The Stacks project

Lemma 29.38.8. Let $f : X \to S$ be a morphism of schemes. Let $\mathcal{L}$ be an invertible $\mathcal{O}_ X$-module. Let $S' \to S$ be a morphism of schemes. Let $f' : X' \to S'$ be the base change of $f$ and denote $\mathcal{L}'$ the pullback of $\mathcal{L}$ to $X'$. If $\mathcal{L}$ is $f$-very ample, then $\mathcal{L}'$ is $f'$-very ample.

Proof. By Definition 29.38.1 there exists there exist a quasi-coherent $\mathcal{O}_ S$-module $\mathcal{E}$ and an immersion $i : X \to \mathbf{P}(\mathcal{E})$ over $S$ such that $\mathcal{L} \cong i^*\mathcal{O}_{\mathbf{P}(\mathcal{E})}(1)$. The base change of $\mathbf{P}(\mathcal{E})$ to $S'$ is the projective bundle associated to the pullback $\mathcal{E}'$ of $\mathcal{E}$ and the pullback of $\mathcal{O}_{\mathbf{P}(\mathcal{E})}(1)$ is $\mathcal{O}_{\mathbf{P}(\mathcal{E}')}(1)$, see Constructions, Lemma 27.16.10. Finally, the base change of an immersion is an immersion (Schemes, Lemma 26.18.2). $\square$

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