The Stacks project

Lemma 34.7.9. Let $S$ be a scheme. Let $\mathit{Sch}_{fppf}$ be a big fppf site containing $S$. Then $(\textit{Aff}/S)_{fppf}$ is a site.

Proof. Let us show that $(\textit{Aff}/S)_{fppf}$ is a site. Reasoning as in the proof of Lemma 34.4.9 it suffices to show that the collection of standard fppf coverings of affines satisfies properties (1), (2) and (3) of Sites, Definition 7.6.2. This is clear since for example, given a standard fppf covering $\{ T_ i \to T\} _{i\in I}$ and for each $i$ we have a standard fppf 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 fppf covering because $\bigcup _{i\in I} J_ i$ is finite and each $T_{ij}$ is affine. $\square$

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