Commutative Algebra/Noether's normalisation lemma

Computational preparation

Lemma 23.1:

Let $R$ be a ring, and let $f\in R[x_{1},\ldots ,x_{n}]$ be a polynomial. Let $N\in \mathbb {N}$ be a number that is strictly larger than the degree of any monomial of $f$ (where the degree of an arbitrary monomial $x_{1}^{k_{1}}x_{2}^{k_{2}}\cdots x_{n}^{k_{n}}$ of $f$ is defined to be $k_{1}+k_{2}+\cdots +k_{n}$). Then the largest monomial (with respect to degree) of the polynomial

$g(x_{1},\ldots ,x_{n}):=f(x_{1}+x_{n}^{N^{n-1}},x_{2}+x_{n}^{N^{n-2}},\ldots ,x_{n-2}+x_{n}^{N^{2}},x_{n-1}+x_{n}^{N},x_{n})$

has the form $x_{n}^{m}$ for a suitable $m\in \mathbb {N}$.

Proof:

Let $x_{1}^{k_{1}}x_{2}^{k_{2}}\cdots x_{n}^{k_{n}}$ be an arbitrary monomial of $f$. Inserting $x_{1}+x_{n}^{N^{n-1}}$ for $x_{1}$, $x_{2}+x_{n}^{N^{n-2}}$ for $x_{2}$ gives

$(x_{1}+x_{n}^{N^{n-1}})^{k_{1}}(x_{2}+x_{n}^{N^{n-2}})^{k_{2}}\cdots (x_{n-1}+x_{n}^{N})^{k_{n-1}}x_{n}^{k_{n}}$.

This is a polynomial, and moreover, by definition $g$ consists of certain coefficients multiplied by polynomials of that form.

We want to find the largest coefficient of $g$. To do so, we first identify the largest monomial of

$(x_{1}+x_{n}^{N^{n-1}})^{k_{1}}(x_{2}+x_{n}^{N^{n-2}})^{k_{2}}\cdots (x_{n-1}+x_{n}^{N})^{k_{n-1}}x_{n}^{k_{n}}$

by multiplying out; it turns out, that always choosing $x_{n}^{N^{j}}$ yields a strictly larger monomial than instead preferring the other variable $x_{j}$. Hence, the strictly largest monomial of that polynomial under consideration is

$(x_{n}^{N^{n-1}})^{k_{1}}(x_{n}^{N^{n-2}})^{k_{2}}\cdots (x_{n}^{N})^{k_{n-1}}x_{n}^{k_{n}}=x_{n}^{k_{1}N^{n-1}+k_{2}N^{n-2}+\cdots +k_{n-1}N+k_{n}}$.

Now $N$ is larger than all the $k_{j}$ involved here, since it's even larger than the degree of any monomial of $f$. Therefore, for $(k_{1},\ldots ,k_{n})$ coming from monomials of $f$, the numbers

$k_{1}N^{n-1}+k_{2}N^{n-2}+\cdots +k_{n-1}N+k_{n}$

represent numbers in the number system base $N$. In particular, no two of them are equal for distinct $(k_{1},\ldots ,k_{n})$, since numbers of base $N$ must have same $N$-cimal places to be equal. Hence, there is a largest of them, call it $m_{1}N^{n-1}+m_{2}N^{n-2}+\cdots +m_{n-1}N+m_{n}$. The largest monomial of

$(x_{1}+x_{n}^{N^{n-1}})^{m_{1}}(x_{2}+x_{n}^{N^{n-2}})^{m_{2}}\cdots (x_{n-1}+x_{n}^{N})^{m_{n-1}}x_{n}^{m_{n}}$

is then

$x_{n}^{m_{1}N^{n-1}+m_{2}N^{n-2}+\cdots +m_{n-1}N+m_{n}}$;

its size dominates certainly all monomials coming from the monomial of $f$ with powers $(m_{1},\ldots ,m_{n})$, and by choice it also dominates the largest monomial of any polynomials generated by any other monomial of $f$. Hence, it is the largest monomial of $g$ measured by degree, and it has the desired form.$\Box$

Algebraic independence in algebras

A notion well-known in the theory of fields extends to algebras.

Theorem 23.2:

Let $R$ be a ring and $A$ an $R$-algebra. Elements $a_{1},\ldots ,a_{n}$ in $A$ are called algebraically independent over $R$ iff there does not exist a polynomial $p\in R[x_{1},\ldots ,x_{n}]$ such that $p(a_{1},\ldots ,a_{n})=0$ (where the polynomial is evaluated as explained in chapter 21).

The theorem

Theorem 23.3 (Noether's normalisation lemma):

Let $D$ be an integral domain, and let $E\supseteq D$ be a ring extension of $D$ that is finitely generated as a $D$-module; in particular, $E$ is a $D$-algebra, where the algebra operations are induced by the ring operations. Then we may pick a $d\in D$ such that there exist $c_{1},\ldots ,c_{n}\in E_{d}$ ($E_{d}$ denoting the localisation of $E_{d}$ at $d$) which are algebraically independent over $E_{d}$ as a $D_{d}$-algebra