Definition 38.21.1. Let $X \to S$ be a morphism of schemes. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. We say that the universal flattening of $\mathcal{F}$ exists if the functor $F_{flat}$ defined in Situation 38.20.13 is representable by a scheme $S'$ over $S$. We say that the universal flattening of $X$ exists if the universal flattening of $\mathcal{O}_ X$ exists.

Comment #4155 by Laurent Moret-Bailly on

Suggested example where the universal flattening does not exist: $X=S=\operatorname{Spec}k[x,y]$, $\mathcal{F}=\widetilde{M}$ where $M=k[x,x^{-1},y]/(y)$. Namely, $F_{flat}(k[x,y]/(x,y)^n)=\lbrace\ast\rbrace$ for all $n$, while $F_{flat}(k[[x,y]])=\emptyset$.

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