The Stacks project

Lemma 36.36.7. Let $X$ be a scheme. If $X$ has an ample family of invertible modules (Morphisms, Definition 29.12.1), then $X$ has the resolution property.

Proof. Since $X$ is quasi-compact, there exists $n$ and pairs $(\mathcal{L}_ i, s_ i)$, $i = 1, \ldots , n$ where $\mathcal{L}_ i$ is an invertible $\mathcal{O}_ X$-module and $s_ i \in \Gamma (X, \mathcal{L}_ i)$ is a section such that the set of points $U_ i \subset X$ where $s_ i$ is nonvanishing is affine and $X = U_1 \cup \ldots \cup U_ n$. Let $\mathcal{I}_ i \subset \mathcal{O}_ X$ be the image of $s_ i : \mathcal{L}_ i^{\otimes -1} \to \mathcal{O}_ X$. Applying Lemma 36.36.6 we find that $X$ has the resolution property. $\square$

Comments (0)

There are also:

  • 2 comment(s) on Section 36.36: The resolution property

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Your email address will not be published.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



View Lemma 36.36.7 as pdf