The Stacks project

Lemma 28.15.5. Let $X$ be a scheme and $x \in X$. Let $X_ i$, $i \in I$ be the irreducible components of $X$ passing through $x$. Then the number of (geometric) branches of $X$ at $x$ is the sum over $i \in I$ of the number of (geometric) branches of $X_ i$ at $x$.

Proof. We view the $X_ i$ as integral closed subschemes of $X$, see Schemes, Definition 26.12.5 and Lemma 28.3.4. Observe that the number of (geometric) branches of $X_ i$ at $x$ is at least $1$ for all $i$ (essentially by definition). Recall that the $X_ i$ correspond $1$-to-$1$ with the minimal prime ideals $\mathfrak p_ i \subset \mathcal{O}_{X, x}$, see Algebra, Lemma 10.26.3. Thus, if $I$ is infinite, then $\mathcal{O}_{X, x}$ has infinitely many minimal primes, whence both $\mathcal{O}_{X, x}^ h$ and $\mathcal{O}_{X, x}^{sh}$ have infinitely many minimal primes (combine Algebra, Lemmas 10.30.5 and 10.30.7 and the injectivity of the maps $\mathcal{O}_{X, x} \to \mathcal{O}_{X, x}^ h \to \mathcal{O}_{X, x}^{sh}$). In this case the number of (geometric) branches of $X$ at $x$ is defined to be $\infty $ which is also true for the sum. Thus we may assume $I$ is finite. Let $A'$ be the integral closure of $\mathcal{O}_{X, x}$ in the total ring of fractions $Q$ of $(\mathcal{O}_{X, x})_{red}$. Let $A'_ i$ be the integral closure of $\mathcal{O}_{X, x}/\mathfrak p_ i$ in the total ring of fractions $Q_ i$ of $\mathcal{O}_{X, x}/\mathfrak p_ i$. By Algebra, Lemma 10.25.4 we have $Q = \prod _{i \in I} Q_ i$. Thus $A' = \prod A'_ i$. Then the equality of the lemma follows from More on Algebra, Lemma 15.106.7 which expresses the number of (geometric) branches in terms of the maximal ideals of $A'$. $\square$

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Your email address will not be published.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 0E20. Beware of the difference between the letter 'O' and the digit '0'.