[III Proposition 2.6.1, EGA]

Lemma 30.17.1. Let $R$ be a Noetherian ring. Let $f : X \to \mathop{\mathrm{Spec}}(R)$ be a proper morphism. Let $\mathcal{L}$ be an invertible $\mathcal{O}_ X$-module. The following are equivalent

1. $\mathcal{L}$ is ample on $X$ (this is equivalent to many other things, see Properties, Proposition 28.26.13 and Morphisms, Lemma 29.39.4),

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 quasi-coherent sheaf of ideals $\mathcal{I} \subset \mathcal{O}_ X$, there exists an $n \geq 1$ such that $H^1(X, \mathcal{I} \otimes \mathcal{L}^{\otimes n}) = 0$.

Proof. The implication (1) $\Rightarrow$ (2) follows from Lemma 30.16.1. The implication (2) $\Rightarrow$ (3) is trivial. The implication (3) $\Rightarrow$ (1) is Lemma 30.3.3. $\square$

Comment #2704 by Zhang on

Reference: EGA, Chapitre III "Étude cohomologique des faisceaux cohérents", Proposition (2.6.1)

Comment #7415 by Arnab Kundu on

Is it possible to remove the noetherian hypothesis on $\spec R$ by a limit argument of schemes when we replace the absolute ampleness condition by a relative one? It is probably true that the higher direct image cohomology commutes with limits of schemes.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).