## Tag `0DQG`

## 90.13. Local irreducibility

We have defined the geometric number of branches of a scheme at a point in Properties, Section 27.15 and for an algebraic space at a point in Properties of Spaces, Section 57.22. 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 15.89.8. As in Properties of Spaces, Remark 57.7.6 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 spaces $X$ at a point $x$ (see Properties of Spaces, Definition 57.7.5) is just the property ''the number of geometric branches of $X$ at $x$ is $n$'', see Properties of Spaces, Definition 57.22.4. Moreover, the property $\mathcal{P}_n$ is smooth local, see Descent, Definition 34.18.1. This follows either from the equivalence displayed above or More on Morphisms, Lemma 36.33.4. Thus Definition 90.7.5 applies and we obtain a notion for algebraic stacks at a point.

Definition 90.13.1. 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 90.7.4 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 $1$.

The code snippet corresponding to this tag is a part of the file `stacks-properties.tex` and is located in lines 3071–3128 (see updates for more information).

```
\section{Local irreducibility}
\label{section-irreducible-local-ring}
\noindent
We have defined the geometric number of branches of a scheme at a point
in Properties, Section \ref{properties-section-unibranch}
and for an algebraic space at a point in Properties of Spaces, Section
\ref{spaces-properties-section-irreducible-local-ring}.
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
\ref{more-algebra-lemma-invariance-number-branches-smooth}.
As in Properties of Spaces, Remark
\ref{spaces-properties-remark-list-properties-local-ring-local-etale-topology}
we may use $P_n$ to define an \'etale 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 spaces $X$ at a point $x$
(see Properties of Spaces, Definition
\ref{spaces-properties-definition-property-at-point})
is just the property
``the number of geometric branches of $X$ at $x$ is $n$'', see
Properties of Spaces, Definition
\ref{spaces-properties-definition-number-of-geometric-branches}.
Moreover, the property $\mathcal{P}_n$ is smooth local, see
Descent, Definition \ref{descent-definition-local-at-point}.
This follows either from the equivalence displayed above
or More on Morphisms, Lemma
\ref{more-morphisms-lemma-number-of-branches-and-smooth}.
Thus Definition \ref{definition-property-at-point}
applies and we obtain a notion for algebraic stacks at a point.
\begin{definition}
\label{definition-number-of-geometric-branches}
Let $\mathcal{X}$ be an algebraic stack. Let $x \in |\mathcal{X}|$.
\begin{enumerate}
\item The {\it number of geometric branches of $\mathcal{X}$ at $x$}
is either $n \in \mathbf{N}$ if the equivalent conditions of
Lemma \ref{lemma-local-source-target-at-point} hold for
$\mathcal{P}_n$ defined above, or else $\infty$.
\item We say $\mathcal{X}$ is {\it geometrically unibranch at $x$}
if the number of geometric branches of $\mathcal{X}$ at $x$ is $1$.
\end{enumerate}
\end{definition}
```

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