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Label a morphism
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aisejohan committed Jul 19, 2021
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Expand Up @@ -2995,13 +2995,13 @@ \section{Relatively ample sheaves}
Lemma \ref{lemma-ample-base-change}.
Assume (2). This implies that $Y \to V$ is
quasi-compact and separated (Lemma \ref{lemma-relatively-ample-properties})
and $Y$ is a scheme. Then we conclude that $X \to U$ is
and $Y$ is a scheme. It follows that the morphism $f : X \to U$ is
quasi-compact and separated
(Morphisms of Spaces, Lemmas
\ref{spaces-morphisms-lemma-quasi-compact-local} and
\ref{spaces-morphisms-lemma-separated-local}).
Set $\mathcal{A} = \bigoplus_{d \geq 0} f_*\mathcal{L}^{\otimes d}$.
Thus is a quasi-coherent sheaf of graded $\mathcal{O}_U$-algebras
This is a quasi-coherent sheaf of graded $\mathcal{O}_U$-algebras
(Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-pushforward}).
By adjunction we have a map
$\psi : f^*\mathcal{A} \to \bigoplus_{d \geq 0} \mathcal{L}^{\otimes d}$.
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