The Stacks project

Lemma 34.3.8. Let $S$ be a scheme. Let $\mathit{Sch}_{Zar}$ be a big Zariski site containing $S$. The structures $S_{Zar}$, $(\textit{Aff}/S)_{Zar}$, and $S_{affine, Zar}$ defined above are sites.

Proof. Let us show that $S_{Zar}$ is a site. It is a category with a given set of families of morphisms with fixed target. Thus we have to show properties (1), (2) and (3) of Sites, Definition 7.6.2. Since $(\mathit{Sch}/S)_{Zar}$ is a site, it suffices to prove that given any covering $\{ U_ i \to U\} $ of $(\mathit{Sch}/S)_{Zar}$ with $U \in \mathop{\mathrm{Ob}}\nolimits (S_{Zar})$ we also have $U_ i \in \mathop{\mathrm{Ob}}\nolimits (S_{Zar})$. This follows from the definitions as the composition of open immersions is an open immersion.

Let us show that $(\textit{Aff}/S)_{Zar}$ is a site. Reasoning as above, it suffices to show that the collection of standard Zariski coverings of affines satisfies properties (1), (2) and (3) of Sites, Definition 7.6.2. Let $R$ be a ring. Let $f_1, \ldots , f_ n \in R$ generate the unit ideal. For each $i \in \{ 1, \ldots , n\} $ let $g_{i1}, \ldots , g_{in_ i} \in R_{f_ i}$ be elements generating the unit ideal of $R_{f_ i}$. Write $g_{ij} = f_{ij}/f_ i^{e_{ij}}$ which is possible. After replacing $f_{ij}$ by $f_ i f_{ij}$ if necessary, we have that $D(f_{ij}) \subset D(f_ i) \cong \mathop{\mathrm{Spec}}(R_{f_ i})$ is equal to $D(g_{ij}) \subset \mathop{\mathrm{Spec}}(R_{f_ i})$. Hence we see that the family of morphisms $\{ D(g_{ij}) \to \mathop{\mathrm{Spec}}(R)\} $ is a standard Zariski covering. From these considerations it follows that (2) holds for standard Zariski coverings. We omit the verification of (1) and (3).

We omit the proof that $S_{affine, Zar}$ is a site. $\square$

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