Lemma 10.36.21. Suppose $R \to S$ is finite. Then the fibres of $\mathop{\mathrm{Spec}}(S) \to \mathop{\mathrm{Spec}}(R)$ are finite.

**Proof.**
By the discussion in Remark 10.17.8 the fibres are the spectra of the rings $S \otimes _ R \kappa (\mathfrak p)$. As $R \to S$ is finite, these fibre rings are finite over $\kappa (\mathfrak p)$ hence Noetherian by Lemma 10.31.1. By Lemma 10.36.20 every prime of $S \otimes _ R \kappa (\mathfrak p)$ is a minimal prime. Hence by Lemma 10.31.6 there are at most finitely many.
$\square$

## 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Your email address will not be published.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.

## Comments (0)

There are also: