## 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

$D$ is an effective Cartier divisor, and

$D \to Y$ 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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