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Tag 0DTH

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

Lemma 90.11.12. Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. Let $x \in |\mathcal{X}|$ with image $y \in |\mathcal{Y}|$. If the residual gerbes $\mathcal{Z}_x \subset \mathcal{X}$ and $\mathcal{Z}_y \subset \mathcal{Y}$ of $x$ and $y$ exist, then $f$ induces a commutative diagram $$ \xymatrix{ \mathcal{X} \ar[d]_f & \mathcal{Z}_x \ar[l] \ar[d] \\ \mathcal{Y} & \mathcal{Z}_y \ar[l] } $$

Proof. Choose a field $k$ and a surjective, flat, locally finitely presented morphism $\mathop{\rm Spec}(k) \to \mathcal{Z}_x$. The morphism $\mathop{\rm Spec}(k) \to \mathcal{Y}$ factors through $\mathcal{Z}_y$ by Lemma 90.11.10. Thus $\mathcal{Z}_x \times_\mathcal{Y} \mathcal{Z}_y$ is a nonempty substack of $\mathcal{Z}_x$ hence equal to $\mathcal{Z}_x$ by Lemma 90.11.4. $\square$

    The code snippet corresponding to this tag is a part of the file stacks-properties.tex and is located in lines 2849–2862 (see updates for more information).

    \begin{lemma}
    \label{lemma-residual-gerbe-functorial}
    Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks.
    Let $x \in |\mathcal{X}|$ with image $y \in |\mathcal{Y}|$.
    If the residual gerbes $\mathcal{Z}_x \subset \mathcal{X}$
    and $\mathcal{Z}_y \subset \mathcal{Y}$ of $x$ and $y$ exist,
    then $f$ induces a commutative diagram
    $$
    \xymatrix{
    \mathcal{X} \ar[d]_f & \mathcal{Z}_x \ar[l] \ar[d] \\
    \mathcal{Y} & \mathcal{Z}_y \ar[l]
    }
    $$
    \end{lemma}
    
    \begin{proof}
    Choose a field $k$ and a surjective, flat, locally finitely presented
    morphism $\Spec(k) \to \mathcal{Z}_x$. The morphism
    $\Spec(k) \to \mathcal{Y}$ factors through $\mathcal{Z}_y$ by
    Lemma \ref{lemma-residual-gerbe-points}.
    Thus $\mathcal{Z}_x \times_\mathcal{Y} \mathcal{Z}_y$
    is a nonempty substack of $\mathcal{Z}_x$
    hence equal to $\mathcal{Z}_x$ by Lemma \ref{lemma-monomorphism-into-point}.
    \end{proof}

    Comments (2)

    Comment #2647 by Daniel Dore on July 16, 2017 a 4:13 am UTC

    Shouldn't the application of Lemma 90.11.12 be to show the morphism $\mathrm{Spec}(k) \rightarrow \mathcal{Y}$ factors through $\mathcal{Z}_y$, not through $\mathcal{Z}_x$?

    Comment #2667 by Johan (site) on July 28, 2017 a 5:27 pm UTC

    Yes and thanks. Fix is here.

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

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