The Stacks project

[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

  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 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

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.

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