The Stacks Project


Tag 06MX

Chapter 90: Properties of Algebraic Stacks > Section 90.11: Residual gerbes

Lemma 90.11.11. Let $\mathcal{X}$ be an algebraic stack. Let $x \in |\mathcal{X}|$. Let $\mathcal{Z}$ be an algebraic stack satisfying the equivalent conditions of Lemma 90.11.3 and let $\mathcal{Z} \to \mathcal{X}$ be a monomorphism such that the image of $|\mathcal{Z}| \to |\mathcal{X}|$ is $x$. Then the residual gerbe $\mathcal{Z}_x$ of $\mathcal{X}$ at $x$ exists and $\mathcal{Z} \to \mathcal{X}$ factors as $\mathcal{Z} \to \mathcal{Z}_x \to \mathcal{X}$ where the first arrow is an equivalence.

Proof. Let $\mathcal{Z}_x \subset \mathcal{X}$ be the full subcategory corresponding to the essential image of the functor $\mathcal{Z} \to \mathcal{X}$. Then $\mathcal{Z} \to \mathcal{Z}_x$ is an equivalence, hence $\mathcal{Z}_x$ is an algebraic stack, see Algebraic Stacks, Lemma 84.12.4. Since $\mathcal{Z}_x$ inherits all the properties of $\mathcal{Z}$ from this equivalence it is clear from the uniqueness in Lemma 90.11.7 that $\mathcal{Z}_x$ is the residual gerbe of $\mathcal{X}$ at $x$. $\square$

    The code snippet corresponding to this tag is a part of the file stacks-properties.tex and is located in lines 2824–2835 (see updates for more information).

    \begin{lemma}
    \label{lemma-residual-gerbe-unique}
    Let $\mathcal{X}$ be an algebraic stack. Let $x \in |\mathcal{X}|$.
    Let $\mathcal{Z}$ be an algebraic stack satisfying the equivalent conditions of
    Lemma \ref{lemma-unique-point-better}
    and let $\mathcal{Z} \to \mathcal{X}$ be a monomorphism such that the image
    of $|\mathcal{Z}| \to |\mathcal{X}|$ is $x$. Then the residual gerbe
    $\mathcal{Z}_x$ of $\mathcal{X}$ at $x$ exists and
    $\mathcal{Z} \to \mathcal{X}$ factors as
    $\mathcal{Z} \to \mathcal{Z}_x \to \mathcal{X}$ where the first arrow
    is an equivalence.
    \end{lemma}
    
    \begin{proof}
    Let $\mathcal{Z}_x \subset \mathcal{X}$ be the full subcategory corresponding
    to the essential image of the functor $\mathcal{Z} \to \mathcal{X}$.
    Then $\mathcal{Z} \to \mathcal{Z}_x$ is an equivalence, hence
    $\mathcal{Z}_x$ is an algebraic stack, see
    Algebraic Stacks, Lemma \ref{algebraic-lemma-equivalent}.
    Since $\mathcal{Z}_x$ inherits all the properties of $\mathcal{Z}$ from
    this equivalence it is clear from the uniqueness in
    Lemma \ref{lemma-residual-gerbe}
    that $\mathcal{Z}_x$ is the residual gerbe of $\mathcal{X}$ at $x$.
    \end{proof}

    Comments (0)

    There are no comments yet for this tag.

    There are also 2 comments on Section 90.11: Properties of Algebraic Stacks.

    Add a comment on tag 06MX

    Your email address will not be published. Required fields are marked.

    In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the lower-right corner).

    All contributions are licensed under the GNU Free Documentation License.




    In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following box. So in case this where tag 0321 you just have to write 0321. Beware of the difference between the letter 'O' and the digit 0.

    This captcha seems more appropriate than the usual illegible gibberish, right?