## 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).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 020M. Beware of the difference between the letter 'O' and the digit '0'.