The Stacks project

Lemma 56.7.6. Let $f : X \to Y$ be a finite type separated morphism of schemes. Let $\mathcal{F}$ be a finite type quasi-coherent module on $X$ with support finite over $Y$ and with $\mathcal{L} = f_*\mathcal{F}$ an invertible $\mathcal{O}_ X$-module. Then there exists a section $s : Y \to X$ such that $\mathcal{F} \cong s_*\mathcal{L}$.

Proof. Looking affine locally this translates into the following algebra problem. Let $A \to B$ be a ring map and let $N$ be a $B$-module which is invertible as an $A$-module. Then the annihilator $J$ of $N$ in $B$ has the property that $A \to B/J$ is an isomorphism. We omit the details. $\square$

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