The Stacks project

Lemma 96.8.1. Let $S$ be a scheme. Let $\mathcal{X}$ be a category fibred in groupoids over $(\mathit{Sch}/S)$. Assume $\mathcal{X}$ is representable by a scheme $X$. For $\tau \in \{ Zar,\linebreak[0] {\acute{e}tale},\linebreak[0] smooth,\linebreak[0] syntomic,\linebreak[0] fppf\} $ there is a canonical equivalence

\[ (\mathcal{X}_\tau , \mathcal{O}_\mathcal {X}) = ((\mathit{Sch}/X)_\tau , \mathcal{O}_ X) \]

of ringed sites.

Proof. This follows by choosing an equivalence $(\mathit{Sch}/X)_\tau \to \mathcal{X}$ of categories fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$ and using the functoriality of the construction $\mathcal{X} \leadsto \mathcal{X}_\tau $. $\square$

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