The Stacks project

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

\[ S \supset S_0 \supset S_1 \supset \ldots \supset S_ t = \emptyset \]

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 $S_ i \setminus S_{i + 1}$.

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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