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Tag: 036O

This tag has label descent-lemma-locally-finite-type-fppf-local-source and it points to

The corresponding content:

Lemma 31.24.2. The property $\mathcal{P}(f)=$''$f$ is locally of finite type'' is fppf local on the source.

Proof. Being locally of finite type is Zariski local on the source and the target, see Morphisms, Lemma 25.16.2. It is a property which is preserved under composition, see Morphisms, Lemma 25.16.3, and a flat morphism locally of finite presentation is locally of finite type, see Morphisms, Lemma 25.22.8. This proves (1), (2) and (3) of Lemma 31.22.3. The final condition (4) is Lemma 31.10.2. Hence we win. $\square$

\begin{lemma}
\label{lemma-locally-finite-type-fppf-local-source}
The property $\mathcal{P}(f)=$$f$ is locally of finite type''
is fppf local on the source.
\end{lemma}

\begin{proof}
Being locally of finite type is Zariski local on the source
and the target, see Morphisms,
Lemma \ref{morphisms-lemma-locally-finite-type-characterize}.
It is a property which is preserved under composition, see
Morphisms, Lemma \ref{morphisms-lemma-composition-finite-type}, and
a flat morphism locally of finite presentation is locally of finite type, see
Morphisms, Lemma \ref{morphisms-lemma-finite-presentation-finite-type}.
This proves
(1), (2) and (3) of Lemma \ref{lemma-properties-morphisms-local-source}.
The final condition (4) is
Lemma \ref{lemma-finite-type-local-source-fppf-algebra}. Hence we win.
\end{proof}


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