# The Stacks Project

## Tag 01TS

Lemma 28.20.4. The base change of a morphism which is locally of finite presentation is locally of finite presentation. The same is true for morphisms of finite presentation.

Proof. In the proof of Lemma 28.20.2 we saw that being of finite presentation is a local property of ring maps. Hence the first statement of the lemma follows from Lemma 28.13.5 combined with the fact that being of finite presentation is a property of ring maps that is stable under base change, see Algebra, Lemma 10.13.2. By the above and the fact that a base change of a quasi-compact, quasi-separated morphism is quasi-compact and quasi-separated, see Schemes, Lemmas 25.19.3 and 25.21.13 we see that the base change of a morphism of finite presentation is a morphism of finite presentation. $\square$

The code snippet corresponding to this tag is a part of the file morphisms.tex and is located in lines 3726–3731 (see updates for more information).

\begin{lemma}
\label{lemma-base-change-finite-presentation}
The base change of a morphism which is locally of finite presentation
is locally of finite presentation. The same is true for morphisms of
finite presentation.
\end{lemma}

\begin{proof}
In the proof of Lemma \ref{lemma-locally-finite-presentation-characterize}
we saw that being of finite presentation is a local property of ring maps.
Hence the first statement of the lemma follows from
Lemma \ref{lemma-composition-type-P} combined
with the fact that being of finite presentation
is a property of ring maps that is
stable under base change, see
Algebra, Lemma \ref{algebra-lemma-base-change-finiteness}.
By the above and the fact that a base change of a
quasi-compact, quasi-separated morphism is quasi-compact
and quasi-separated, see
Schemes, Lemmas \ref{schemes-lemma-quasi-compact-preserved-base-change}
and \ref{schemes-lemma-separated-permanence}
we see that the base change of a morphism of finite presentation is
a morphism of finite presentation.
\end{proof}

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