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Tag 03J7

Chapter 28: Morphisms of Schemes > Section 28.53: Universally bounded fibres

Lemma 28.53.6. A base change of a morphism with universally bounded fibres is a morphism with universally bounded fibres. More precisely, if $n$ bounds the degrees of the fibres of $f : X \to Y$ and $Y' \to Y$ is any morphism, then the degrees of the fibres of the base change $f' : Y' \times_Y X \to Y'$ is also bounded by $n$.

Proof. This is clear from the result of Lemma 28.53.2. $\square$

    The code snippet corresponding to this tag is a part of the file morphisms.tex and is located in lines 13604–13611 (see updates for more information).

    \begin{lemma}
    \label{lemma-base-change-universally-bounded}
    A base change of a morphism with universally bounded fibres is
    a morphism with universally bounded fibres. More precisely, if
    $n$ bounds the degrees of the fibres of $f : X \to Y$ and $Y' \to Y$
    is any morphism, then the degrees of the fibres of the base change
    $f' : Y' \times_Y X \to Y'$ is also bounded by $n$.
    \end{lemma}
    
    \begin{proof}
    This is clear from the result of
    Lemma \ref{lemma-characterize-universally-bounded}.
    \end{proof}

    Comments (2)

    Comment #2347 by Eric Ahlqvist on January 11, 2017 a 8:48 am UTC

    There is one "$\to Y'$" too many in the end of the lemma.

    Comment #2416 by Johan (site) on February 17, 2017 a 1:49 pm UTC

    Thanks. Fixed here.

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