The Stacks project

Lemma 38.27.6. Let $f : X \to S$ be a morphism of schemes. 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 $S$, then there exists a universal flattening $S' \to S$ of $\mathcal{F}$. Moreover $S' \to S$ is a monomorphism of finite presentation.

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

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