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aisejohan committed May 7, 2024
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Expand Up @@ -27173,8 +27173,8 @@ \section{The dimension formula}
\end{lemma}

\begin{proof}
Suppose that $R \subset S' \subset S$ is a finitely generated $R$-subalgebra
of $S$. In this case set $\mathfrak q' = S' \cap \mathfrak q$.
Suppose that $R \subset S' \subset S$, where $S'$ is a finitely generated
$R$-subalgebra of $S$. In this case set $\mathfrak q' = S' \cap \mathfrak q$.
The lemma for the ring maps $R \to S'$ and $S' \to S$ implies the
lemma for $R \to S$ by additivity of transcendence degree in towers
of fields (Fields, Lemma \ref{fields-lemma-transcendence-degree-tower}).
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