Relative ampleness over an arbitrary base

Let $f:X\to S$ be a morphism of schemes and let $L$ be an invertible ${\mathcal{O}}_X$-module (Invertible sheaves). Suppose first that $f$ is quasi-compact (Quasi-compact and quasi-separated morphisms). Then $L$ is $f$-ample, or ample relative to $S$, when for every affine open subscheme $U\subseteq S$ the restriction $L{\mathrm{\mid }}_{f^{-1}(U)}$ is an ample invertible sheaf on the scheme $f^{-1}(U)$ in the absolute sense of Absolute ampleness by affine section opens.

The definition is well posed: $f^{-1}(U)$ is quasi-compact because $f$ is quasi-compact and $U$ is quasi-compact (being affine), so the scheme $f^{-1}(U)$ satisfies the quasi-compactness clause required of an ample invertible sheaf, and ampleness of the restriction is then a condition on the affine nonvanishing loci of its positive powers. The empty scheme is allowed on both sides: if $f^{-1}(U)=\mathrm{\varnothing }$ the restriction is ample vacuously, and if $S=\mathrm{\varnothing }$ the condition is vacuous.

• Affine base. If $S=\mathrm{Spec}R$ is affine, then $f^{-1}(S)=X$ and the single condition on $U=S$ says that $L$ is ample on $X$ in the absolute sense; so for an affine base $f$-ampleness is exactly ampleness of Absolute ampleness by affine section opens, with no extra hypothesis beyond quasi-compactness of $f$, which for an affine base is the quasi-compactness of $X$.
• Nonaffine base. The definition is stated for every affine open of $S$ precisely because over a nonaffine base a globally ample ${\mathcal{O}}_X$-module need not have ample restriction to every open subscheme; relative ampleness is a hypothesis on the restrictions, not on $L$ itself.
• Relation to relative very ampleness. Under the Axiom of Choice (The Axiom of Choice), H-very ampleness relative to $S$ is defined at Relative very ampleness in the finite projective-space convention and implies $f$-ampleness by Relative very ampleness implies relative ampleness; the converse fails in general, and the sufficient criterion in the proper, finite-type, Noetherian case is High powers of an ample line bundle embed a proper scheme.