Lemma 70.14.6. 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. The following are equivalent

$\mathcal{L}$ is ample on $X/Y$,

for every scheme $Z$ and every morphism $Z \to Y$ the algebraic space $X_ Z = Z \times _ Y X$ is a scheme and the pullback $\mathcal{L}_ Z$ is ample on $X_ Z/Z$,

for every affine scheme $Z$ and every morphism $Z \to Y$ the algebraic space $X_ Z = Z \times _ Y X$ is a scheme and the pullback $\mathcal{L}_ Z$ is ample on $X_ Z/Z$,

there exists a scheme $V$ and a surjective étale morphism $V \to Y$ such that the algebraic space $X_ V = V \times _ Y X$ is a scheme and the pullback $\mathcal{L}_ V$ is ample on $X_ V/V$.

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