The Stacks project

Lemma 69.21.1. 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

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

  2. 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

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

Proof. The implication (1) $\Rightarrow $ (2) follows from Cohomology of Schemes, Lemma 30.17.1. The implication (2) $\Rightarrow $ (3) is trivial. The implication (3) $\Rightarrow $ (1) is Lemma 69.16.9. $\square$

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