Lemma 68.21.1. Let $R$ be a Noetherian ring. Let $X$ be a proper algebraic space over $R$. Let $\mathcal{L}$ be an invertible $\mathcal{O}_ X$-module. The following are equivalent

$X$ is a scheme and $\mathcal{L}$ is ample on $X$,

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 $p > 0$, and

for every coherent $\mathcal{O}_ X$-module $\mathcal{F}$ there exists an $n \geq 1$ such that $H^1(X, \mathcal{F} \otimes \mathcal{L}^{\otimes n}) = 0$.

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