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
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Comment #9566 by Erhard Neher on
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