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
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