## 99.13 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 65.23. Let $n \in \mathbf{N}$. For a local ring $A$ set

\[ P_ n(A) = \text{the number of geometric branches of }A\text{ is }n \]

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

\[ P_ n(A_\mathfrak p) \Leftrightarrow P_ n(B_\mathfrak q) \]

by More on Algebra, Lemma 15.106.8. As in Properties of Spaces, Remark 65.7.6 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 65.7.5) is just the property “the number of geometric branches of $X$ at $x$ is $n$”, see Properties of Spaces, Definition 65.23.4. 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 99.7.5 applies and we obtain a notion for algebraic stacks at a point.

Definition 99.13.1. Let $\mathcal{X}$ be an algebraic stack. Let $x \in |\mathcal{X}|$.

The *number of geometric branches of $\mathcal{X}$ at $x$* is either $n \in \mathbf{N}$ if the equivalent conditions of Lemma 99.7.4 hold for $\mathcal{P}_ n$ defined above, or else $\infty $.

We say $\mathcal{X}$ is *geometrically unibranch at $x$* if the number of geometric branches of $\mathcal{X}$ at $x$ is $1$.

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