Lemma 33.38.14. Let $f : X \to Y$ be a morphism of schemes. Assume $Y$ is Noetherian of dimension $\leq 1$, $f$ is finite, and there exists a dense open $V \subset Y$ such that $f^{-1}(V) \to V$ is a closed immersion. Then every invertible $\mathcal{O}_ X$-module is the pullback of an invertible $\mathcal{O}_ Y$-module.

Proof. We factor $f$ as $X \to Z \to Y$ where $Z$ is the scheme theoretic image of $f$. Then $X \to Z$ is an isomorphism over $V \cap Z$ and Lemma 33.38.7 applies. On the other hand, Lemma 33.38.11 applies to $Z \to Y$. Some details omitted. $\square$

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