Lemma 100.36.10. Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. The following are equivalent
$f$ is étale, and
$f$ is locally of finite presentation, flat, and unramified,
$f$ is locally of finite presentation, flat, and its diagonal is étale.
Proof. The equivalence of (2) and (3) follows immediately from Lemma 100.36.9. Thus in each case the morphism $f$ is DM. Then we can choose Then we can choose algebraic spaces $U$, $V$, a smooth surjective morphism $V \to \mathcal{Y}$ and a surjective étale morphism $U \to \mathcal{X} \times _\mathcal {Y} V$ (Lemma 100.21.7). To finish the proof we have to show that $U \to V$ is étale if and only if it is locally of finite presentation, flat, and unramified. This follows from Morphisms of Spaces, Lemma 66.39.12 (and the more trivial Morphisms of Spaces, Lemmas 66.39.10, 66.39.8, and 66.39.7). $\square$
Your email address will not be published.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)