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Tag 01TS

Chapter 28: Morphisms of Schemes > Section 28.20: Morphisms of finite presentation

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 3723–3728 (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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