Open Subfunctors and Big Zariski Sheaves

Setup: Fix a base scheme $\ S$.  Let $\ \mbox{Sch}_S$  be the category of schemes over $\ S$.  Write $\ X/S$ for $\ X \in \mbox{Sch}_S$. Define $\ \mathcal{P}(\mbox{Sch}_S)$ to be the category of functors from $\ \mbox{Sch}_S^{op}$ to SETS.  As usual we have a functor $\ h.$:

$\ \mbox{Sch}_S \to \mathcal{P}(\mbox{Sch}_S)$

$\ X \mapsto \hom(-, X)$

The motivating question is, for general $\ F \in \mathcal{P}(\mbox{Sch}_S)$, when is $\ F$ representable, i.e. when is $\ F \cong \hom(-, X)$?

Big Zariski Sheaf

Note if $\ F$ is a functor as above, then for every $\ X/S$ the open subsets of $\ X$ are also schemes over $\ S$.  So I can ask if $\ F$ restricted to the open subsets of $\ X$ for a sheaf on $\ X$.  If this holds for every $\ X$, then $\ F$ is a big Zariski sheaf.

Representable functors (in this context) are sheafs.  This is basically the statement that morphisms of schemes glue.  Thus a necessary condition for functor to be representable is that it be a sheaf.

If $\ F$ is functor as a above that is a sheaf, then its sheaf structure is determined by its restriction to affine schemes.

Open Subfunctors and the main claim

I should define what an open subfunctor so that this makes sense:

THM1

$\ F$ is rep. iff $\ F$ is a sheaf and is covered by open representable subfunctors.

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Given a morphism of functors $\ F' \to F$, and another morphism into $\ F$ by something representable: $\ \hom(-,X) \to F$, the fibered product can be formed:

$\ F' \times_F \hom(-,X)$

This is the functor that takes $\ Y/S$ to $\ F'(Y)\times_{F(Y)} \hom(Y,X)$ in SETS.  From this construction, for every $\ Y/S$, we get a map

$\ F' \times_F \hom(-,Y) \to \hom(-,Y)$

$\ F'$ is declared to be an open subfunctor if for all $\ Y$ the above map is represented by a open subset of $\ Y$, i.e.

$\ F' \times_F \hom(-,Y) \cong \hom(-, U)$

where $\ U\subset Y$ is open.

Given a collection $\ V = \{F_i\}$ of open subfunctors of $\ F$ there is a presheaf $\ U \mapsto \sqcup F_i(U)$.  Set $\ F_V$ to be the functor which for any scheme $\ T/S$ acts as the sheaf associated to the aforementioned presheaf. Since there is a map $\ F_i \to F$ for each i, there is a natural morphism

$\ F_V \to F$

For every $\ T/S$, the above map is a map of sheaves.  Declare $\ V$ to cover $\ F$ is the map of sheaves is surjective (i.e. on stalks) for every $\ T/S$.

Results without proofs

Lemma1: Let $\ V$ be an collection of open subfunctors of $\ F$.  TFAE

1. $\ V$ covers $\ F$
2. For all $\ T/S$ and maps $\ f\colon \hom(-,T) \to F$, there exists covering $\ \mathcal{V}$ of $\ T$ s.t. for all $\ U \in \mathcal{V}$  there is i, and $\ f_{i,U}\colon \hom(-,U) \to F_i$ s.t. $\ f_{i,U} = f|_U$.
3. For all fields $\ K$, the map $\ \cup F_i(\mbox{Spec} K) \to F(\mbox{Spec} K)$ is surjective.

Lemma2: For $\ F,F_V$ as above consider the two projections $\ F_V\times_F F_V \rightrightarrows F_V$.  Then $\ F_V \to F$ is the coequalizer of the previous two maps in the category of (functor) sheaves.

These results are the basic tools to prove the THM1 above.  As an application, Ogus used this theorem to show the equivalence of the category of qcoh $\ \mathcal{O}_X$-algebras and the category of affine morphisms over $\ X$.  More specifically, given a qcoh sheaf of $\mathcal{O}_X$ algebras $\mathcal{A}$, consider the functor

$\mbox{Sch}_X \to \mbox{SETS}$

$T \mapsto \hom_{\mathcal{O}_X}(\mathcal{A}, p_*\mathcal{O}_T)$

Where $p \colon T \to X$ is the map to the base.  Then the assumptions of Thm1 can be checked to conclude that the above functor is represented by a scheme which denoted $\mbox{Spec}_X \mathcal{A}$, said relative spec over X of $\mathcal{A}$.  The proof of Thm1 (which is not given here) can be used to see what $\mbox{Spec}_X \mathcal{A}$ really is.  Hartshorne ex. II.5.17 gives an explicit construction of $\mbox{Spec}_X \mathcal{A}$.

Then there were more theorems.

Thm2: If $\ F_1, F_2, F$ are rep. by objects of $\ \mbox{Sch}_S$, then so is $\ F_1 \times_F F_2$.

Which is proved by use of thm1 and:

Lemma3: If $\ U_i \to F_i$ for $\ i = 1,2$ and $\ V \to \hom(-, S)$ are open then so is

$\ U_1 \times_V U_2 \to F_1 \times_{\hom(-,S)} F_2$.

That’s all for functors.  Next, $\ \mathbb{V}E$ and $\ \mathbb{P}E$.