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The Stacks project

Lemma 71.5.7. 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 71.5.6. 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 étale localization. \square


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