Lemma 71.15.1. Let $R$ be a Noetherian ring. Let $X$ be an algebraic space over $R$ such that the structure morphism $f : X \to \mathop{\mathrm{Spec}}(R)$ is proper. Let $\mathcal{L}$ be an invertible $\mathcal{O}_ X$-module. The following are equivalent
$\mathcal{L}$ is ample on $X/R$ (Definition 71.14.1),
for every coherent $\mathcal{O}_ X$-module $\mathcal{F}$ there exists an $n_0 \geq 0$ such that $H^ p(X, \mathcal{F} \otimes \mathcal{L}^{\otimes n}) = 0$ for all $n \geq n_0$ and $p > 0$.
Proof. The implication (1) $\Rightarrow $ (2) follows from Cohomology of Schemes, Lemma 30.16.1 because assumption (1) implies that $X$ is a scheme. The implication (2) $\Rightarrow $ (1) is Cohomology of Spaces, Lemma 69.16.9. $\square$
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