The Stacks project

Lemma 28.26.3. Let $X$ be a scheme. Let $\mathcal{L}$ be an ample invertible $\mathcal{O}_ X$-module. For any closed subscheme $Z \subset X$ the restriction of $\mathcal{L}$ to $Z$ is ample.

Proof. This is clear since a closed subset of a quasi-compact space is quasi-compact and a closed subscheme of an affine scheme is affine (see Schemes, Lemma 26.8.2). $\square$

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