The Stacks project

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. $\square$