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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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