# Open Subfunctors and Big Zariski Sheaves

Setup: Fix a base scheme . Let be the category of schemes over . Write for . Define to be the category of functors from to **SETS**. As usual we have a functor :

The motivating question is, for general , when is representable, i.e. when is ?

#### Big Zariski Sheaf

Note if is a functor as above, then for every the open subsets of are also schemes over . So I can ask if restricted to the open subsets of for a sheaf on . If this holds for every , then 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 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

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

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Given a morphism of functors , and another morphism into by something representable: , the fibered product can be formed:

This is the functor that takes to in **SETS**. From this construction, for every , we get a map

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

where is open.

Given a collection of open subfunctors of there is a presheaf . Set to be the functor which for any scheme acts as the sheaf associated to the aforementioned presheaf. Since there is a map for each i, there is a natural morphism

For every , the above map is a map of sheaves. Declare to cover is the map of sheaves is surjective (i.e. on stalks) for every .

#### Results without proofs

Lemma1: Let be an collection of open subfunctors of . TFAE

- covers
- For all and maps , there exists covering of s.t. for all there is i, and s.t. .
- For all fields , the map is surjective.

Lemma2: For as above consider the two projections . Then 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 -algebras and the category of affine morphisms over . More specifically, given a qcoh sheaf of algebras , consider the functor

Where 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 , said relative spec over X of . The proof of Thm1 (which is not given here) can be used to see what really is. Hartshorne ex. II.5.17 gives an explicit construction of .

Then there were more theorems.

Thm2: If are rep. by objects of , then so is .

Which is proved by use of thm1 and:

Lemma3: If for and are open then so is

.

That’s all for functors. Next, and .

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You’re currently reading “Open Subfunctors and Big Zariski Sheaves,” an entry on Math Meandering

- Published:
- June 6, 2009 / 11:16 pm

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- Ogus Excerpts

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