The Stacks project

[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. $\square$

Comments (3)

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 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.

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