[II, Proposition 4.6.2, EGA]

Lemma 29.38.2. Let $f : X \to S$ be a morphism of schemes. Let $\mathcal{L}$ be an invertible $\mathcal{O}_ X$-module. If $f$ is quasi-compact and $\mathcal{L}$ is a relatively very ample invertible sheaf, then $\mathcal{L}$ is a relatively ample invertible sheaf.

Proof. By definition there exists quasi-coherent $\mathcal{O}_ S$-module $\mathcal{E}$ and an immersion $i : X \to \mathbf{P}(\mathcal{E})$ over $S$ such that $\mathcal{L} \cong i^*\mathcal{O}_{\mathbf{P}(\mathcal{E})}(1)$. Set $\mathcal{A} = \text{Sym}(\mathcal{E})$, so $\mathbf{P}(\mathcal{E}) = \underline{\text{Proj}}_ S(\mathcal{A})$ by definition. The graded $\mathcal{O}_ S$-algebra $\mathcal{A}$ comes equipped with a map

$\psi : \mathcal{A} \to \bigoplus \nolimits _{n \geq 0} \pi _*\mathcal{O}_{\mathbf{P}(\mathcal{E})}(n) \to \bigoplus \nolimits _{n \geq 0} f_*\mathcal{L}^{\otimes n}$

where the second arrow uses the identification $\mathcal{L} \cong i^*\mathcal{O}_{\mathbf{P}(\mathcal{E})}(1)$. By adjointness of $f_*$ and $f^*$ we get a morphism $\psi : f^*\mathcal{A} \to \bigoplus _{n \geq 0}\mathcal{L}^{\otimes n}$. We omit the verification that the morphism $r_{\mathcal{L}, \psi }$ associated to this map is exactly the immersion $i$. Hence the result follows from part (6) of Lemma 29.37.4. $\square$

Comment #2723 by Matt Stevenson on

This is EGA II Prop 4.6.2.

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