Lemma 70.15.1. 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 70.14.1),

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 $p > 0$.

## Comments (0)