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 and an important baby case.
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 86.33.3). 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.
We end this section showing that if we do not insist on a canonical stratification, then we can use generic flatness to construct some stratification such that our sheaf is flat over the strata.
Lemma 38.21.4 (Generic flatness stratification). Let $f : X \to S$ be a morphism of finite presentation between quasi-compact and quasi-separated schemes. Let $\mathcal{F}$ be an $\mathcal{O}_ X$-module of finite presentation. Then there exists a $t \geq 0$ and closed subschemes such that $S_ i \to S$ is defined by a finite type ideal sheaf, $S_0 \subset S$ is a thickening, and $\mathcal{F}$ pulled back to $X \times _ S (S_ i \setminus S_{i + 1})$ is flat over $S_ i \setminus S_{i + 1}$.
\[ S \supset S_0 \supset S_1 \supset \ldots \supset S_ t = \emptyset \]
Proof. We can find a cartesian diagram
\[ \xymatrix{ X \ar[d] \ar[r] & X_0 \ar[d] \\ S \ar[r] & S_0 } \]
and a finitely presented $\mathcal{O}_{X_0}$-module $\mathcal{F}_0$ which pulls back to $\mathcal{F}$ such that $X_0$ and $S_0$ are of finite type over $\mathbf{Z}$. See Limits, Proposition 32.5.4 and Lemmas 32.10.1 and 32.10.2. Thus we may assume $X$ and $S$ are of finite type over $\mathbf{Z}$ and $\mathcal{F}$ is a coherent $\mathcal{O}_ X$-module.
Assume $X$ and $S$ are of finite type over $\mathbf{Z}$ and $\mathcal{F}$ is a coherent $\mathcal{O}_ X$-module. In this case every quasi-coherent ideal is of finite type, hence we do not have to check the condition that $S_ i$ is cut out by a finite type ideal. Set $S_0 = S_{red}$ equal to the reduction of $S$. By generic flatness as stated in Morphisms, Proposition 29.27.2 there is a dense open $U_0 \subset S_0$ such that $\mathcal{F}$ pulled back to $X \times _ S U_0$ is flat over $U_0$. Let $S_1 \subset S_0$ be the reduced closed subscheme whose underlying closed subset is $S \setminus U_0$. We continue in this way, provided $S_1 \not= \emptyset $, to find $S_0 \supset S_1 \supset \ldots $. Because $S$ is Noetherian any descending chain of closed subsets stabilizes hence we see that $S_ t = \emptyset $ for some $t \geq 0$. $\square$
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