The Stacks project

Definition 34.3.5. A big Zariski site is any site $\mathit{Sch}_{Zar}$ as in Sites, Definition 7.6.2 constructed as follows:

  1. Choose any set of schemes $S_0$, and any set of Zariski coverings $\text{Cov}_0$ among these schemes.

  2. As underlying category of $\mathit{Sch}_{Zar}$ take any category $\mathit{Sch}_\alpha $ constructed as in Sets, Lemma 3.9.2 starting with the set $S_0$.

  3. As coverings of $\mathit{Sch}_{Zar}$ choose any set of coverings as in Sets, Lemma 3.11.1 starting with the category $\mathit{Sch}_\alpha $ and the class of Zariski coverings, and the set $\text{Cov}_0$ chosen above.

Comments (3)

Comment #79 by Keenan Kidwell on

Probably I'm misunderstanding something, but it seems to me that one could choose the empty set of coverings in the first \item and end up with the same topos (by the result in the Sites chapter showing that the topos is independent of the way we shrink a proper class of coverings satisfying the axioms of a site to a set of coverings). Is this true, or is there something that prohibits it? I admit I haven't read the relevant proof in the Set Theory chapter where a set of coverings is constructed from a proper class of coverings and an initial set , but no matter what is, the set constructed still satisfies the property that every covering in the proper class is combinatorially equivalent to one in the set, and this seems to be the property that matters for the resulting topos.

Comment #86 by on

Yes, that is completely correct. The reason for stating it the way it is done here is that the reader may have (for example) an a priori given collection of cocycle calculations which uses a particular set of coverings and they may want to make sure that those particular coverings are part of the site constructed here.

Comment #87 by Keenan Kidwell on

Got it. Thanks for explaining.

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