The Stacks project

Lemma 34.5.9. Let $S$ be a scheme. Let $\mathit{Sch}_{smooth}$ be a big smooth site containing $S$. The functor $(\textit{Aff}/S)_{smooth} \to (\mathit{Sch}/S)_{smooth}$ is special cocontinuous and induces an equivalence of topoi from $\mathop{\mathit{Sh}}\nolimits ((\textit{Aff}/S)_{smooth})$ to $\mathop{\mathit{Sh}}\nolimits ((\mathit{Sch}/S)_{smooth})$.

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 $u : (\textit{Aff}/S)_{smooth} \to (\mathit{Sch}/S)_{smooth}$. Being cocontinuous just means that any smooth covering of $T/S$, $T$ affine, can be refined by a standard smooth covering of $T$. This is the content of Lemma 34.5.4. Hence (1) holds. We see $u$ is continuous simply because a standard smooth covering is a smooth 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 (2)

Comment #7124 by Zhenbing Shang on

Here is a typo, "Let be a big smooth..." should be .

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