Lemma 97.4.1. Let $\mathcal{X} \to \mathcal{Y} \to \mathcal{Z}$ be $1$-morphisms of categories fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$. If $\mathcal{X} \to \mathcal{Z}$ and $\mathcal{Y} \to \mathcal{Z}$ are representable by algebraic spaces and étale so is $\mathcal{X} \to \mathcal{Y}$.
Proof. Let $\mathcal{U}$ be a representable category fibred in groupoids over $S$. Let $f : \mathcal{U} \to \mathcal{Y}$ be a $1$-morphism. We have to show that $\mathcal{X} \times _\mathcal {Y} \mathcal{U}$ is representable by an algebraic space and étale over $\mathcal{U}$. Consider the composition $h : \mathcal{U} \to \mathcal{Z}$. Then
\[ \mathcal{X} \times _\mathcal {Z} \mathcal{U} \longrightarrow \mathcal{Y} \times _\mathcal {Z} \mathcal{U} \]
is a $1$-morphism between categories fibres in groupoids which are both representable by algebraic spaces and both étale over $\mathcal{U}$. Hence by Properties of Spaces, Lemma 66.16.6 this is represented by an étale morphism of algebraic spaces. Finally, we obtain the result we want as the morphism $f$ induces a morphism $\mathcal{U} \to \mathcal{Y} \times _\mathcal {Z} \mathcal{U}$ and we have
\[ \mathcal{X} \times _\mathcal {Y} \mathcal{U} = (\mathcal{X} \times _\mathcal {Z} \mathcal{U}) \times _{(\mathcal{Y} \times _\mathcal {Z} \mathcal{U})} \mathcal{U}. \]
$\square$
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Your email address will not be published.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).
All contributions are licensed under the GNU Free Documentation License.
Comments (0)