The Stacks project

100.13 Local irreducibility

We have defined the geometric number of branches of a scheme at a point in Properties, Section 28.15 and for an algebraic space at a point in Properties of Spaces, Section 66.23. Let $n \in \mathbf{N}$. For a local ring $A$ set

For a smooth ring map $A \to B$ and a prime ideal $\mathfrak q$ of $B$ lying over $\mathfrak p$ of $A$ we have

by More on Algebra, Lemma 15.106.8. As in Properties of Spaces, Remark 66.7.6 we may use $P_ n$ to define an étale local property $\mathcal{P}_ n$ of germs $(U, u)$ of schemes by setting $\mathcal{P}_ n(U, u) = P_ n(\mathcal{O}_{U, u})$. The corresponding property $\mathcal{P}_ n$ of an algebraic space $X$ at a point $x$ (see Properties of Spaces, Definition 66.7.5) is just the property “the number of geometric branches of $X$ at $x$ is $n$”, see Properties of Spaces, Definition 66.23.4. Moreover, the property $\mathcal{P}_ n$ is smooth local, see Descent, Definition 35.21.1. This follows either from the equivalence displayed above or More on Morphisms, Lemma 37.36.4. Thus Definition 100.7.5 applies and we obtain a notion for algebraic stacks at a point.