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Fix two places where birational is used incorrectly
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Thanks to Elías Guisado Villalgordo
https://stacks.math.columbia.edu/tag/035Q#comment-8535
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aisejohan committed Jun 27, 2023
1 parent b6ec398 commit 0292797
Showing 1 changed file with 3 additions and 2 deletions.
5 changes: 3 additions & 2 deletions morphisms.tex
Expand Up @@ -14214,7 +14214,8 @@ \section{Normalization}
$X'$ satisfies the assumptions under which the normalization
is defined. Let $f' : Y' \to X'$ be the morphism
(\ref{equation-generic-points}) constructed starting with $X'$.
As $\alpha$ is birational it is clear that $Y' = Y$ and $f = \alpha \circ f'$.
As $\alpha$ is locally birational it is clear that
$Y' = Y$ and $f = \alpha \circ f'$.
Hence the factorization $X^\nu \to X' \to X$ exists
and $X^\nu \to X'$ is the normalization of $X'$ by
Lemma \ref{lemma-characterize-normalization}. This proves (3).
Expand Down Expand Up @@ -14262,7 +14263,7 @@ \section{Normalization}
Then the lemma follows either from the local description in
Lemma \ref{lemma-description-normalization}
or from Lemma \ref{lemma-normalization-normal} part (3) because
$\coprod Z_i \to X$ is integral and birational (as $X$ is reduced
$\coprod Z_i \to X$ is integral and locally birational (as $X$ is reduced
and has locally finitely many irreducible components).
\end{proof}

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