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The Stacks project

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

Lemma 41.21.3. Let X be a locally Noetherian scheme. Let D \subset X be a strict normal crossings divisor. If f : Y \to X is a smooth morphism of schemes, then the pullback f^*D is a strict 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 a regular system of parameters x_1, \ldots , x_ d \in \mathfrak m_ p and 1 \leq r \leq d as in Definition 41.21.1. Since f is smooth the local ring homomorphism \mathcal{O}_{X, p} \to \mathcal{O}_{Y, q} is flat and the fibre ring

\mathcal{O}_{Y, q}/\mathfrak m_ p \mathcal{O}_{Y, q} = \mathcal{O}_{Y_ p, q}

is a regular local ring (see for example Algebra, Lemma 10.140.3). Pick y_1, \ldots , y_ n \in \mathfrak m_ q which map to a regular system of parameters in \mathcal{O}_{Y_ p, q}. Then x_1, \ldots , x_ d, y_1, \ldots , y_ n generate the maximal ideal \mathfrak m_ q. Hence \mathcal{O}_{Y, q} is a regular local ring of dimension d + n by Algebra, Lemma 10.112.7 and x_1, \ldots , x_ d, y_1, \ldots , y_ n is a regular system of parameters. Since f^*D is cut out by x_1 \ldots x_ r in \mathcal{O}_{Y, q} we conclude that the lemma is true. \square


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