## Tag `0A40`

Chapter 55: Morphisms of Algebraic Spaces > Section 55.43: Valuative criterion of properness

Lemma 55.43.1 (Valuative criterion for properness). Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Assume $f$ is of finite type and quasi-separated. Then the following are equivalent

- $f$ is proper,
- the valuative criterion holds as in Definition 55.40.1,
- 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 unique dotted arrow, i.e., $f$ satisfies the valuative criterion as in Schemes, Definition 25.20.3.

Proof.Formal consequence of Lemma 55.42.3 and the definitions. $\square$

The code snippet corresponding to this tag is a part of the file `spaces-morphisms.tex` and is located in lines 8930–8951 (see updates for more information).

```
\begin{lemma}[Valuative criterion for properness]
\label{lemma-characterize-proper}
Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces
over $S$. Assume $f$ is of finite type and quasi-separated. Then the
following are equivalent
\begin{enumerate}
\item $f$ is proper,
\item the valuative criterion holds 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 unique dotted arrow, i.e., $f$ satisfies the valuative
criterion as in
Schemes, Definition \ref{schemes-definition-valuative-criterion}.
\end{enumerate}
\end{lemma}
\begin{proof}
Formal consequence of
Lemma \ref{lemma-characterize-separated-and-universally-closed}
and the definitions.
\end{proof}
```

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