# The Stacks Project

## Tag 0CLK

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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