Lemma 38.27.3. Assume that $X \to S$ is a smooth morphism of affine schemes with geometrically irreducible fibres of dimension $d$ and that $\mathcal{F}$ is a quasi-coherent $\mathcal{O}_ X$-module of finite presentation. Then $F_ d = \coprod _{p = 0, \ldots , c} H_ p$ for some $c \geq 0$ with $F_ d$ as in (38.20.11.1) and $H_ p$ as in (38.20.7.2).

Proof. As $X$ is affine and $\mathcal{F}$ is quasi-coherent of finite presentation we know that $\mathcal{F}$ can be generated by $c \geq 0$ elements. Then $\dim _{\kappa (x)}(\mathcal{F}_ x \otimes \kappa (x))$ in any point $x \in X$ never exceeds $c$. In particular $H_ p = \emptyset$ for $p > c$. Moreover, note that there certainly is an inclusion $\coprod H_ p \to F_ d$. Having said this the content of the lemma is that, if a base change $\mathcal{F}_ T$ is flat in dimensions $\geq d$ over $T$ and if $t \in T$, then $\mathcal{F}_ T$ is free of some rank $r$ in an open neighbourhood $U \subset X_ T$ of the unique generic point $\xi$ of $X_ t$. Namely, then $H_ r$ contains the image of $U$ which is an open neighbourhood of $t$. The existence of $U$ follows from More on Morphisms, Lemma 37.16.7. $\square$

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