# The Stacks Project

## Tag 0CLZ

Lemma 91.42.1. 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

1. $f$ is proper, and
2. $f$ 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 91.40.1, 91.40.2, 91.41.1, and 91.41.2. $\square$

The code snippet corresponding to this tag is a part of the file stacks-morphisms.tex and is located in lines 9327–9337 (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}

There are no comments yet for this tag.

## Add a comment on tag 0CLZ

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the lower-right corner).