## 34.2 The general procedure

In this section we explain a general procedure for producing the sites we will be working with. Suppose we want to study sheaves over schemes with respect to some topology $\tau$. In order to get a site, as in Sites, Definition 7.6.2, of schemes with that topology we have to do some work. Namely, we cannot simply say “consider all schemes with the Zariski topology” since that would give a “big” category. Instead, in each section of this chapter we will proceed as follows:

1. We define a class $\text{Cov}_\tau$ of coverings of schemes satisfying the axioms of Sites, Definition 7.6.2. It will always be the case that a Zariski open covering of a scheme is a covering for $\tau$.

2. We single out a notion of standard $\tau$-covering within the category of affine schemes.

3. We define what is an “absolute” big $\tau$-site $\mathit{Sch}_\tau$. These are the sites one gets by appropriately choosing a set of schemes and a set of coverings.

4. For any object $S$ of $\mathit{Sch}_\tau$ we define the big $\tau$-site $(\mathit{Sch}/S)_\tau$ and for suitable $\tau$ the small1 $\tau$-site $S_\tau$.

5. In addition there is a site $(\textit{Aff}/S)_\tau$ using the notion of standard $\tau$-covering of affines2 whose category of sheaves is equivalent to the category of sheaves on $(\mathit{Sch}/S)_\tau$.

The above is a little clumsy in that we do not end up with a canonical choice for the big $\tau$-site of a scheme, or even the small $\tau$-site of a scheme. If you are willing to ignore set theoretic difficulties, then you can work with classes and end up with canonical big and small sites...

[1] The words big and small here do not relate to bigness/smallness of the corresponding categories.
[2] In the case of the ph topology we deviate very slightly from this approach, see Definition 34.8.11 and the surrounding discussion.

Comment #2588 by Ingo Blechschmidt on

I have seen people define the big toposes not using the site $\mathrm{Aff}/S$ (consisting of $S$-schemes which are affine as absolute schemes) but by using the site of affine $S$-schemes ($S$-schemes whose structural morphism to $S$ is affine). The resulting sheaf toposes are equivalent in case that the diagonal morphism $S \to S \times S$ is affine, since in this case morphisms of the form $\operatorname{Spec} A \to S$ are affine. If the diagonal is not affine, then the two toposes probably differ.

Should we add a remark along these lines?

Comment #2620 by on

Hi Ingo Blechschmidt! OK, I am trying to become more like my idol Linus Torvalds. As you may know he isn't the nicest person in the world. In that spirit, I would just say: the people who do that are wrong.

Comment #3042 by Kannappan on

If not for your remark, I'd have understood "affine U/S" used in the subsequent sections to mean the structure map U --> S is affine. To be sure, are the objects of the category (Aff/S) just U --> S with U affine? Thank you in advance!

Comment #3043 by Kannappan on

If not for your remark, I'd have understood "affine U/S" used in the subsequent sections to mean the structure map U --> S is affine. To be sure, are the objects of the category (Aff/S) just U --> S with U affine? Thank you in advance! (Yes. And some of the proofs clue in on what's going on.)

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).