The Stacks Project


Tag 06MW

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

Lemma 90.11.10. Let $\mathcal{X}$ be an algebraic stack. Let $x \in |\mathcal{X}|$. Assume that the residual gerbe $\mathcal{Z}_x$ of $\mathcal{X}$ exists. Let $f : \mathop{\rm Spec}(K) \to \mathcal{X}$ be a morphism where $K$ is a field in the equivalence class of $x$. Then $f$ factors through the inclusion morphism $\mathcal{Z}_x \to \mathcal{X}$.

Proof. Choose a field $k$ and a surjective flat locally finite presentation morphism $\mathop{\rm Spec}(k) \to \mathcal{Z}_x$. Set $T = \mathop{\rm Spec}(K) \times_\mathcal{X} \mathcal{Z}_x$. By Lemma 90.4.3 we see that $T$ is nonempty. As $\mathcal{Z}_x \to \mathcal{X}$ is a monomorphism we see that $T \to \mathop{\rm Spec}(K)$ is a monomorphism. Hence by Morphisms of Spaces, Lemma 58.10.8 we see that $T = \mathop{\rm Spec}(K)$ which proves the lemma. $\square$

    The code snippet corresponding to this tag is a part of the file stacks-properties.tex and is located in lines 2802–2809 (see updates for more information).

    \begin{lemma}
    \label{lemma-residual-gerbe-points}
    Let $\mathcal{X}$ be an algebraic stack. Let $x \in |\mathcal{X}|$.
    Assume that the residual gerbe $\mathcal{Z}_x$ of $\mathcal{X}$ exists.
    Let $f : \Spec(K) \to \mathcal{X}$ be a morphism where $K$ is a field
    in the equivalence class of $x$. Then $f$ factors through the inclusion
    morphism $\mathcal{Z}_x \to \mathcal{X}$.
    \end{lemma}
    
    \begin{proof}
    Choose a field $k$ and a surjective flat locally finite presentation
    morphism $\Spec(k) \to \mathcal{Z}_x$. Set
    $T = \Spec(K) \times_\mathcal{X} \mathcal{Z}_x$. By
    Lemma \ref{lemma-points-cartesian}
    we see that $T$ is nonempty. As $\mathcal{Z}_x \to \mathcal{X}$
    is a monomorphism we see that $T \to \Spec(K)$ is a monomorphism.
    Hence by
    Morphisms of Spaces, Lemma
    \ref{spaces-morphisms-lemma-monomorphism-toward-field}
    we see that $T = \Spec(K)$ which proves the lemma.
    \end{proof}

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