The Stacks project

Lemma 37.3.2. Let $(f, f') : (X \subset X') \to (S \subset S')$ be a morphism of thickenings. Let $\mathcal{L}'$ be an invertible sheaf on $X'$ and denote $\mathcal{L}$ the restriction to $X$. Then $\mathcal{L}'$ is $f'$-ample if and only if $\mathcal{L}$ is $f$-ample.

Proof. Recall that being relatively ample is a condition for each affine open in the base, see Morphisms, Definition 29.37.1. By Lemma 37.2.3 there is a 1-to-1 correspondence between affine opens of $S$ and $S'$. Thus we may assume $S$ and $S'$ are affine and we reduce to proving that $\mathcal{L}'$ is ample if and only if $\mathcal{L}$ is ample. This is Limits, Lemma 32.11.4. $\square$

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