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Tag 0CLG

Chapter 91: Morphisms of Algebraic Stacks > Section 91.38: Valuative criteria

Definition 91.38.6. Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. We say $f$ satisfies the uniqueness part of the valuative criterion if for every diagram (91.38.1.1) and $\gamma$ as in Definition 91.38.1 the category of dotted arrows is either empty or a setoid with exactly one isomorphism class.

    The code snippet corresponding to this tag is a part of the file stacks-morphisms.tex and is located in lines 8816–8824 (see updates for more information).

    \begin{definition}
    \label{definition-uniqueness}
    Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks.
    We say $f$ satisfies the {\it uniqueness part of the valuative criterion}
    if for every diagram (\ref{equation-diagram}) and $\gamma$
    as in Definition \ref{definition-fill-in-diagram}
    the category of dotted arrows is either empty or
    a setoid with exactly one isomorphism class.
    \end{definition}

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