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fix tizkcd figure

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Maximilian Keßler 2022-02-16 01:32:03 +01:00
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2021_Algebra_I.tex
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@ -851,14 +851,12 @@ The following will lead to another proof of the Nullstellensatz, which uses the
\label{artintate} \label{artintate}
Let $A$ be a subalgebra of the $R$-algebra $B$, where $R$ is Noetherian. If $ B / R$ is of finite type and $B / A$ is finite, then $A / R$ is also of finite type. Let $A$ be a subalgebra of the $R$-algebra $B$, where $R$ is Noetherian. If $ B / R$ is of finite type and $B / A$ is finite, then $A / R$ is also of finite type.
\begin{figure}[H] \[
\centering
\begin{tikzcd} \begin{tikzcd}
A \arrow[hookrightarrow]{rr}{\subseteq}& & B \\ A \arrow[hookrightarrow]{rr}{\subseteq}& & B \\
&R \arrow{ul}{\alpha} \arrow{ur}{\alpha} \text{~(Noeth.)} &R \arrow{ul}{\alpha} \arrow{ur}{\alpha} \text{~(Noeth.)}
\end{tikzcd} \end{tikzcd}
\end{figure} \]
\end{proposition} \end{proposition}
\begin{proof} \begin{proof}
Let $(b_i)_{i=1}^{m}$ generate $B$ as an $A$-module and $(\beta_j)_{j=1}^m$ as an $R$-algebra. Let $(b_i)_{i=1}^{m}$ generate $B$ as an $A$-module and $(\beta_j)_{j=1}^m$ as an $R$-algebra.