Lemma 28.26.5 (0890)—The Stacks project

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. $\square$

Comments (3)

Comment #2694 by Matt Stevenson on August 01, 2017 at 23:17

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

Comment #2722 by Takumi Murayama on August 01, 2017 at 23:47

Added the reference; thanks!

Comment #2840 by Johan on October 04, 2017 at 13:34

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.

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5 comment(s) on Section 28.26: Ample invertible sheaves

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