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unramified => loc qf for stacks
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aisejohan committed Jan 16, 2023
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Expand Up @@ -8166,6 +8166,21 @@ \section{Unramified morphisms}
This finishes the proof.
\end{proof}

\begin{lemma}
\label{lemma-unramified-quasi-finite}
An unramified morphism of algebraic stacks is locally quasi-finite.
\end{lemma}

\begin{proof}
This follows from
Lemma \ref{lemma-unramified} (characterizing unramified morphisms),
Lemma \ref{lemma-characterize-locally-quasi-finite} (characterizing
locally quasi-finite morphisms), and
Morphisms of Spaces, Lemma
\ref{spaces-morphisms-lemma-unramified-quasi-finite} (the corresponding
result for algebraic spaces).
\end{proof}

\begin{lemma}
\label{lemma-permanence-unramified}
Let $\mathcal{X} \to \mathcal{Y} \to \mathcal{Z}$ be
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