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Tag 0CLZ

Chapter 91: Morphisms of Algebraic Stacks > Section 91.42: Valuative criterion for properness

Lemma 91.42.1. Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. Assume $f$ is of finite type and and quasi-separated. Then the following are equivalent

  1. $f$ is proper, and
  2. $f$ satisfies both the uniqueness and existence parts of the valuative criterion.

Proof. A proper morphism is the same thing as a separated, finite type, and universally closed morphism. Thus this lemma follows from Lemmas 91.40.2, 91.40.3, 91.41.1, and 91.41.2. $\square$

    The code snippet corresponding to this tag is a part of the file stacks-morphisms.tex and is located in lines 9393–9403 (see updates for more information).

    \begin{lemma}
    \label{lemma-criterion-proper}
    Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks.
    Assume $f$ is of finite type and and quasi-separated.
    Then the following are equivalent
    \begin{enumerate}
    \item $f$ is proper, and
    \item $f$ satisfies both the uniqueness and existence parts
    of the valuative criterion.
    \end{enumerate}
    \end{lemma}
    
    \begin{proof}
    A proper morphism is the same thing as a separated, finite type, and
    universally closed morphism. Thus this lemma follows from Lemmas
    \ref{lemma-uniqueness-and-diagonal},
    \ref{lemma-converse-uniqueness-and-diagonal},
    \ref{lemma-quasi-compact-existence-universally-closed}, and
    \ref{lemma-converse-existence-universally-closed}.
    \end{proof}

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