Lemma 56.7.7 (0FZQ)—The Stacks project

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Table of contents
Part 3: Topics in Scheme Theory
Chapter 56: Functors and Morphisms
Section 56.7: Functors between categories of coherent modules
Lemma 56.7.7 (cite)

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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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