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Tag 07XN

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

Lemma 88.11.3. Let $S$ be a scheme. Let $\mathcal{X}$ be an algebraic stack over $S$. Then the following are equivalent

  1. $\mathcal{X}$ is a stack in setoids and $\mathcal{X} \to (\textit{Sch}/S)_{fppf}$ is limit preserving on objects,
  2. $\mathcal{X}$ is a stack in setoids and limit preserving,
  3. $\mathcal{X}$ is representable by an algebraic space locally of finite presentation.

Proof. Under each of the three assumptions $\mathcal{X}$ is representable by an algebraic space $X$ over $S$, see Algebraic Stacks, Proposition 84.13.3. It is clear that (1) and (2) are equivalent as a functor between setoids is an equivalence if and only if it is surjective on isomorphism classes. Finally, (1) and (3) are equivalent by Limits of Spaces, Proposition 61.3.9. $\square$

    The code snippet corresponding to this tag is a part of the file artin.tex and is located in lines 1374–1385 (see updates for more information).

    \begin{lemma}
    \label{lemma-limit-preserving-algebraic-space}
    Let $S$ be a scheme. Let $\mathcal{X}$ be an algebraic stack over $S$.
    Then the following are equivalent
    \begin{enumerate}
    \item $\mathcal{X}$ is a stack in setoids and
    $\mathcal{X} \to (\Sch/S)_{fppf}$ is limit preserving on objects,
    \item $\mathcal{X}$ is a stack in setoids and limit preserving,
    \item $\mathcal{X}$ is representable by an algebraic space
    locally of finite presentation.
    \end{enumerate}
    \end{lemma}
    
    \begin{proof}
    Under each of the three assumptions $\mathcal{X}$ is representable
    by an algebraic space $X$ over $S$, see Algebraic Stacks, Proposition
    \ref{algebraic-proposition-algebraic-stack-no-automorphisms}.
    It is clear that (1) and (2) are equivalent as a functor between
    setoids is an equivalence if and only if it is surjective on isomorphism
    classes. Finally, (1) and (3) are equivalent by
    Limits of Spaces, Proposition
    \ref{spaces-limits-proposition-characterize-locally-finite-presentation}.
    \end{proof}

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