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Tag 0DQG

89.13. Local irreducibility

We have defined the geometric number of branches of a scheme at a point in Properties, Section 27.15 and for na 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 15.88.8. 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 89.7.5 applies and we obtain a notion for algebraic stacks at a point.

Definition 89.13.1. Let $\mathcal{X}$ be an algebraic stack. Let $x \in |\mathcal{X}|$.

  1. The number of geometric branches of $\mathcal{X}$ at $x$ is either $n \in \mathbf{N}$ if the equivalent conditions of Lemma 89.7.4 hold for $\mathcal{P}_n$ defined above, or else $\infty$.
  2. 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 na 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}

    Comments (1)

    Comment #2604 by Matthieu Romagny on June 18, 2017 a 8:02 pm UTC

    Typo : replace 'na' by 'an' in first sentence of this section.

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