The Stacks project

70.9 Relative effective Cartier divisors

The following lemma shows that an effective Cartier divisor which is flat over the base is really a “family of effective Cartier divisors” over the base. For example the restriction to any fibre is an effective Cartier divisor.

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 étale local. Thus this lemmma immediately reduces to the case of schemes which is Divisors, Lemma 31.18.1. $\square$

This lemma is the motivation for the following definition.

Definition 70.9.2. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. A relative effective Cartier divisor on $X/Y$ is an effective Cartier divisor $D \subset X$ such that $D \to Y$ is a flat morphism of algebraic spaces.

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