Lemma 29.39.5. Let $f : X \to S$ be a morphism of schemes. Let $\mathcal{L}$ be an invertible $\mathcal{O}_ X$-module. Assume $S$ quasi-compact and $f$ of finite type. The following are equivalent
$\mathcal{L}$ is $f$-ample,
$\mathcal{L}^{\otimes d}$ is $f$-very ample for some $d \geq 1$,
$\mathcal{L}^{\otimes d}$ is $f$-very ample for all $d \gg 1$.
Proof. Trivially (3) implies (2). Lemma 29.38.2 guarantees that (2) implies (1) since a morphism of finite type is quasi-compact by definition. Assume that $\mathcal{L}$ is $f$-ample. Choose a finite affine open covering $S = V_1 \cup \ldots \cup V_ m$. Write $X_ i = f^{-1}(V_ i)$. By Lemma 29.39.4 above we see there exists a $d_0$ such that $\mathcal{L}^{\otimes d}$ is relatively very ample on $X_ i/V_ i$ for all $d \geq d_0$. Hence we conclude (1) implies (3) by Lemma 29.38.7. $\square$
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