The Stacks project

Situation 38.20.7. Let $f : X \to S$ be a smooth morphism with geometrically irreducible fibres. 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 smooth with geometrically irreducible fibres, see Morphisms, Lemma 29.34.5 and More on Morphisms, Lemma 37.26.2. Let $p \geq 0$ be an integer. Given a point $t \in T$ consider the condition

38.20.7.1
\begin{equation} \label{flat-equation-free-at-generic-point-fibre} \mathcal{F}_ T \text{ is free of rank }p\text{ in a neighbourhood of }\xi _ t \end{equation}

where $\xi _ t$ is the generic point of the fibre $X_ t$. This condition for all $t \in T$ is stable under base change, and hence we obtain a functor

38.20.7.2
\begin{equation} \label{flat-equation-free-at-generic-points} H_ p : (\mathit{Sch}/S)^{opp} \longrightarrow \textit{Sets}, \quad T \longrightarrow \left\{ \begin{matrix} \{ *\} & \text{if }\mathcal{F}_ T\text{ satisfies (05MQ) }\forall t\in T, \\ \emptyset & \text{else.} \end{matrix} \right. \end{equation}

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