Lemma 33.40.1. Let $X$ be a scheme. Assume every quasi-compact open of $X$ has finitely many irreducible components. Let $\nu : X^\nu \to X$ be the normalization of $X$. Let $x \in X$.

The number of branches of $X$ at $x$ is the number of inverse images of $x$ in $X^\nu $.

The number of geometric branches of $X$ at $x$ is $\sum _{\nu (x^\nu ) = x} [\kappa (x^\nu ) : \kappa (x)]_ s$.

## Comments (0)