The Stacks project

Lemma 64.11.6. Let $S$ be a scheme contained in $\mathit{Sch}_{fppf}$. Let $F, G$ be algebraic spaces over $S$. Let $a : F \to G$ be a morphism. Given any $V \in \mathop{\mathrm{Ob}}\nolimits ((\mathit{Sch}/S)_{fppf})$ and a surjective étale morphism $q : V \to G$ there exists a $U \in \mathop{\mathrm{Ob}}\nolimits ((\mathit{Sch}/S)_{fppf})$ and a commutative diagram

\[ \xymatrix{ U \ar[d]_ p \ar[r]_\alpha & V \ar[d]^ q \\ F \ar[r]^ a & G } \]

with $p$ surjective and étale.

Proof. First choose $W \in \mathop{\mathrm{Ob}}\nolimits ((\mathit{Sch}/S)_{fppf})$ with surjective étale morphism $W \to F$. Next, put $U = W \times _ G V$. Since $G$ is an algebraic space we see that $U$ is isomorphic to an object of $(\mathit{Sch}/S)_{fppf}$. As $q$ is surjective étale, we see that $U \to W$ is surjective étale (see Lemma 64.5.5). Thus $U \to F$ is surjective étale as a composition of surjective étale morphisms (see Lemma 64.5.4). $\square$

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Your email address will not be published.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 02X1. Beware of the difference between the letter 'O' and the digit '0'.