The Stacks project

Lemma 31.17.4. Let $\pi : X \to Y$ be a finite morphism of schemes. Assume $X$ has an ample invertible sheaf and there exists a norm of degree $d$ for $\pi $. Then $Y$ has an ample invertible sheaf.

Proof. Let $\mathcal{L}$ be the ample invertible sheaf on $X$ given to us by assumption. We will prove that $\mathcal{N} = \text{Norm}_\pi (\mathcal{L})$ is ample on $Y$.

Since $X$ is quasi-compact (Properties, Definition 28.26.1) and $X \to Y$ surjective (by the existence of $\text{Norm}_\pi $) we see that $Y$ is quasi-compact. Let $y \in Y$ be a point. To finish the proof we will show that there exists a section $t$ of some positive tensor power of $\mathcal{N}$ which does not vanish at $y$ such that $Y_ t$ is affine. To do this, choose an affine open neighbourhood $V \subset Y$ of $y$. Choose $n \gg 0$ and a section $s \in \Gamma (X, \mathcal{L}^{\otimes n})$ such that

\[ \pi ^{-1}(\{ y\} ) \subset X_ s \subset \pi ^{-1}V \]

by Properties, Lemma 28.29.6. Then $t = \text{Norm}_\pi (s)$ is a section of $\mathcal{N}^{\otimes n}$ which does not vanish at $x$ and with $Y_ t \subset V$, see Lemma 31.17.3. Then $Y_ t$ is affine by Properties, Lemma 28.26.4. $\square$

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