## Tag `0CLZ`

Chapter 87: Morphisms of Algebraic Stacks > Section 87.42: Valuative criterion for properness

Lemma 87.42.1. Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. Assume $f$ is of finite type and and quasi-separated. Then the following are equivalent

- $f$ is proper, and
- $f$ satisfies both the uniqueness and existence parts of the valuative criterion.

Proof.A proper morphism is the same thing as a separated, finite type, and universally closed morphism. Thus this lemma follows from Lemmas 87.40.1, 87.40.2, 87.41.1, and 87.41.2. $\square$

The code snippet corresponding to this tag is a part of the file `stacks-morphisms.tex` and is located in lines 9283–9293 (see updates for more information).

```
\begin{lemma}
\label{lemma-criterion-proper}
Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks.
Assume $f$ is of finite type and and quasi-separated.
Then the following are equivalent
\begin{enumerate}
\item $f$ is proper, and
\item $f$ satisfies both the uniqueness and existence parts
of the valuative criterion.
\end{enumerate}
\end{lemma}
\begin{proof}
A proper morphism is the same thing as a separated, finite type, and
universally closed morphism. Thus this lemma follows from Lemmas
\ref{lemma-uniqueness-and-diagonal},
\ref{lemma-converse-uniqueness-and-diagonal},
\ref{lemma-quasi-compact-existence-universally-closed}, and
\ref{lemma-converse-existence-universally-closed}.
\end{proof}
```

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