# The Stacks Project

## Tag 03J7

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 13586–13593 (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}

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.

## Add a comment on tag 03J7

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the lower-right corner).

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following box. So in case this where tag 0321 you just have to write 0321. Beware of the difference between the letter 'O' and the digit 0.

This captcha seems more appropriate than the usual illegible gibberish, right?