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

  1. 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

  2. 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. $\square$

Comments (0)

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 03FP. Beware of the difference between the letter 'O' and the digit '0'.



View Lemma 65.15.1 as pdf