Situation 38.20.11. Let $f : X \to S$ be a morphism of schemes which is locally of finite type. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module of finite type. For any scheme $T$ over $S$ we will denote $\mathcal{F}_ T$ the base change of $\mathcal{F}$ to $T$, in other words, $\mathcal{F}_ T$ is the pullback of $\mathcal{F}$ via the projection morphism $X_ T = X \times _ S T \to X$. Note that $X_ T \to T$ is of finite type and that $\mathcal{F}_ T$ is an $\mathcal{O}_{X_ T}$-module of finite type (Morphisms, Lemma 29.15.4 and Modules, Lemma 17.9.2). Let $n \geq 0$. By Definition 38.20.10 and Lemma 38.20.9 we obtain a functor

38.20.11.1
$$\label{flat-equation-flat-dimension-n} F_ n : (\mathit{Sch}/S)^{opp} \longrightarrow \textit{Sets}, \quad T \longrightarrow \left\{ \begin{matrix} \{ *\} & \text{if }\mathcal{F}_ T\text{ is flat over }T\text{ in }\dim \geq n, \\ \emptyset & \text{else.} \end{matrix} \right.$$

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