The Stacks project

Lemma 70.5.7. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\mathcal{F}$ be an $\mathcal{O}_ X$-module of finite presentation. Let $X = Z_{-1} \subset Z_0 \subset Z_1 \subset \ldots $ be as in Lemma 70.5.6. Set $X_ r = Z_{r - 1} \setminus Z_ r$. Then $X' = \coprod _{r \geq 0} X_ r$ represents the functor

\[ F_{flat} : \mathit{Sch}/X \longrightarrow \textit{Sets},\quad \quad T \longmapsto \left\{ \begin{matrix} \{ *\} & \text{if }\mathcal{F}_ T\text{ flat over }T \\ \emptyset & \text{otherwise} \end{matrix} \right. \]

Moreover, $\mathcal{F}|_{X_ r}$ is locally free of rank $r$ and the morphisms $X_ r \to X$ and $X' \to X$ are of finite presentation.

Proof. Reduces to Divisors, Lemma 31.9.7 by ├ętale localization. $\square$

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