Lemma 63.16.6. Let $S$ be a scheme. Let $X, Y, Z$ be algebraic spaces. Let $g : X \to Z$, $h : Y \to Z$ be étale morphisms and let $f : X \to Y$ be a morphism such that $h \circ f = g$. Then $f$ is étale.
Proof. Choose a commutative diagram
\[ \xymatrix{ U \ar[d] \ar[r]_\chi & V \ar[d] \\ X \ar[r] & Y } \]
where $U \to X$ and $V \to Y$ are surjective and étale, see Spaces, Lemma 62.11.6. By assumption the morphisms $\varphi : U \to X \to Z$ and $\psi : V \to Y \to Z$ are étale. Moreover, $\psi \circ \chi = \varphi $ by our assumption on $f, g, h$. Hence $U \to V$ is étale by Lemma 63.16.1 part (2). $\square$
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Your email address will not be published.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)