The Stacks project

Situation 38.20.13. Let $f : X \to S$ be a morphism of schemes. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. 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 $X_ T = X \times _ S T \to X$. Since the base change of a flat module is flat we obtain a functor

38.20.13.1
\begin{equation} \label{flat-equation-flat} F_{flat} : (\mathit{Sch}/S)^{opp} \longrightarrow \textit{Sets}, \quad T \longrightarrow \left\{ \begin{matrix} \{ *\} & \text{if } \mathcal{F}_ T \text{ is flat over }T, \\ \emptyset & \text{else.} \end{matrix} \right. \end{equation}

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