# The Stacks Project

## Tag 032J

Lemma 10.155.6. Let $R$ be a Noetherian domain. If $R[z, z^{-1}]$ is N-1, then so is $R$.

Proof. Let $R'$ be the integral closure of $R$ in its field of fractions $K$. Let $S'$ be the integral closure of $R[z, z^{-1}]$ in its field of fractions. Clearly $R' \subset S'$. Since $K[z, z^{-1}]$ is a normal domain we see that $S' \subset K[z, z^{-1}]$. Suppose that $f_1, \ldots, f_n \in S'$ generate $S'$ as $R[z, z^{-1}]$-module. Say $f_i = \sum a_{ij}z^j$ (finite sum), with $a_{ij} \in K$. For any $x \in R'$ we can write $$x = \sum h_i f_i$$ with $h_i \in R[z, z^{-1}]$. Thus we see that $R'$ is contained in the finite $R$-submodule $\sum Ra_{ij} \subset K$. Since $R$ is Noetherian we conclude that $R'$ is a finite $R$-module. $\square$

The code snippet corresponding to this tag is a part of the file algebra.tex and is located in lines 42631–42635 (see updates for more information).

\begin{lemma}
\label{lemma-Laurent-ring-N-1}
Let $R$ be a Noetherian domain.
If $R[z, z^{-1}]$ is N-1, then so is $R$.
\end{lemma}

\begin{proof}
Let $R'$ be the integral closure of $R$ in its field of fractions $K$.
Let $S'$ be the integral closure of $R[z, z^{-1}]$ in its field of fractions.
Clearly $R' \subset S'$.
Since $K[z, z^{-1}]$ is a normal domain we see that $S' \subset K[z, z^{-1}]$.
Suppose that $f_1, \ldots, f_n \in S'$ generate $S'$ as $R[z, z^{-1}]$-module.
Say $f_i = \sum a_{ij}z^j$ (finite sum), with $a_{ij} \in K$.
For any $x \in R'$ we can write
$$x = \sum h_i f_i$$
with $h_i \in R[z, z^{-1}]$. Thus we see that $R'$ is contained in the
finite $R$-submodule $\sum Ra_{ij} \subset K$. Since $R$ is Noetherian
we conclude that $R'$ is a finite $R$-module.
\end{proof}

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