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

Definition 91.38.10. Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. We say $f$ satisfies the existence part of the valuative criterion if for every diagram (91.38.1.1) and $\gamma$ as in Definition 91.38.1 there exists an extension $K'/K$ of fields, a valuation ring $A' \subset K'$ dominating $A$ such that the category of dotted arrows for the outer rectangle of the diagram $$ \xymatrix{ \mathop{\rm Spec}(K') \ar[r] \ar@/^2em/[rr]_{x'} \ar[d]_{j'} & \mathop{\rm Spec}(K) \ar[d]_j \ar[r]_-x & \mathcal{X} \ar[d]^f \\ \mathop{\rm Spec}(A') \ar[r] \ar@/_2em/[rr]^{y'} & \mathop{\rm Spec}(A) \ar[r]^-y & \mathcal{Y} } $$ with induced $2$-arrow $\gamma' : y' \circ j' \to f \circ x'$ is nonempty.

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    \begin{definition}
    \label{definition-existence}
    Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks.
    We say $f$ satisfies the {\it existence part of the valuative criterion}
    if for every diagram (\ref{equation-diagram}) and $\gamma$
    as in Definition \ref{definition-fill-in-diagram}
    there exists an extension $K'/K$ of fields, a valuation ring $A' \subset K'$
    dominating $A$ such that the category of dotted arrows for the
    outer rectangle of the diagram
    $$
    \xymatrix{
    \Spec(K') \ar[r] \ar@/^2em/[rr]_{x'} \ar[d]_{j'} &
    \Spec(K) \ar[d]_j \ar[r]_-x &
    \mathcal{X} \ar[d]^f \\
    \Spec(A') \ar[r] \ar@/_2em/[rr]^{y'} &
    \Spec(A) \ar[r]^-y &
    \mathcal{Y}
    }
    $$
    with induced $2$-arrow $\gamma' : y' \circ j' \to f \circ x'$ is nonempty.
    \end{definition}

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