The Stacks project

\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

10.114 Noether normalization

In this section we prove variants of the Noether normalization lemma. The key ingredient we will use is contained in the following two lemmas.

Lemma 10.114.1. Let $n \in \mathbf{N}$. Let $N$ be a finite nonempty set of multi-indices $\nu = (\nu _1, \ldots , \nu _ n)$. Given $e = (e_1, \ldots , e_ n)$ we set $e \cdot \nu = \sum e_ i\nu _ i$. Then for $e_1 \gg e_2 \gg \ldots \gg e_{n-1} \gg e_ n$ we have: If $\nu , \nu ' \in N$ then

\[ (e \cdot \nu = e \cdot \nu ') \Leftrightarrow (\nu = \nu ') \]

Proof. Say $N = \{ \nu _ j\} $ with $\nu _ j = (\nu _{j1}, \ldots , \nu _{jn})$. Let $A_ i = \max _ j \nu _{ji} - \min _ j \nu _{ji}$. If for each $i$ we have $e_{i - 1} > A_ ie_ i + A_{i + 1}e_{i + 1} + \ldots + A_ ne_ n$ then the lemma holds. For suppose that $e \cdot (\nu - \nu ') = 0$. Then for $n \ge 2$,

\[ e_1(\nu _1 - \nu '_1) = \sum \nolimits _{i = 2}^ n e_ i(\nu '_ i - \nu _ i). \]

We may assume that $(\nu _1 - \nu '_1) \ge 0$. If $(\nu _1 - \nu '_1) > 0$, then

\[ e_1(\nu _1 - \nu '_1) \ge e_1 > A_2e_2 + \ldots + A_ ne_ n \ge \sum \nolimits _{i = 2}^ n e_ i|\nu '_ i - \nu _ i| \ge \sum \nolimits _{i = 2}^ n e_ i(\nu '_ i - \nu _ i). \]

This contradiction implies that $\nu '_1 = \nu _1$. By induction, $\nu '_ i = \nu _ i$ for $2 \le i \le n$. $\square$

Lemma 10.114.2. Let $R$ be a ring. Let $g \in R[x_1, \ldots , x_ n]$ be an element which is nonconstant, i.e., $g \not\in R$. For $e_1 \gg e_2 \gg \ldots \gg e_{n-1} \gg e_ n = 1$ the polynomial

\[ g(x_1 + x_ n^{e_1}, x_2 + x_ n^{e_2}, \ldots , x_{n - 1} + x_ n^{e_{n - 1}}, x_ n) = ax_ n^ d + \text{lower order terms in }x_ n \]

where $d > 0$ and $a \in R$ is one of the nonzero coefficients of $g$.

Proof. Write $g = \sum _{\nu \in N} a_\nu x^\nu $ with $a_\nu \in R$ not zero. Here $N$ is a finite set of multi-indices as in Lemma 10.114.1 and $x^\nu = x_1^{\nu _1} \ldots x_ n^{\nu _ n}$. Note that the leading term in

\[ (x_1 + x_ n^{e_1})^{\nu _1} \ldots (x_{n-1} + x_ n^{e_{n-1}})^{\nu _{n-1}} x_ n^{\nu _ n} \quad \text{is}\quad x_ n^{e_1\nu _1 + \ldots + e_{n-1}\nu _{n-1} + \nu _ n}. \]

Hence the lemma follows from Lemma 10.114.1 which guarantees that there is exactly one nonzero term $a_\nu x^\nu $ of $g$ which gives rise to the leading term of $g(x_1 + x_ n^{e_1}, x_2 + x_ n^{e_2}, \ldots , x_{n - 1} + x_ n^{e_{n - 1}}, x_ n)$, i.e., $a = a_\nu $ for the unique $\nu \in N$ such that $e \cdot \nu $ is maximal. $\square$

Lemma 10.114.3. Let $k$ be a field. Let $S = k[x_1, \ldots , x_ n]/I$ for some proper ideal $I$. If $I \not= 0$, then there exist $y_1, \ldots , y_{n-1} \in k[x_1, \ldots , x_ n]$ such that $S$ is finite over $k[y_1, \ldots , y_{n-1}]$. 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. Pick $f \in I$, $f\not= 0$. It suffices to show the lemma for $k[x_1, \ldots , x_ n]/(f)$ since $S$ is a quotient of that ring. We will take $y_ i = x_ i - x_ n^{e_ i}$, $i = 1, \ldots , n-1$ for suitable integers $e_ i$. When does this work? It suffices to show that $\overline{x_ n} \in k[x_1, \ldots , x_ n]/(f)$ is integral over the ring $k[y_1, \ldots , y_{n-1}]$. The equation for $\overline{x_ n}$ over this ring is

\[ f(y_1 + x_ n^{e_1}, \ldots , y_{n-1} + x_ n^{e_{n-1}}, x_ n) = 0. \]

Hence we are done if we can show there exists integers $e_ i$ such that the leading coefficient with respect to $x_ n$ of the equation above is a nonzero element of $k$. This can be achieved for example by choosing $e_1 \gg e_2 \gg \ldots \gg e_{n-1}$, see Lemma 10.114.2. $\square$

Lemma 10.114.4. Let $k$ be a field. Let $S = k[x_1, \ldots , x_ n]/I$ for some ideal $I$. If $I \neq (1)$, 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. $\square$

Lemma 10.114.5. Let $k$ be a field. Let $S$ be a finite type $k$ algebra and denote $X = \mathop{\mathrm{Spec}}(S)$. Let $\mathfrak q$ be a prime of $S$, and let $x \in X$ be the corresponding point. There exists a $g \in S$, $g \not\in \mathfrak q$ such that $\dim (S_ g) = \dim _ x(X) =: d$ and such that there exists a finite injective map $k[y_1, \ldots , y_ d] \to S_ g$.

Proof. Note that by definition $\dim _ x(X)$ is the minimum of the dimensions of $S_ g$ for $g \in S$, $g \not\in \mathfrak q$, i.e., the minimum is attained. Thus the lemma follows from Lemma 10.114.4. $\square$

Lemma 10.114.6. Let $k$ be a field. Let $\mathfrak q \subset k[x_1, \ldots , x_ n]$ be a prime ideal. Set $r = \text{trdeg}_ k\ \kappa (\mathfrak q)$. Then there exists a finite ring map $\varphi : k[y_1, \ldots , y_ n] \to k[x_1, \ldots , x_ n]$ such that $\varphi ^{-1}(\mathfrak q) = (y_{r + 1}, \ldots , y_ n)$.

Proof. By induction on $n$. The case $n = 0$ is clear. Assume $n > 0$. If $r = n$, then $\mathfrak q = (0)$ and the result is clear. Choose a nonzero $f \in \mathfrak q$. Of course $f$ is nonconstant. After applying an automorphism of the form

\[ k[x_1, \ldots , x_ n] \longrightarrow k[x_1, \ldots , x_ n], \quad x_ n \mapsto x_ n, \quad x_ i \mapsto x_ i + x_ n^{e_ i}\ (i < n) \]

we may assume that $f$ is monic in $x_ n$ over $k[x_1, \ldots , x_ n]$, see Lemma 10.114.2. Hence the ring map

\[ k[y_1, \ldots , y_ n] \longrightarrow k[x_1, \ldots , x_ n], \quad y_ n \mapsto f, \quad y_ i \mapsto x_ i\ (i < n) \]

is finite. Moreover $y_ n \in \mathfrak q \cap k[y_1, \ldots , y_ n]$ by construction. Thus $\mathfrak q \cap k[y_1, \ldots , y_ n] = \mathfrak pk[y_1, \ldots , y_ n] + (y_ n)$ where $\mathfrak p \subset k[y_1, \ldots , y_{n - 1}]$ is a prime ideal. Note that $\kappa (\mathfrak p) \subset \kappa (\mathfrak q)$ is finite, and hence $r = \text{trdeg}_ k\ \kappa (\mathfrak p)$. Apply the induction hypothesis to the pair $(k[y_1, \ldots , y_{n - 1}], \mathfrak p)$ and we obtain a finite ring map $k[z_1, \ldots , z_{n - 1}] \to k[y_1, \ldots , y_{n - 1}]$ such that $\mathfrak p \cap k[z_1, \ldots , z_{n - 1}] = (z_{r + 1}, \ldots , z_{n - 1})$. We extend the ring map $k[z_1, \ldots , z_{n - 1}] \to k[y_1, \ldots , y_{n - 1}]$ to a ring map $k[z_1, \ldots , z_ n] \to k[y_1, \ldots , y_ n]$ by mapping $z_ n$ to $y_ n$. The composition of the ring maps

\[ k[z_1, \ldots , z_ n] \to k[y_1, \ldots , y_ n] \to k[x_1, \ldots , x_ n] \]

solves the problem. $\square$

Lemma 10.114.7. Let $R \to S$ be an injective finite type map of domains. Then there exists an integer $d$ and a factorization

\[ R \to R[y_1, \ldots , y_ d] \to S' \to S \]

by injective maps such that $S'$ is finite over $R[y_1, \ldots , y_ d]$ and such that $S'_ f \cong S_ f$ for some nonzero $f \in R$.

Proof. Pick $x_1, \ldots , x_ n \in S$ which generate $S$ over $R$. Let $K$ be the fraction field of $R$ and $S_ K = S \otimes _ R K$. By Lemma 10.114.4 we can find $y_1, \ldots , y_ d \in S$ such that $K[y_1, \ldots , y_ d] \to S_ K$ is a finite injective map. Note that $y_ i \in S$ because we may pick the $y_ j$ in the $\mathbf{Z}$-algebra generated by $x_1, \ldots , x_ n$. As a finite ring map is integral (see Lemma 10.35.3) we can find monic $P_ i \in K[y_1, \ldots , y_ d][T]$ such that $P_ i(x_ i) = 0$ in $S_ K$. Let $f \in R$ be a nonzero element such that $fP_ i \in R[y_1, \ldots , y_ d][T]$ for all $i$. Set $x_ i' = fx_ i$ and let $S' \subset S$ be the subalgebra generated by $y_1, \ldots , y_ d$ and $x'_1, \ldots , x'_ n$. Note that $x'_ i$ is integral over $R[y_1, \ldots , y_ d]$ as we have $Q_ i(x_ i') = 0$ where $Q_ i = f^{\deg _ T(P_ i)}P_ i(T/f)$ which is a monic polynomial in $T$ with coefficients in $R[y_1, \ldots , y_ d]$ by our choice of $f$. Hence $R[y_1, \ldots , y_ n] \subset S'$ is finite by Lemma 10.35.5. By construction $S'_ f \cong S_ f$ and we win. $\square$


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