Lemma 28.26.5. Let $X$ be a scheme. Let $\mathcal{L}$ and $\mathcal{M}$ be invertible $\mathcal{O}_ X$-modules. If
$\mathcal{L}$ is ample, and
the open sets $X_ t$ where $t \in \Gamma (X, \mathcal{M}^{\otimes m})$ for $m > 0$ cover $X$,
[II Proposition 4.5.6(ii), EGA]
Lemma 28.26.5. Let $X$ be a scheme. Let $\mathcal{L}$ and $\mathcal{M}$ be invertible $\mathcal{O}_ X$-modules. If
$\mathcal{L}$ is ample, and
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 28.26.1. 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 28.26.4. $\square$
Your email address will not be published.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).
All contributions are licensed under the GNU Free Documentation License.
Comments (3)
Comment #2694 by Matt Stevenson on
Comment #2722 by Takumi Murayama on
Comment #2840 by Johan on
There are also: