Lemma 65.15.1 (03FP)—The Stacks project

Lemma 65.15.1. Suppose given big sites $\mathit{Sch}_{fppf}$ and $\mathit{Sch}'_{fppf}$ . Assume that $\mathit{Sch}_{fppf}$ is contained in $\mathit{Sch}'_{fppf}$ , see Topologies, Section 34.12. Let $S$ be an object of $\mathit{Sch}_{fppf}$ . Let

$\begin{align*} g : \mathop{\mathit{Sh}}\nolimits ((\mathit{Sch}/S)_{fppf}) \longrightarrow \mathop{\mathit{Sh}}\nolimits ((\mathit{Sch}'/S)_{fppf}), \\ f : \mathop{\mathit{Sh}}\nolimits ((\mathit{Sch}'/S)_{fppf}) \longrightarrow \mathop{\mathit{Sh}}\nolimits ((\mathit{Sch}/S)_{fppf}) \end{align*}$

be the morphisms of topoi of Topologies, Lemma 34.12.2. Let $F$ be a sheaf of sets on $(\mathit{Sch}/S)_{fppf}$ . Then

if $F$ is representable by a scheme $X \in \mathop{\mathrm{Ob}}\nolimits ((\mathit{Sch}/S)_{fppf})$ over $S$ , then $f^{-1}F$ is representable too, in fact it is representable by the same scheme $X$ , now viewed as an object of $(\mathit{Sch}'/S)_{fppf}$ , and
if $F$ is an algebraic space over $S$ , then $f^{-1}F$ is an algebraic space over $S$ also.

Proof. Let $X \in \mathop{\mathrm{Ob}}\nolimits ((\mathit{Sch}/S)_{fppf})$ . Let us write $h_ X$ for the representable sheaf on $(\mathit{Sch}/S)_{fppf}$ associated to $X$ , and $h'_ X$ for the representable sheaf on $(\mathit{Sch}'/S)_{fppf}$ associated to $X$ . By the description of $f^{-1}$ in Topologies, Section 34.12 we see that $f^{-1}h_ X = h'_ X$ . This proves (1).

Next, suppose that $F$ is an algebraic space over $S$ . By Lemma 65.9.1 this means that $F = h_ U/h_ R$ for some étale equivalence relation $R \to U \times _ S U$ in $(\mathit{Sch}/S)_{fppf}$ . Since $f^{-1}$ is an exact functor we conclude that $f^{-1}F = h'_ U/h'_ R$ . Hence $f^{-1}F$ is an algebraic space over $S$ by Theorem 65.10.5. $\square$

The Stacks project

Comments (0)

Post a comment