The Stacks project

Lemma 34.12.2. Let $\tau \in \{ Zariski, {\acute{e}tale}, smooth, syntomic, fppf\} $. Suppose given big sites $\mathit{Sch}_\tau $ and $\mathit{Sch}'_\tau $. Assume that $\mathit{Sch}_\tau $ is contained in $\mathit{Sch}'_\tau $. The inclusion functor $\mathit{Sch}_\tau \to \mathit{Sch}'_\tau $ satisfies the assumptions of Sites, Lemma 7.21.8. There are morphisms of topoi

\begin{eqnarray*} g : \mathop{\mathit{Sh}}\nolimits (\mathit{Sch}_\tau ) & \longrightarrow & \mathop{\mathit{Sh}}\nolimits (\mathit{Sch}'_\tau ) \\ f : \mathop{\mathit{Sh}}\nolimits (\mathit{Sch}'_\tau ) & \longrightarrow & \mathop{\mathit{Sh}}\nolimits (\mathit{Sch}_\tau ) \end{eqnarray*}

such that $f \circ g \cong \text{id}$. For any object $S$ of $\mathit{Sch}_\tau $ the inclusion functor $(\mathit{Sch}/S)_\tau \to (\mathit{Sch}'/S)_\tau $ satisfies the assumptions of Sites, Lemma 7.21.8 also. Hence similarly we obtain morphisms

\begin{eqnarray*} g : \mathop{\mathit{Sh}}\nolimits ((\mathit{Sch}/S)_\tau ) & \longrightarrow & \mathop{\mathit{Sh}}\nolimits ((\mathit{Sch}'/S)_\tau ) \\ f : \mathop{\mathit{Sh}}\nolimits ((\mathit{Sch}'/S)_\tau ) & \longrightarrow & \mathop{\mathit{Sh}}\nolimits ((\mathit{Sch}/S)_\tau ) \end{eqnarray*}

with $f \circ g \cong \text{id}$.

Proof. Assumptions (b), (c), and (e) of Sites, Lemma 7.21.8 are immediate for the functors $\mathit{Sch}_\tau \to \mathit{Sch}'_\tau $ and $(\mathit{Sch}/S)_\tau \to (\mathit{Sch}'/S)_\tau $. Property (a) holds by Lemma 34.3.6, 34.4.7, 34.5.7, 34.6.7, or 34.7.7. Property (d) holds because fibre products in the categories $\mathit{Sch}_\tau $, $\mathit{Sch}'_\tau $ exist and are compatible with fibre products in the category of schemes. $\square$

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