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.
38.21 Flattening stratifications
Just the definitions. The reader looking for a “generic flatness stratification”, should consult More on Morphisms, Section 37.54.
Note that if the universal flattening $S'$1 of $\mathcal{F}$ exists, then the morphism $S' \to S$ is a surjective monomorphism of schemes such that $\mathcal{F}_{S'}$ is flat over $S'$ and such that a morphism $T \to S$ factors through $S'$ if and only if $\mathcal{F}_ T$ is flat over $T$.
Example 38.21.2. Let $X = S = \mathop{\mathrm{Spec}}(k[x, y])$ where $k$ is a field. Let $\mathcal{F} = \widetilde{M}$ where $M = k[x, x^{-1}, y]/(y)$. For a $k[x, y]$-algebra $A$ set $F_{flat}(A) = F_{flat}(\mathop{\mathrm{Spec}}(A))$. Then $F_{flat}(k[x, y]/(x, y)^ n) = \{ *\} $ for all $n$, while $F_{flat}(k[[x, y]]) = \emptyset $. This means that $F_{flat}$ isn't representable (even by an algebraic space, see Formal Spaces, Lemma 87.33.3). Thus the universal flattening does not exist in this case.
We define (compare with Topology, Remark 5.28.5) a (locally finite, scheme theoretic) stratification of a scheme $S$ to be given by closed subschemes $Z_ i \subset S$ indexed by a partially ordered set $I$ such that $S = \bigcup Z_ i$ (set theoretically), such that every point of $S$ has a neighbourhood meeting only a finite number of $Z_ i$, and such that
\[ Z_ i \cap Z_ j = \bigcup \nolimits _{k \leq i, j} Z_ k. \]
Setting $S_ i = Z_ i \setminus \bigcup _{j < i} Z_ j$ the actual stratification is the decomposition $S = \coprod S_ i$ into locally closed subschemes. We often only indicate the strata $S_ i$ and leave the construction of the closed subschemes $Z_ i$ to the reader. Given a stratification we obtain a monomorphism
\[ S' = \coprod \nolimits _{i \in I} S_ i \longrightarrow S. \]
We will call this the monomorphism associated to the stratification. With this terminology we can define what it means to have a flattening stratification.
Definition 38.21.3. Let $X \to S$ be a morphism of schemes. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. We say that $\mathcal{F}$ has a flattening stratification if the functor $F_{flat}$ defined in Situation 38.20.13 is representable by a monomorphism $S' \to S$ associated to a stratification of $S$ by locally closed subschemes. We say that $X$ has a flattening stratification if $\mathcal{O}_ X$ has a flattening stratification.
When a flattening stratification exists, it is often important to understand the index set labeling the strata and its partial ordering. This often has to do with ranks of modules. For example if $X = S$ and $\mathcal{F}$ is a finitely presented $\mathcal{O}_ S$-module, then the flattening stratification exists and is given by the Fitting ideals of $\mathcal{F}$, see Divisors, Lemma 31.9.7.
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