The Stacks project

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

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 $Y$.

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. $\square$

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