The Stacks project

Lemma 34.8.12. Let $S$ be a scheme. Let $\mathit{Sch}_{ph}$ be a big ph site containing $S$. Then $(\textit{Aff}/S)_{ph}$ is a site.

Proof. Reasoning as in the proof of Lemma 34.4.9 it suffices to show that the collection of finite ph coverings $\{ U_ i \to U\} $ with $U$, $U_ i$ affine satisfies properties (1), (2) and (3) of Sites, Definition 7.6.2. This is clear since for example, given a finite ph covering $\{ T_ i \to T\} _{i\in I}$ with $T_ i, T$ affine, and for each $i$ a finite ph covering $\{ T_{ij} \to T_ i\} _{j\in J_ i}$ with $T_{ij}$ affine , then $\{ T_{ij} \to T\} _{i \in I, j\in J_ i}$ is a ph covering (Lemma 34.8.8), $\bigcup _{i\in I} J_ i$ is finite and each $T_{ij}$ is affine. $\square$

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