The Stacks project

Lemma 71.9.1. 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 71.6.2 the property of being an effective Cartier divisor is étale local. Thus this lemmma immediately reduces to the case of schemes which is Divisors, Lemma 31.18.1. $\square$

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