The Stacks project

Lemma 34.7.11. Let $S$ be a scheme. Let $\mathit{Sch}_{fppf}$ be a big fppf site containing $S$. The functor $(\textit{Aff}/S)_{fppf} \to (\mathit{Sch}/S)_{fppf}$ is cocontinuous and induces an equivalence of topoi from $\mathop{\mathit{Sh}}\nolimits ((\textit{Aff}/S)_{fppf})$ to $\mathop{\mathit{Sh}}\nolimits ((\mathit{Sch}/S)_{fppf})$.

Proof. The notion of a special cocontinuous functor is introduced in Sites, Definition 7.29.2. Thus we have to verify assumptions (1) – (5) of Sites, Lemma 7.29.1. Denote the inclusion functor $u : (\textit{Aff}/S)_{fppf} \to (\mathit{Sch}/S)_{fppf}$. Being cocontinuous just means that any fppf covering of $T/S$, $T$ affine, can be refined by a standard fppf covering of $T$. This is the content of Lemma 34.7.4. Hence (1) holds. We see $u$ is continuous simply because a standard fppf covering is a fppf covering. Hence (2) holds. Parts (3) and (4) follow immediately from the fact that $u$ is fully faithful. And finally condition (5) follows from the fact that every scheme has an affine open covering. $\square$

Comments (1)

Comment #9566 by Erhard Neher on

The statement of Lemma 021V could be more precise: the proof of the lemma shows that the functor is special cocontinuous, not only cocontinuous. The same remark applies to Lemma 0DBP for the ph topology

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  • 2 comment(s) on Section 34.7: The fppf topology

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