## Tag `0A3W`

Chapter 58: Morphisms of Algebraic Spaces > Section 58.40: Valuative criteria

Lemma 58.40.5. Let $S$ be a scheme. Let $f : X \to Y$ be a separated morphism of algebraic spaces over $S$. The following are equivalent

- $f$ satisfies the existence part of the valuative criterion as in Definition 58.40.1,
- given any commutative solid diagram $$ \xymatrix{ \mathop{\rm Spec}(K) \ar[r] \ar[d] & X \ar[d] \\ \mathop{\rm Spec}(A) \ar[r] \ar@{-->}[ru] & Y } $$ where $A$ is a valuation ring with field of fractions $K$, there exists a dotted arrow, i.e., $f$ satisfies the existence part of the valuative criterion as in Schemes, Definition 25.20.3.

Proof.We have to show that (1) implies (2). Suppose given a commutative diagram $$ \xymatrix{ \mathop{\rm Spec}(K) \ar[r] \ar[d] & X \ar[d] \\ \mathop{\rm Spec}(A) \ar[r] & Y } $$ as in part (2). By (1) there exists a commutative diagram $$ \xymatrix{ \mathop{\rm Spec}(K') \ar[r] \ar[d] & \mathop{\rm Spec}(K) \ar[r] & X \ar[d] \\ \mathop{\rm Spec}(A') \ar[r] \ar[rru] & \mathop{\rm Spec}(A) \ar[r] & Y } $$ as in Definition 58.40.1 with $K \subset K'$ arbitrary. By Lemma 58.40.4 we can find a morphism $\mathop{\rm Spec}(A) \to X$ fitting into the diagram, i.e., (2) holds. $\square$

The code snippet corresponding to this tag is a part of the file `spaces-morphisms.tex` and is located in lines 8276–8295 (see updates for more information).

```
\begin{lemma}
\label{lemma-usual-enough}
Let $S$ be a scheme. Let $f : X \to Y$ be a separated morphism of
algebraic spaces over $S$. The following are equivalent
\begin{enumerate}
\item $f$ satisfies the existence part of the valuative criterion
as in Definition \ref{definition-valuative-criterion},
\item given any commutative solid diagram
$$
\xymatrix{
\Spec(K) \ar[r] \ar[d] & X \ar[d] \\
\Spec(A) \ar[r] \ar@{-->}[ru] & Y
}
$$
where $A$ is a valuation ring with field of fractions $K$, there exists
a dotted arrow, i.e., $f$ satisfies the existence part of the valuative
criterion as in
Schemes, Definition \ref{schemes-definition-valuative-criterion}.
\end{enumerate}
\end{lemma}
\begin{proof}
We have to show that (1) implies (2). Suppose given a commutative diagram
$$
\xymatrix{
\Spec(K) \ar[r] \ar[d] & X \ar[d] \\
\Spec(A) \ar[r] & Y
}
$$
as in part (2). By (1) there exists a commutative diagram
$$
\xymatrix{
\Spec(K') \ar[r] \ar[d] & \Spec(K) \ar[r] & X \ar[d] \\
\Spec(A') \ar[r] \ar[rru] & \Spec(A) \ar[r] & Y
}
$$
as in Definition \ref{definition-valuative-criterion} with $K \subset K'$
arbitrary. By Lemma \ref{lemma-push-down-solution} we can find a morphism
$\Spec(A) \to X$ fitting into the diagram, i.e., (2) holds.
\end{proof}
```

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