Lemma 29.39.4. Let $f : X \to S$ be a morphism of schemes. Let $\mathcal{L}$ be an invertible $\mathcal{O}_ X$-module. Assume $S$ affine and $f$ of finite type. The following are equivalent

1. $\mathcal{L}$ is ample on $X$,

2. $\mathcal{L}$ is $f$-ample,

3. $\mathcal{L}^{\otimes d}$ is $f$-very ample for some $d \geq 1$,

4. $\mathcal{L}^{\otimes d}$ is $f$-very ample for all $d \gg 1$,

5. for some $d \geq 1$ there exist $n \geq 1$ and an immersion $i : X \to \mathbf{P}^ n_ S$ such that $\mathcal{L}^{\otimes d} \cong i^*\mathcal{O}_{\mathbf{P}^ n_ S}(1)$, and

6. for all $d \gg 1$ there exist $n \geq 1$ and an immersion $i : X \to \mathbf{P}^ n_ S$ such that $\mathcal{L}^{\otimes d} \cong i^*\mathcal{O}_{\mathbf{P}^ n_ S}(1)$.

Proof. The equivalence of (1) and (2) is Lemma 29.37.5. The implication (2) $\Rightarrow$ (6) is Lemma 29.39.3. Trivially (6) implies (5). As $\mathbf{P}^ n_ S$ is a projective bundle over $S$ (see Constructions, Lemma 27.21.5) we see that (5) implies (3) and (6) implies (4) from the definition of a relatively very ample sheaf. Trivially (4) implies (3). To finish we have to show that (3) implies (2) which follows from Lemma 29.38.2 and Lemma 29.37.2. $\square$

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