The Stacks project

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 ( and $H_ p$ as in (

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. $\square$

Comments (0)

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 05UE. Beware of the difference between the letter 'O' and the digit '0'.