The Stacks project

Lemma 70.14.2. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $\mathcal{L}$ be an invertible $\mathcal{O}_ X$-module. Assume $Y$ is a scheme. The following are equivalent

  1. $\mathcal{L}$ is ample on $X/Y$ in the sense of Definition 70.14.1, and

  2. $X$ is a scheme and $\mathcal{L}$ is ample on $X/Y$ in the sense of Morphisms, Definition 29.37.1.

Proof. This follows from the definitions and Morphisms, Lemma 29.37.9 (which says that being relatively ample for schemes is preserved under base change). $\square$

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