Let $f: X \to S$ be a quasi-compact morphism of schemes and $\mathcal{L}$ an invertible $\mathcal{O}_X$-module. Is it true that $\mathcal{L}$ is $f$-very ample if and only if $\mathcal{L}$ is $f$-ample and the $\mathbb{Z}_{\geq 0}$-graded commutative $\mathcal{O}_S$-algebra $\bigoplus_{m ∈ \mathbb{Z}_{\geq 0}} f_* \mathcal{L}^{\otimes m}$ is generated over $\mathcal{O_S}$ in degree 1?
This is similar but not quite the same as the descriptions given in the Stacks project.