The Stacks project

Lemma 56.7.7. Let $f : X \to Y$ be a finite type separated morphism of schemes with a section $s : Y \to X$. Let $\mathcal{F}$ be a finite type quasi-coherent module on $X$, set theoretically supported on $s(Y)$ with $\mathcal{L} = f_*\mathcal{F}$ an invertible $\mathcal{O}_ X$-module. If $Y$ is reduced, then $\mathcal{F} \cong s_*\mathcal{L}$.

Proof. By Lemma 56.7.6 there exists a section $s' : Y \to X$ such that $\mathcal{F} = s'_*\mathcal{L}$. Since $s'(Y)$ and $s(Y)$ have the same underlying closed subset and since both are reduced closed subschemes of $X$, they have to be equal. Hence $s = s'$ and the lemma holds. $\square$

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