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Tag 0CXI

Chapter 88: Artin's axioms > Section 88.11: Limit preserving

Lemma 88.11.4. Let $S$ be a scheme. Let $\mathcal{X}$ be a category fibred in groupoids over $(\textit{Sch}/S)_{fppf}$. Assume $\Delta : \mathcal{X} \to \mathcal{X} \times \mathcal{X}$ is representable by algebraic spaces and $\mathcal{X}$ is limit preserving. Then $\Delta$ is locally of finite type.

Proof. We apply Criteria for Representability, Lemma 87.5.6. Let $V$ be an affine scheme $V$ of finite type over $S$ and let $\theta$ be an object of $\mathcal{X} \times \mathcal{X}$ over $V$. Let $F_\theta$ be an algebraic space representing $\mathcal{X} \times_{\Delta, \mathcal{X} \times \mathcal{X}, \theta} (\textit{Sch}/V)_{fppf}$ and let $f_\theta : F_\theta \to V$ be the canonical morphism (see Algebraic Stacks, Section 84.9). It suffices to show that $F_\theta \to V$ has the corresponding properties. By Lemmas 88.11.2 and 88.11.3 we see that $F_\theta \to S$ is locally of finite presentation. It follows that $F_\theta \to V$ is locally of finite type by Morphisms of Spaces, Lemma 58.23.6. $\square$

    The code snippet corresponding to this tag is a part of the file artin.tex and is located in lines 1398–1405 (see updates for more information).

    \begin{lemma}
    \label{lemma-diagonal}
    Let $S$ be a scheme. Let $\mathcal{X}$ be a category fibred
    in groupoids over $(\Sch/S)_{fppf}$. Assume
    $\Delta : \mathcal{X} \to \mathcal{X} \times \mathcal{X}$ is
    representable by algebraic spaces and $\mathcal{X}$ is limit preserving.
    Then $\Delta$ is locally of finite type.
    \end{lemma}
    
    \begin{proof}
    We apply Criteria for Representability, Lemma
    \ref{criteria-lemma-check-property-limit-preserving}.
    Let $V$ be an affine scheme $V$ of finite type over $S$
    and let $\theta$ be an object of $\mathcal{X} \times \mathcal{X}$
    over $V$. Let $F_\theta$ be an algebraic space representing
    $\mathcal{X} \times_{\Delta, \mathcal{X} \times \mathcal{X}, \theta}
    (\Sch/V)_{fppf}$ and let $f_\theta : F_\theta \to V$ be the canonical morphism
    (see Algebraic Stacks, Section
    \ref{algebraic-section-morphisms-representable-by-algebraic-spaces}).
    It suffices to show that
    $F_\theta \to V$ has the corresponding properties. By
    Lemmas \ref{lemma-fibre-product-limit-preserving} and
    \ref{lemma-limit-preserving-algebraic-space}
    we see that $F_\theta \to S$ is locally of finite presentation.
    It follows that $F_\theta \to V$ is locally of finite type
    by Morphisms of Spaces, Lemma
    \ref{spaces-morphisms-lemma-permanence-finite-type}.
    \end{proof}

    Comments (2)

    Comment #2644 by Xiaowen on July 12, 2017 a 9:22 am UTC

    $Y$ in the proof should be $F_\theta$.

    Comment #2664 by Johan (site) on July 28, 2017 a 5:12 pm UTC

    Thanks, fixed here.

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