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Tag 02LA

Lemma 34.20.23. The property $\mathcal{P}(f) =$''$f$ is finite'' is fpqc local on the base.

Proof. An finite morphism is the same thing as an integral morphism which is locally of finite type. See Morphisms, Lemma 28.42.4. Hence the lemma follows on combining Lemmas 34.20.10 and 34.20.22. $\square$

The code snippet corresponding to this tag is a part of the file descent.tex and is located in lines 5168–5172 (see updates for more information).

\begin{lemma}
\label{lemma-descending-property-finite}
The property $\mathcal{P}(f) =$$f$ is finite''
is fpqc local on the base.
\end{lemma}

\begin{proof}
An finite morphism is the same thing as an integral
morphism which is locally of finite type. See
Morphisms, Lemma \ref{morphisms-lemma-finite-integral}.
Hence the lemma follows on combining
Lemmas \ref{lemma-descending-property-locally-finite-type}
and \ref{lemma-descending-property-integral}.
\end{proof}

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