Lemma 34.12.1. Any set of big Zariski sites is contained in a common big Zariski site. The same is true, mutatis mutandis, for big fppf and big étale sites.
34.12 Change of big sites
In this section we explain what happens on changing the big Zariski/fppf/étale sites.
Let $\tau , \tau ' \in \{ Zariski, {\acute{e}tale}, smooth, syntomic, fppf\} $. Given two big sites $\mathit{Sch}_\tau $ and $\mathit{Sch}'_{\tau '}$ we say that $\mathit{Sch}_\tau $ is contained in $\mathit{Sch}'_{\tau '}$ if $\mathop{\mathrm{Ob}}\nolimits (\mathit{Sch}_\tau ) \subset \mathop{\mathrm{Ob}}\nolimits (\mathit{Sch}'_{\tau '})$ and $\text{Cov}(\mathit{Sch}_\tau ) \subset \text{Cov}(\mathit{Sch}'_{\tau '})$. In this case $\tau $ is stronger than $\tau '$, for example, no fppf site can be contained in an étale site.
Proof. This is true because the union of a set of sets is a set, and the constructions in Sets, Lemmas 3.9.2 and 3.11.1 allow one to start with any initially given set of schemes and coverings. $\square$
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 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 with $f \circ g \cong \text{id}$.
\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*}
\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*}
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. $\square$
Discussion: The functor $g^{-1} = f_*$ is simply the restriction functor which associates to a sheaf $\mathcal{G}$ on $\mathit{Sch}'_\tau $ the restriction $\mathcal{G}|_{\mathit{Sch}_\tau }$. Hence this lemma simply says that given any sheaf of sets $\mathcal{F}$ on $\mathit{Sch}_\tau $ there exists a canonical sheaf $\mathcal{F}'$ on $\mathit{Sch}'_\tau $ such that $\mathcal{F}|_{\mathit{Sch}'_\tau } = \mathcal{F}'$. In fact the sheaf $\mathcal{F}'$ has the following description: it is the sheafification of the presheaf
\[ \mathit{Sch}'_\tau \longrightarrow \textit{Sets}, \quad V \longmapsto \mathop{\mathrm{colim}}\nolimits _{V \to U} \mathcal{F}(U) \]
where $U$ is an object of $\mathit{Sch}_\tau $. This is true because $\mathcal{F}' = f^{-1}\mathcal{F} = (u_ p\mathcal{F})^\# $ according to Sites, Lemmas 7.21.5 and 7.21.8.
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