The Stacks project

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...