrm{Aff}_{/T}^{op} \rightarrow \mathrm{Sets}, \quad T' \mapsto \mathrm{Isom}( (x_1)_{T'}, (x_2)_{T'})$

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 ${F}$, we can use the cover ${Y}$.

So we want to find conditions under which ${\mathrm{Coh}_{X/S}}$ is an Artin stack (relative to ${S}$).

2. The isomorphism objects

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

$\displaystyle \mathrm{Coh}_{X/S} \rightarrow \mathrm{Coh}_{X/S} \times_S \mathrm{Coh}_{X/S}$

is representable, separated, and quasi-compact. In other words, we have to show that if ${T}$ is an ${S}$-scheme and ${\mathcal{F}_1, \mathcal{F}_2}$ are ${T}$-flat finitely presented sheaves on ${X \times_S T}$, then the functor ${\mathrm{Aff}_{/T}^{op} \rightarrow \mathrm{Grpd}}$,

$\displaystyle T' \mapsto \mathrm{Isom}_{X \times_S T'}( \mathcal{F}_{1T'}, \mathcal{F}_{2T'} )$

is representable by a quasi-compact, separated scheme over ${T}$.

Reduction: We can assume that ${T}$ is finitely presented over ${S}$. In fact, ${T = \mathrm{Spec} A}$ and there is a given map ${T \rightarrow S}$. Now ${T}$ is a filtered limit of ${\mathrm{Spec} A'}$, ${A' \subset A}$ finitely presented over ${S}$. Moreover, ${\mathcal{F}_{1}, \mathcal{F}_2}$ “descend” to a finitely presented flat sheaves over affine ${\widetilde{T} \rightarrow S}$ where ${\widetilde{T}}$ is of finite presentation over ${S}$. 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 ${\widetilde{T}}$, then we’ll be done.

Anyway, the real point of all this is that we can assume ${T}$ is noetherian. In fact, once we’ve said all this, we can even assume that ${T = S}$ (by replacing ${T}$ by ${S}$). So let’s assume we have ${S}$-flat f.p sheaves ${\mathcal{F}_1, \mathcal{F}_2}$ on ${X}$ itself. And now, we can appeal to the following fact.

Theorem 3 Let ${S}$ be noetherian, and let ${X \rightarrow S}$ be projective. Let ${\mathcal{E}, \mathcal{F}}$ be coherent sheaves on ${X}$, with ${\mathcal{F}}$ flat over ${S}$. Then the functor

$\displaystyle T \mapsto \hom_{X \times_S T}(\mathcal{E}_{T}, \mathcal{F}_T), \quad \mathrm{Sch}_{/S}^{op} \rightarrow \mathrm{Sets}$

is representable by a scheme affine and finite type over ${S}$ (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

$\displaystyle T \mapsto \mathrm{Isom}_{X \times_S T}(\mathcal{E}_{T}, \mathcal{F}_T),$

is representable. The idea is that ${\mathrm{Isom}_{X \times_S T}(\mathcal{E}_T, \mathcal{F}_T) }$ sits isomorphically as a subfunctor

$\displaystyle \mathrm{Sub} \subset \mathrm{Hom}_{X \times_S T}(\mathcal{E}_T, \mathcal{F}_T) \times \mathrm{Hom}_{X \times_S T}(\mathcal{F}_T, \mathcal{E}_T)$

where ${\mathrm{Sub}(T)}$ consists of pairs ${(\alpha, \beta)}$ such that ${\alpha \circ \beta = \beta \circ \alpha = \mathrm{1}}$. So, we just need to show that ${\mathrm{Sub}}$ is a closed subfunctor (hence representable).

But that’s because of the following. Fix a noetherian scheme ${U}$ and a projective morphism ${X \rightarrow U}$, two ${S}$-flat coherent sheaves ${\mathcal{E}, \mathcal{F} }$ over ${X}$, and two morphisms ${\alpha, \beta: \mathcal{E} \rightarrow \mathcal{F}}$.

Then the following lemma implies that ${\mathrm{Sub}}$ is relatively representable:

Proposition 4 There exists a maximal closed subscheme ${U' \subset U}$ such that ${\alpha|_{U'} = \beta_{U'}}$ and ${U'}$ is terminal among morphisms to ${U}$ with this property.

In fact, this is because the scheme of maps ${\mathcal{E} \rightarrow \mathcal{F}}$ is a linear scheme, and we take the zero section—without loss of generality, take ${\beta =0}$.

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

Proposition 5 If ${X \rightarrow S}$ is projective, then the map ${\mathrm{Coh}_{X/S} \rightarrow \mathrm{Coh}_{X/S} \times_S \mathrm{Coh}_{X/S}}$ is representable, separated, and quasi-compact.

Of course, to see that ${\mathrm{Coh}_{X/S}}$ is an Artin stack we need a smooth presentation; we will need another hypothesis for this.

3. The ${\mathrm{Quot}}$ scheme

In order to get a smooth presentation of ${\mathrm{Coh}_{X/S}}$, we will use ${\mathrm{Quot}}$ schemes.

As before, let’s assume the map ${\pi: X \rightarrow S}$ is projective, and let ${\mathcal{O}(1)}$ be an ample line bundle on ${X}$, and we denote twisting by ${\mathcal{O}(m) = \mathcal{O}(1)^{\otimes m}}$ the usual way (by ${(m)}$).

Let ${\mathcal{F}}$ be a coherent sheaf on ${X}$ which is flat over ${S}$. Then the sheaves

$\displaystyle \pi_* (\mathcal{F}(m))$

are coherent on ${S}$, and are flat (i.e., locally free) on ${S}$ for ${m \gg 0}$. Moreover, the maps ${\pi^* \pi_* (\mathcal{F}(m)) \rightarrow \mathcal{F}(m)}$ are surjections for ${m \gg 0}$: this means that ${\mathcal{F}(m)}$ is a quotient of a locally free sheaf for ${m \gg 0}$.

A key observation is that, given a sheaf on ${X}$, the space of (flat) quotients of ${\mathcal{F}}$ is parametrized by a scheme.

Definition 6 Let ${\pi: X \rightarrow S}$ and let ${\mathcal{F}}$ be a coherent sheaf on ${X}$. We define the functor

$\displaystyle \mathrm{Quot}_{\mathcal{F}}: \mathrm{Sch}_{/S}^{op} \rightarrow \mathrm{Sets}$

which sends a ${S}$-scheme ${T}$ to the set of isomorphism classes of pairs ${\mathcal{F}_T \twoheadrightarrow \mathcal{E}}$, where ${\mathcal{E}}$ is a quotient of ${\mathcal{F}_T}$ flat over ${T}$.

Observe that this is actually a functor taking values in sets, rather than groupoids: the act of requiring a specific quotient map ${\mathcal{F}_T \twoheadrightarrow \mathcal{E}}$ “rigidifies” the moduli problem.

Theorem 7 (Grothendieck) If ${\pi}$ is projective, then ${\mathrm{Quot}_{\mathcal{F}}}$ is representable by a scheme.

When ${\mathcal{F} = \mathcal{O}_X}$, this scheme parametrizes families of subschemes of ${X}$, and is called the Hilbert scheme; hopefully I’ll have more to say about this in the future.

4. A cover of ${\mathrm{Coh}_{X/S}}$

Let ${\pi: X \rightarrow S}$ be a projective morphism. Assume moreover that ${\pi_*(\mathcal{O}_X) \simeq \mathcal{O}_S}$, and that this holds universally: that is, after any base change ${S' \rightarrow S}$.

For each ${N}$, there is a map

$\displaystyle \mathrm{Quot}_{\mathcal{O}_X^N} \rightarrow \mathrm{Coh}_{X/S},$

which sends a flat quotient ${\mathcal{O}_{X \times_S T}^N \twoheadrightarrow \mathcal{E}}$ of ${\mathcal{O}_X^N}$ to the underlying ${\mathcal{E}}$. Together, this induces a map

$\displaystyle \bigsqcup_N \mathrm{Quot}_{\mathcal{O}_X^N} \rightarrow \mathrm{Coh}_{X/S},$

which still isn’t necessarily what we want, because a coherent sheaf on ${X}$ might not be a quotient (even étale locally) of a trivial vector bundle. (This isn’t even true when ${S = \mathrm{Spec} k}$: for instance, ${\mathcal{O}(-1)}$ is not such a quotient.) In other words, this map is not a surjection.

Instead, the strategy is to define a map

$\displaystyle \bigsqcup_{N, n} \mathrm{Quot}_{\mathcal{O}_X(-n)^N} \rightarrow \mathrm{Coh}_{X/S},$

which sends a flat quotient ${\mathcal{O}_X(-n)^N \rightarrow \mathcal{E}}$ to ${\mathcal{E}}$. Zariski locally, ${\mathcal{E}}$ is a quotient of ${\mathcal{O}_X(-n)}$, for ${n \gg 0}$ 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 ${T}$, a map ${T \rightarrow \mathrm{Coh}_{X/S}}$ classifying a coherent sheaf ${\mathcal{F}}$ on ${X \times_S T}$, and analyze the ${T}$-scheme ${T \times_{\mathrm{Coh}_{X/S}} \mathrm{Quot}_{\mathcal{O}_X(-n)^N}}$. What functor does this represent? For a ${T}$-scheme ${T'}$, we have that

$\displaystyle (T \times_{\mathrm{Coh}_{X/S}} \mathrm{Quot}_{\mathcal{O}_X(-n)^N}) ( T')$

consists of the set of surjective maps ${\mathcal{O}_{X \times_S T'}^N(-n) \twoheadrightarrow \mathcal{F}_{T'}}$. In other words, the set of surjective maps

$\displaystyle \mathcal{O}_{X \times_S T'}^N \twoheadrightarrow \mathcal{F}_{T'}(n),$

or equivalently the set of maps

$\displaystyle \mathcal{O}_{T'}^N \rightarrow \pi_{T'*} (\mathcal{F}_{T'}(n))$

such that the adjoint map is surjective. (Here we have implicitly used the fact that ${\pi_* \mathcal{O}_X \simeq \mathcal{O}_S}$ universally.)

Goal: The functor that sends ${T'}$ to the set of maps ${\mathcal{O}_{T'}^N \rightarrow \pi_{T'*} (\mathcal{F}_{T'}(n)) }$ such that the adjoint is surjective should be representable by a smooth ${T}$-scheme.

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

5. The smooth cover of ${\mathrm{Coh}_{X/S}}$

Let us define a subfunctor of ${\mathrm{Quot}_{\mathcal{O}_X(-n)^N}}$ as follows.

Definition 8 ${\mathrm{Quot}^{\circ}_{\mathcal{O}_X(-n)^N}}$ sends a scheme ${T}$ to the set of flat quotients ${\mathcal{O}_{X \times_S T}(-n)^N \twoheadrightarrow \mathcal{F}}$ of ${\mathcal{O}_{X\times_S T}(-n)^N}$ such that ${H^p(X_t, \mathcal{F}(n)|_{X_t}) = 0}$ for ${p > 0}$ and such that the adjoint map

$\displaystyle \mathcal{O}_{ T}^N \rightarrow \pi_{T*}( \mathcal{F}(n)),$

is an isomorphism.

Let us note that this is literally an open subfunctor of ${\mathrm{Quot}}$. The first condition (that ${H^p(X_t, \mathcal{F}(n)|_{X_t}) = 0}$ for ${p > 0}$) is an open condition by the semicontinuity theorem. It implies that ${\pi_{T*}}$ commutes with base change, and in particular, that the second condition after the first is imposed is an open condition.

Proposition 9 The map ${\mathrm{Quot}^\circ_{\mathcal{O}_X(-n)^N} \rightarrow \mathrm{Coh}_{X/S}}$ is a smooth morphism.

Proof: In fact, we need to show that if ${T \rightarrow S}$ and ${\mathcal{F}}$ is coherent and ${T}$-flat on ${X \times_S T}$, then the associated fiber product ${T \times_{\mathrm{Coh}_{X/S}} \mathrm{Quot}^\circ_{\mathcal{O}_X(-n)^N}}$ is representable by a smooth ${T}$-scheme. Let ${U \subset T}$ be the open subscheme defined by the conditions ${H^p(X_t, \mathcal{F}(n)) = 0}$ for ${p > 0}$ and such that ${H^0(X_t, \mathcal{F}(n))}$ has rank ${N}$ and ${\pi^* \pi_* \mathcal{F}(n) \rightarrow \mathcal{F}(n)}$ is surjective. The second is an open condition, since ${\pi_{T*} ( \mathcal{F}(n))}$ is locally free on ${U}$—this is implied by the other conditions.

So then ${T \times_{\mathrm{Coh}_{X/S}} \mathrm{Quot}^\circ_{\mathcal{O}_X(-n)^N}}$ maps into ${U}$. Over ${U}$, this fibered product is in fact a ${\mathrm{GL}_N}$-torsor: it is

$\displaystyle \mathrm{Isom}(\mathcal{O}_U^N, \pi_{U*}(\mathcal{F}(n)).$

In fact, it consists of all maps ${\mathcal{O}_U^N \rightarrow \pi_{U*}(\mathcal{F}(n))}$ 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 ${U}$. $\Box$

The way we get a smooth cover of ${\mathrm{Coh}_{X/S}}$ is to take the map

$\displaystyle \bigsqcup_{N, n} \mathrm{Quot}^{\circ}_{\mathcal{O}_X(-n)^N} \rightarrow \mathrm{Coh}_{X/S},$

which we have just seen is smooth. The last thing is to see that it is surjective. Let’s say we have an object ${\mathcal{F}}$, a coherent sheaf on ${X}$ flat over ${S}$. (We would have to do this for any ${T}$ mapping to ${S}$, but the argument is the same.) But this is Serre’s theorem: for ${n \gg 0}$, ${\mathcal{F}(n)}$ has no cohomology on fibers, ${\pi_* \mathcal{F}(n)}$ is a vector bundle, and ${\pi^* \pi_* \mathcal{F}(n) \rightarrow \mathcal{F}(n)}$ is a surjection. Restricting to an open neighborhood, we can assume ${\pi_*\mathcal{F}(n)}$ is a trivial bundle of rank ${N}$: in that case, ${\mathcal{F}}$ is in the image of ${\mathrm{Quot}^{\circ}_{\mathcal{O}_X(-n)^N}}$.

Finally, this proves:

Theorem: If $\pi_* (\mathcal{O}_X) \simeq \mathcal{O}_S$ holds universally and $\pi$ is projective, then $\mathrm{Coh}_{X/S}$ is an Artin stack.