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
Comment #4365 by Johan on
Comment #5344 by Lucas das Dores on
Comment #5586 by Johan on