This tag has label descent-lemma-locally-finite-presentation-fppf-local-source and it points to
The corresponding content:
Lemma 31.24.1. The property $\mathcal{P}(f)=$''$f$ is locally of finite presentation'' is fppf local on the source.Proof. Being locally of finite presentation is Zariski local on the source and the target, see Morphisms, Lemma 25.22.2. It is a property which is preserved under composition, see Morphisms, Lemma 25.22.3. This proves (1), (2) and (3) of Lemma 31.22.3. The final condition (4) is Lemma 31.10.1. Hence we win. $\square$
\begin{lemma}
\label{lemma-locally-finite-presentation-fppf-local-source}
The property $\mathcal{P}(f)=$``$f$ is locally of finite presentation''
is fppf local on the source.
\end{lemma}
\begin{proof}
Being locally of finite presentation is Zariski local on the source
and the target, see Morphisms,
Lemma \ref{morphisms-lemma-locally-finite-presentation-characterize}.
It is a property which is preserved under composition, see
Morphisms, Lemma \ref{morphisms-lemma-composition-finite-presentation}.
This proves
(1), (2) and (3) of Lemma \ref{lemma-properties-morphisms-local-source}.
The final condition (4) is
Lemma \ref{lemma-flat-finitely-presented-permanence-algebra}. Hence we win.
\end{proof}
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