Commutative Algebra/Noether's normalisation lemma

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Computational preparation[edit]

Lemma 23.1:

Let be a ring, and let be a polynomial. Let be a number that is strictly larger than the degree of any monomial of (where the degree of an arbitrary monomial of is defined to be ). Then the largest monomial (with respect to degree) of the polynomial

has the form for a suitable .

Proof:

Let be an arbitrary monomial of . Inserting for , for gives

.

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

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

by multiplying out; it turns out, that always choosing yields a strictly larger monomial than instead preferring the other variable . Hence, the strictly largest monomial of that polynomial under consideration is

.

Now is larger than all the involved here, since it's even larger than the degree of any monomial of . Therefore, for coming from monomials of , the numbers

represent numbers in the number system base . In particular, no two of them are equal for distinct , since numbers of base must have same -cimal places to be equal. Hence, there is a largest of them, call it . The largest monomial of

is then

;

its size dominates certainly all monomials coming from the monomial of with powers , and by choice it also dominates the largest monomial of any polynomials generated by any other monomial of . Hence, it is the largest monomial of measured by degree, and it has the desired form.

Algebraic independence in algebras[edit]

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

Theorem 23.2:

Let be a ring and an -algebra. Elements in are called algebraically independent over iff there does not exist a polynomial such that (where the polynomial is evaluated as explained in chapter 21).

Transitivity of localisation[edit]

The theorem[edit]

Theorem 23.3 (Noether's normalisation lemma):

Let be an integral domain, and let be a ring extension of that is finitely generated as a -module; in particular, is a -algebra, where the algebra operations are induced by the ring operations. Then we may pick a such that there exist ( denoting the localisation of at ) which are algebraically independent over as a -algebra

Localisation of fields[edit]