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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.

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.

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

\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. $\square$

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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