## Tag `0CLK`

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 criterionif 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.

The code snippet corresponding to this tag is a part of the file `stacks-morphisms.tex` and is located in lines 8849–8869 (see updates for more information).

```
\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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