The Stacks project

Lemma 37.54.2. Let $f : X \to S$ be a morphism of finite presentation between quasi-compact and quasi-separated schemes. 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 $X \times _ S (S_ i \setminus S_{i + 1})$ is flat over $S_ i \setminus S_{i + 1}$.

Proof. Apply Lemma 37.54.1 with $\mathcal{F} = \mathcal{O}_ X$. $\square$

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