The Stacks project

Lemma 70.9.1. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $D \subset X$ be a closed subspace. Assume

  1. $D$ is an effective Cartier divisor, and

  2. $D \to Y$ is a flat morphism.

Then for every morphism of schemes $g : Y' \to Y$ the pullback $(g')^{-1}D$ is an effective Cartier divisor on $X' = Y' \times _ Y X$ where $g' : X' \to X$ is the projection.

Proof. Using Lemma 70.6.2 the property of being an effective Cartier divisor is ├ętale local. Thus this lemmma immediately reduces to the case of schemes which is Divisors, Lemma 31.18.1. $\square$

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