The Stacks project

Lemma 33.39.2. Let $k$ be a field. Let $X$ be a locally algebraic $k$-scheme. Let $K/k$ be an extension of fields. Let $y \in X_ K$ be a point with image $x$ in $X$. Then the number of geometric branches of $X$ at $x$ is the number of geometric branches of $X_ K$ at $y$.

Proof. Write $Y = X_ K$ and let $X^\nu $, resp. $Y^\nu $ be the normalization of $X$, resp. $Y$. Consider the commutative diagram

\[ \xymatrix{ Y^\nu \ar[r] \ar[d] & X^\nu _ K \ar[r] \ar[d]_{\nu _ K} & X^\nu \ar[d]_\nu \\ Y \ar@{=}[r] & Y \ar[r] & X } \]

By Lemma 33.27.5 we see that the left top horizontal arrow is a universal homeomorphism. Hence it induces purely inseparable residue field extensions, see Morphisms, Lemmas 29.45.5 and 29.10.2. Thus the number of geometric branches of $Y$ at $y$ is $\sum _{\nu _ K(y') = y} [\kappa (y') : \kappa (y)]_ s$ by Lemma 33.39.1. Similarly $\sum _{\nu (x') = x} [\kappa (x') : \kappa (x)]_ s$ is the number of geometric branches of $X$ at $x$. Using Schemes, Lemma 26.17.5 our statement follows from the following algebra fact: given a field extension $l/\kappa $ and an algebraic field extension $m/\kappa $, then

\[ \sum \nolimits _{m \otimes _\kappa l \to m'} [m' : l']_ s = [m : \kappa ]_ s \]

where the sum is over the quotient fields of $m \otimes _\kappa l$. One can prove this in an elementary way, or one can use Lemma 33.7.6 applied to

\[ \mathop{\mathrm{Spec}}(m \otimes _\kappa l) \times _{\mathop{\mathrm{Spec}}(l)} \mathop{\mathrm{Spec}}(\overline{l}) = \mathop{\mathrm{Spec}}(m) \otimes _{\mathop{\mathrm{Spec}}(\kappa )} \mathop{\mathrm{Spec}}(\overline{l}) \longrightarrow \mathop{\mathrm{Spec}}(m) \times _{\mathop{\mathrm{Spec}}(\kappa )} \mathop{\mathrm{Spec}}(\overline{\kappa }) \]

because one can interpret $[m : \kappa ]_ s$ as the number of connected components of the right hand side and the sum $\sum _{m \otimes _\kappa l \to m'} [m' : l']_ s$ as the number of connected components of the left hand side. $\square$


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Your email address will not be published.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 0C40. Beware of the difference between the letter 'O' and the digit '0'.