## Tag `00OY`

Chapter 10: Commutative Algebra > Section 10.114: Noether normalization

Lemma 10.114.4. Let $k$ be a field. Let $S = k[x_1, \ldots, x_n]/I$ for some ideal $I$. There exist $r\geq 0$, and $y_1, \ldots, y_r \in k[x_1, \ldots, x_n]$ such that (a) the map $k[y_1, \ldots, y_r] \to S$ is injective, and (b) the map $k[y_1, \ldots, y_r] \to S$ is finite. In this case the integer $r$ is the dimension of $S$. Moreover we may choose $y_i$ to be in the $\mathbf{Z}$-subalgebra of $k[x_1, \ldots, x_n]$ generated by $x_1, \ldots, x_n$.

Proof.By induction on $n$, with $n = 0$ being trivial. If $I = 0$, then take $r = n$ and $y_i = x_i$. If $I \not = 0$, then choose $y_1, \ldots, y_{n-1}$ as in Lemma 10.114.3. Let $S' \subset S$ be the subring generated by the images of the $y_i$. By induction we can choose $r$ and $z_1, \ldots, z_r \in k[y_1, \ldots, y_{n-1}]$ such that (a), (b) hold for $k[z_1, \ldots, z_r] \to S'$. Since $S' \to S$ is injective and finite we see (a), (b) hold for $k[z_1, \ldots, z_r] \to S$. The last assertion follows from Lemma 10.111.4. $\square$

The code snippet corresponding to this tag is a part of the file `algebra.tex` and is located in lines 26589–26599 (see updates for more information).

```
\begin{lemma}
\label{lemma-Noether-normalization}
Let $k$ be a field. Let $S = k[x_1, \ldots, x_n]/I$ for some ideal $I$.
There exist $r\geq 0$, and $y_1, \ldots, y_r \in k[x_1, \ldots, x_n]$
such that (a) the map $k[y_1, \ldots, y_r] \to S$ is injective,
and (b) the map $k[y_1, \ldots, y_r] \to S$ is finite.
In this case the integer $r$ is the dimension of $S$.
Moreover we may choose $y_i$ to be in the
$\mathbf{Z}$-subalgebra of $k[x_1, \ldots, x_n]$
generated by $x_1, \ldots, x_n$.
\end{lemma}
\begin{proof}
By induction on $n$, with $n = 0$ being trivial.
If $I = 0$, then take $r = n$ and $y_i = x_i$.
If $I \not = 0$, then choose $y_1, \ldots, y_{n-1}$
as in Lemma \ref{lemma-one-relation}. Let
$S' \subset S$ be the subring generated by
the images of the $y_i$. By induction we can
choose $r$ and $z_1, \ldots, z_r \in k[y_1, \ldots, y_{n-1}]$
such that (a), (b) hold for $k[z_1, \ldots, z_r]
\to S'$. Since $S' \to S$ is injective and finite
we see (a), (b) hold for $k[z_1, \ldots, z_r]
\to S$. The last assertion follows from Lemma
\ref{lemma-integral-sub-dim-equal}.
\end{proof}
```

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