Lemma 53.20.7. Let $S' \to S$ be an étale morphism of schemes. Let $f : X \to S'$ be at-worst-nodal of relative dimension $1$. Then the composition $g : X \to S$ is at-worst-nodal of relative dimension $1$.
Proof. Observe that $g$ is flat and locally of finite presentation as a composition of morphisms which are flat and locally of finite presentation (use Morphisms, Lemmas 29.36.11, 29.36.12, 29.21.3, and 29.25.6). Thus it suffices to prove the fibres of $g$ have at-worst-nodal singularities. This follows from Lemma 53.19.14 and the analogous result for smooth points. $\square$
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Your email address will not be published.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.