## 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 13637–13644 (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}
```

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