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 $X$.

Proof. These statements follow immediately from Theorem 38.27.5 applied to $F_0 = F_{flat}$ and the fact that if $f$ is proper then $\mathcal{F}$ is automatically pure over the base, see Lemma 38.17.1. $\square$

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