The Stacks Project

Tag 00GL

Lemma 10.7.3. Suppose that $R \to S$ and $S \to T$ are finite ring maps. Then $R \to T$ is finite.

Proof. If $t_i$ generate $T$ as an $S$-module and $s_j$ generate $S$ as an $R$-module, then $t_i s_j$ generate $T$ as an $R$-module. (Also follows from Lemma 10.7.2.) $\square$

The code snippet corresponding to this tag is a part of the file algebra.tex and is located in lines 649–653 (see updates for more information).

\begin{lemma}
\label{lemma-finite-transitive}
Suppose that $R \to S$ and $S \to T$ are finite ring maps.
Then $R \to T$ is finite.
\end{lemma}

\begin{proof}
If $t_i$ generate $T$ as an $S$-module and $s_j$ generate $S$ as an
$R$-module, then $t_i s_j$ generate $T$ as an $R$-module.
(Also follows from
Lemma \ref{lemma-finite-module-over-finite-extension}.)
\end{proof}

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