Lemma 28.26.2 (01PT)—The Stacks project

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Part 2: Schemes
Chapter 28: Properties of Schemes
Section 28.26: Ample invertible sheaves
Lemma 28.26.2 (cite)

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[II Proposition 4.5.6(i), EGA]

Lemma 28.26.2. Let $X$ be a scheme. Let $\mathcal{L}$ be an invertible $\mathcal{O}_ X$ -module. Let $n \geq 1$ . Then $\mathcal{L}$ is ample if and only if $\mathcal{L}^{\otimes n}$ is ample.

Proof. This follows from the fact that $X_{s^ n} = X_ s$ . $\square$

Comments (2)

Comment #2692 by Matt Stevenson on August 01, 2017 at 23:14

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

Comment #2721 by Takumi Murayama on August 01, 2017 at 23:46

Added the reference; thanks!

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