The Stacks project

Lemma 30.17.7. Let $X$ be a scheme. Let $\mathcal{L}$ be an ample invertible $\mathcal{O}_ X$-module. Let $n_0$ be an integer. If $H^ p(X, \mathcal{L}^{\otimes -n}) = 0$ for $n \geq n_0$ and $p > 0$, then $X$ is affine.

Proof. We claim $H^ p(X, \mathcal{F}) = 0$ for every quasi-coherent $\mathcal{O}_ X$-module and $p > 0$. Since $X$ is quasi-compact by Properties, Definition 28.26.1 the claim finishes the proof by Lemma 30.3.1. The scheme $X$ is separated by Properties, Lemma 28.26.8. Say $X$ is covered by $e + 1$ affine opens. Then $H^ p(X, \mathcal{F}) = 0$ for $p > e$, see Lemma 30.4.2. Thus we may use descending induction on $p$ to prove the claim. Writing $\mathcal{F}$ as a filtered colimit of finite type quasi-coherent modules (Properties, Lemma 28.22.3) and using Cohomology, Lemma 20.19.1 we may assume $\mathcal{F}$ is of finite type. Then we can choose $n > n_0$ such that $\mathcal{F} \otimes _{\mathcal{O}_ X} \mathcal{L}^{\otimes n}$ is globally generated, see Properties, Proposition 28.26.13. This means there is a short exact sequence

\[ 0 \to \mathcal{F}' \to \bigoplus \nolimits _{i \in I} \mathcal{L}^{\otimes -n} \to \mathcal{F} \to 0 \]

for some set $I$ (in fact we can choose $I$ finite). By induction hypothesis we have $H^{p + 1}(X, \mathcal{F}') = 0$ and by assumption (combined with the already used commutation of cohomology with colimits) we have $H^ p(X, \bigoplus _{i \in I} \mathcal{L}^{\otimes -n}) = 0$. From the long exact cohomology sequence we conclude that $H^ p(X, \mathcal{F}) = 0$ as desired. $\square$

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Your email address will not be published.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 0EBD. Beware of the difference between the letter 'O' and the digit '0'.