Lemma 41.21.5. Let $X$ be a locally Noetherian scheme. Let $D \subset X$ be a normal crossings divisor. If $f : Y \to X$ is a smooth morphism of schemes, then the pullback $f^*D$ is a normal crossings divisor on $Y$.

** Pullback of a normal crossings divisor by a smooth morphism is a normal crossings divisor. **

**Proof.**
As $f$ is flat the pullback is defined by Divisors, Lemma 31.13.13 hence the statement makes sense. Let $q \in f^*D$ map to $p \in D$. Choose an étale morphism $U \to X$ whose image contains $p$ such that $D \times _ X U \subset U$ is a strict normal crossings divisor as in Definition 41.21.4. Set $V = Y \times _ X U$. Then $V \to Y$ is étale as a base change of $U \to X$ (Morphisms, Lemma 29.36.4) and the pullback $D \times _ X V$ is a strict normal crossings divisor on $V$ by Lemma 41.21.3. Thus we have checked the condition of Definition 41.21.4 for $q \in f^*D$ and we conclude.
$\square$

## 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Your email address will not be published.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).

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.

## Comments (0)