The Stacks project

Lemma 31.9.7. Let $S$ be a scheme. Let $\mathcal{F}$ be an $\mathcal{O}_ S$-module of finite presentation. Let $S = Z_{-1} \supset Z_0 \supset Z_1 \supset \ldots $ be as in Lemma 31.9.6. Set $S_ r = Z_{r - 1} \setminus Z_ r$. Then $S' = \coprod _{r \geq 0} S_ r$ represents the functor

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

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

Proof. Suppose that $g : T \to S$ is a morphism of schemes such that the pullback $\mathcal{F}_ T = g^*\mathcal{F}$ is flat. Then $\mathcal{F}_ T$ is a flat $\mathcal{O}_ T$-module of finite presentation. Hence $\mathcal{F}_ T$ is finite locally free, see Properties, Lemma 28.20.2. Thus $T = \coprod _{r \geq 0} T_ r$, where $\mathcal{F}_ T|_{T_ r}$ is locally free of rank $r$. This implies that

\[ F_{flat} = \coprod \nolimits _{r \geq 0} F_ r \]

in the category of Zariski sheaves on $\mathit{Sch}/S$ where $F_ r$ is as in Lemma 31.9.6. It follows that $F_{flat}$ is represented by $\coprod _{r \geq 0} (Z_{r - 1} \setminus Z_ r)$ where $Z_ r$ is as in Lemma 31.9.6. The other statements also follow from the lemma. $\square$

Comments (4)

Comment #4156 by Laurent Moret-Bailly on

Suggested "counterexample" for finite type modules: absolutely flat, residue field of a nonisolated point.

Comment #5344 by Lucas das Dores on

Typo on the statement: the inclusions are reversed.

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