68.24 Local irreducibility
We have already defined the geometric number of branches of an algebraic space at a point in Properties of Spaces, Section 66.23. The number of branches of an algebraic space at a point can only be defined for decent algebraic spaces.
Lemma 68.24.1. Let $S$ be a scheme. Let $X$ be a decent algebraic space over $S$. Let $x \in |X|$ be a point. The following are equivalent
for any elementary étale neighbourhood $(U, u) \to (X, x)$ the local ring $\mathcal{O}_{U, u}$ has a unique minimal prime,
for any elementary étale neighbourhood $(U, u) \to (X, x)$ there is a unique irreducible component of $U$ through $u$,
for any elementary étale neighbourhood $(U, u) \to (X, x)$ the local ring $\mathcal{O}_{U, u}$ is unibranch,
the henselian local ring $\mathcal{O}_{X, x}^ h$ has a unique minimal prime.
Proof.
The equivalence of (1) and (2) follows from the fact that irreducible components of $U$ passing through $u$ are in $1$-$1$ correspondence with minimal primes of the local ring of $U$ at $u$. The ring $\mathcal{O}_{X, x}^ h$ is the henselization of $\mathcal{O}_{U, u}$, see discussion following Definition 68.11.7. In particular (3) and (4) are equivalent by More on Algebra, Lemma 15.106.3. The equivalence of (2) and (3) follows from More on Morphisms, Lemma 37.36.2.
$\square$
Definition 68.24.2. Let $S$ be a scheme. Let $X$ be a decent algebraic space over $S$. Let $x \in |X|$. We say that $X$ is unibranch at $x$ if the equivalent conditions of Lemma 68.24.1 hold. We say that $X$ is unibranch if $X$ is unibranch at every $x \in |X|$.
This is consistent with the definition for schemes (Properties, Definition 28.15.1).
Lemma 68.24.3. Let $S$ be a scheme. Let $X$ be a decent algebraic space over $S$. Let $x \in |X|$ be a point. Let $n \in \{ 1, 2, \ldots \} $ be an integer. The following are equivalent
for any elementary étale neighbourhood $(U, u) \to (X, x)$ the number of minimal primes of the local ring $\mathcal{O}_{U, u}$ is $\leq n$ and for at least one choice of $(U, u)$ it is $n$,
for any elementary étale neighbourhood $(U, u) \to (X, x)$ the number irreducible components of $U$ passing through $u$ is $\leq n$ and for at least one choice of $(U, u)$ it is $n$,
for any elementary étale neighbourhood $(U, u) \to (X, x)$ the number of branches of $U$ at $u$ is $\leq n$ and for at least one choice of $(U, u)$ it is $n$,
the number of minimal prime ideals of $\mathcal{O}_{X, x}^ h$ is $n$.
Proof.
The equivalence of (1) and (2) follows from the fact that irreducible components of $U$ passing through $u$ are in $1$-$1$ correspondence with minimal primes of the local ring of $U$ at $u$. The ring $\mathcal{O}_{X, x}$ is the henselization of $\mathcal{O}_{U, u}$, see discussion following Definition 68.11.7. In particular (3) and (4) are equivalent by More on Algebra, Lemma 15.106.3. The equivalence of (2) and (3) follows from More on Morphisms, Lemma 37.36.2.
$\square$
Definition 68.24.4. Let $S$ be a scheme. Let $X$ be a decent algebraic space over $S$. Let $x \in |X|$. The number of branches of $X$ at $x$ is either $n \in \mathbf{N}$ if the equivalent conditions of Lemma 68.24.3 hold, or else $\infty $.
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