The Stacks project

Lemma 38.34.15. Let $S$ be a scheme. Let $\mathit{Sch}_ h$ be a big h site containing $S$. The functor $(\textit{Aff}/S)_ h \to (\mathit{Sch}/S)_ h$ is cocontinuous and induces an equivalence of topoi from $\mathop{\mathit{Sh}}\nolimits ((\textit{Aff}/S)_ h)$ to $\mathop{\mathit{Sh}}\nolimits ((\mathit{Sch}/S)_ h)$.

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 $u : (\textit{Aff}/S)_ h \to (\mathit{Sch}/S)_ h$. Being cocontinuous follows because any h covering of $T/S$, $T$ affine, can be refined by a standard h covering for example by Lemma 38.34.4. Hence (1) holds. We see $u$ is continuous simply because a standard h covering is a h 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 (which is a h covering). $\square$

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