The Stacks project

Lemma 76.36.2. In Lemma 76.36.1 assume in addition that $f$ is locally of finite type and $Y$ affine. Then for $y \in Y$ the fibre $\pi ^{-1}(\{ y\} ) = \{ y_1, \ldots , y_ n\} $ is finite and the field extensions $\kappa (y_ i)/\kappa (y)$ are finite.

Proof. Recall that there are no specializations among the points of $\pi ^{-1}(\{ y\} )$, see Algebra, Lemma 10.36.20. As $f'$ is surjective, we find that $|X_ y| \to \pi ^{-1}(\{ y\} )$ is surjective. Observe that $X_ y$ is a quasi-separated algebraic space of finite type over a field (quasi-compactness was shown in the proof of the referenced lemma). Thus $|X_ y|$ is a Noetherian topological space (Morphisms of Spaces, Lemma 67.28.6). A topological argument (omitted) now shows that $\pi ^{-1}(\{ y\} )$ is finite. For each $i$ we can pick a finite type point $x_ i \in |X_ y|$ mapping to $y_ i$ (Morphisms of Spaces, Lemma 67.25.6). We conclude that $\kappa (y_ i)/\kappa (y)$ is finite: $x_ i$ can be represented by a morphism $\mathop{\mathrm{Spec}}(k_ i) \to X_ y$ of finite type (by our definition of finite type points) and hence $\mathop{\mathrm{Spec}}(k_ i) \to y = \mathop{\mathrm{Spec}}(\kappa (y))$ is of finite type (as a composition of finite type morphisms), hence $k_ i/\kappa (y)$ is finite (Morphisms, Lemma 29.16.1). $\square$


Comments (1)

Comment #8715 by Torsten on

Is the condition that is affine really needed?


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Your email address will not be published.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 0E1C. Beware of the difference between the letter 'O' and the digit '0'.