The Stacks project

Lemma 77.11.8. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module.

1. If $f$ is of finite presentation, $\mathcal{F}$ is an $\mathcal{O}_ X$-module of finite presentation, and $\mathcal{F}$ is pure relative to $Y$, then there exists a universal flattening $Y' \to Y$ of $\mathcal{F}$. Moreover $Y' \to Y$ is a monomorphism of finite presentation.

2. If $f$ is of finite presentation and $X$ is pure relative to $Y$, then there exists a universal flattening $Y' \to Y$ of $X$. Moreover $Y' \to Y$ is a monomorphism of finite presentation.

3. If $f$ is proper and of finite presentation and $\mathcal{F}$ is an $\mathcal{O}_ X$-module of finite presentation, then there exists a universal flattening $Y' \to Y$ of $\mathcal{F}$. Moreover $Y' \to Y$ is a monomorphism of finite presentation.

4. If $f$ is proper and of finite presentation then there exists a universal flattening $Y' \to Y$ of $X$.