The Stacks project

Lemma 34.4.9. Let $S$ be a scheme. Let $\mathit{Sch}_{\acute{e}tale}$ be a big étale site containing $S$. Both $S_{\acute{e}tale}$ and $(\textit{Aff}/S)_{\acute{e}tale}$ are sites.

Proof. Let us show that $S_{\acute{e}tale}$ 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 $(\mathit{Sch}/S)_{\acute{e}tale}$ is a site, it suffices to prove that given any covering $\{ U_ i \to U\} $ of $(\mathit{Sch}/S)_{\acute{e}tale}$ with $U \in \mathop{\mathrm{Ob}}\nolimits (S_{\acute{e}tale})$ we also have $U_ i \in \mathop{\mathrm{Ob}}\nolimits (S_{\acute{e}tale})$. This follows from the definitions as the composition of étale morphisms is an étale morphism.

Let us show that $(\textit{Aff}/S)_{\acute{e}tale}$ is a site. Reasoning as above, it suffices to show that the collection of standard étale coverings of affines satisfies properties (1), (2) and (3) of Sites, Definition 7.6.2. This is clear since for example, given a standard étale covering $\{ T_ i \to T\} _{i\in I}$ and for each $i$ we have a standard étale covering $\{ T_{ij} \to T_ i\} _{j\in J_ i}$, then $\{ T_{ij} \to T\} _{i \in I, j\in J_ i}$ is a standard étale covering because $\bigcup _{i\in I} J_ i$ is finite and each $T_{ij}$ is affine. $\square$


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