Proposition 31.17.9 (0BD4)—The Stacks project

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Table of contents
Part 2: Schemes
Chapter 31: Divisors
Section 31.17: Norms
Proposition 31.17.9 (cite)

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Proposition 31.17.9. Let $\pi : X \to Y$ be a finite surjective morphism of schemes. Assume that $X$ has an ample invertible $\mathcal{O}_ X$ -module. If

$\pi$ is finite locally free, or
$Y$ is an integral normal scheme, or
$Y$ is Noetherian, $p\mathcal{O}_ Y = 0$ , and $X = Y_{red}$ ,

then $Y$ has an ample invertible $\mathcal{O}_ Y$ -module.

Proof. Case (1) follows from a combination of Lemmas 31.17.6 and 31.17.4. Case (3) follows from a combination of Lemmas 31.17.8 and 31.17.4. In case (2) we first replace $X$ by an irreducible component of $X$ which dominates $Y$ (viewed as a reduced closed subscheme of $X$ ). Then we can apply Lemma 31.17.7. $\square$

Comments (2)

Comment #3764 by Zhiyu Zhang on November 01, 2018 at 00:48

Maybe a little typo, $f$ shall be $\pi$ in "Let $f:X \rightarrow Y$ be a finite ..." , same typo occurs in Lemma 0BD5.

Comment #3895 by Johan on January 22, 2019 at 10:08

Thanks and fixed here.

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