The Stacks project

96.8 Representable categories

In this short section we compare our definitions with what happens in case the algebraic stacks in question are representable.

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 $. $\square$

Lemma 96.8.2. Let $S$ be a scheme. Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of categories fibred in groupoids over $S$. Assume $\mathcal{X}$, $\mathcal{Y}$ are representable by schemes $X$, $Y$. Let $f : X \to Y$ be the morphism of schemes corresponding to $f$. For $\tau \in \{ Zar,\linebreak[0] {\acute{e}tale},\linebreak[0] smooth,\linebreak[0] syntomic,\linebreak[0] fppf\} $ the morphism of ringed topoi $f : (\mathop{\mathit{Sh}}\nolimits (\mathcal{X}_\tau ), \mathcal{O}_\mathcal {X}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{Y}_\tau ), \mathcal{O}_\mathcal {Y})$ agrees with the morphism of ringed topoi $f : (\mathop{\mathit{Sh}}\nolimits ((\mathit{Sch}/X)_\tau ), \mathcal{O}_ X) \to (\mathop{\mathit{Sh}}\nolimits ((\mathit{Sch}/Y)_\tau ), \mathcal{O}_ Y)$ via the identifications of Lemma 96.8.1.

Proof. Follows by unwinding the definitions. $\square$

Comments (2)

Comment #2259 by Neeraj Deshmukh on

There seems to be a minor typo in lemma 80.8.2. The morphism f is given on the same ringed topos.

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