The Stacks project

Situation 77.2.1. Let $S$ be a scheme. Let $f : X \to Y$ be a finite type, decent1 morphism of algebraic spaces over $S$. Also, $\mathcal{F}$ is a finite type quasi-coherent $\mathcal{O}_ X$-module. Finally $y \in |Y|$ is a point of $Y$.

[1] Quasi-separated morphisms are decent, see Decent Spaces, Lemma 68.17.2. For any morphism $\mathop{\mathrm{Spec}}(k) \to Y$ where $k$ is a field, the algebraic space $X_ k$ is of finite presentation over $k$ because it is of finite type over $k$ and quasi-separated by Decent Spaces, Lemma 68.14.1.

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