The Stacks project

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

  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 68.16.9. $\square$

Comments (0)

Post a comment

Your email address will not be published. Required fields are marked.

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).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0GFA. Beware of the difference between the letter 'O' and the digit '0'.