The Stacks project

Lemma 27.26.14. Let $X$ be a scheme. Let $\mathcal{L}$ be an ample invertible $\mathcal{O}_ X$-module. For any quasi-compact immersion $i : X' \to X$ the pullback $i^*\mathcal{L}$ is ample on $X'$.

Proof. For $s \in \Gamma (X, \mathcal{L}^{\otimes d})$ denote $s' = i^*s$ the restriction to $X'$. By Proposition 27.26.13 the opens $X_ s$, for $s \in \Gamma (X, \mathcal{L}^{\otimes d})$, form a basis for the topology on $X$. Since $X'_{s'} = X' \cap X_ s$ and since $i(X') \subset X$ is locally closed, we conclude the same thing is true for the opens $X'_{s'}$. Hence the lemma is a consequence of Proposition 27.26.13. $\square$


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