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[II Definition 4.6.1, EGA]

Definition 29.37.1. Let $f : X \to S$ be a morphism of schemes. Let $\mathcal{L}$ be an invertible $\mathcal{O}_ X$-module. We say $\mathcal{L}$ is relatively ample, or $f$-relatively ample, or ample on $X/S$, or $f$-ample if $f : X \to S$ is quasi-compact, and if for every affine open $V \subset S$ the restriction of $\mathcal{L}$ to the open subscheme $f^{-1}(V)$ of $X$ is ample.

Comments (3)

Comment #2699 by Matt Stevenson on

This is closely related to EGA II Def 4.6.1 (in EGA, the morphism f is assumed to be quasi-compact).

Comment #2841 by on

It is true that the definition in EGA is formulated in such a way as to only apply to morphisms which are quasi-compact. This leaves the possibility open to have a later definition of a relatively ample sheaf even in cases where the morphism is not quasi-compact. However, as far as I know, there is no such extension in EGA, and hence I think we are safe and our definition (which requires the morphism to be qc) is consistent with EGA.

There are also:

  • 3 comment(s) on Section 29.37: Relatively ample sheaves

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View Definition 29.37.1 as pdf