[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 $X_{s^ n} = X_ s$. $\square$

Comment #2692 by Matt Stevenson on

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

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