The Stacks project

[II Proposition 4.5.6(ii), EGA]

Lemma 27.26.5. Let $X$ be a scheme. Let $\mathcal{L}$ and $\mathcal{M}$ be invertible $\mathcal{O}_ X$-modules. If

  1. $\mathcal{L}$ is ample, and

  2. the open sets $X_ t$ where $t \in \Gamma (X, \mathcal{M}^{\otimes m})$ for $m > 0$ cover $X$,

then $\mathcal{L} \otimes \mathcal{M}$ is ample.

Proof. We check the conditions of Definition 27.26.1. As $\mathcal{L}$ is ample we see that $X$ is quasi-compact. Let $x \in X$. Choose $n \geq 1$, $m \geq 1$, $s \in \Gamma (X, \mathcal{L}^{\otimes n})$, and $t \in \Gamma (X, \mathcal{M}^{\otimes m})$ such that $x \in X_ s$, $x \in X_ t$ and $X_ s$ is affine. Then $s^ mt^ n \in \Gamma (X, (\mathcal{L} \otimes \mathcal{M})^{\otimes nm})$, $x \in X_{s^ mt^ n}$, and $X_{s^ mt^ n}$ is affine by Lemma 27.26.4. $\square$


Comments (3)

Comment #2694 by Matt Stevenson on

This is closely related to EGA II Prop 4.5.6 (ii) (in EGA, the scheme X is assumed to be quasi-compact).

Comment #2840 by on

To avoid misunderstandings, the assumption that L is ample already forces X to be quasi-compact, hence this really is the same thing as in EGA.


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Your email address will not be published.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 0890. Beware of the difference between the letter 'O' and the digit '0'.