Noether Normalization

Noether Normalization and Hilbert's Nullstellensatz

Introduction

The course concludes with a glimpse into algebraic geometry, where the algebraic tools we have developed are used to study geometric objects. Two foundational theorems establish a profound dictionary between the algebra of polynomial rings and the geometry of shapes defined by polynomial equations.

Noether Normalization Lemma

Statement

Theorem 22.1 (Noether Normalization Lemma): Let $k$ be a field, and let $A$ be a finitely generated $k$ -algebra that is an integral domain. Then there exist algebraically independent elements $y_{1}, \dots, y_{d} \in A$ such that $A$ is a finitely generated module over the polynomial ring $k [y_{1}, \dots, y_{d}]$ .

Geometric Interpretation

Geometrically, this means that any affine algebraic variety is a finite-to-one branched cover of an affine space. This theorem is a key technical tool for proving Hilbert's Nullstellensatz.

Examples

Example 1: Consider the ring $A = k [x, y] / (x y - 1)$ . This ring is isomorphic to $k [x, x^{- 1}]$ , and we can take $y_{1} = x + y$ as the algebraically independent element.

Example 2: For the ring $A = k [x, y] / (x^{2} + y^{2} - 1)$ , we can take $y_{1} = x$ as the algebraically independent element.

Hilbert's Nullstellensatz

Statement

Theorem 22.2 (Hilbert's Nullstellensatz): Let $k$ be an algebraically closed field. Let $I$ be an ideal in the polynomial ring $k [x_{1}, \dots, x_{n}]$ . If a polynomial $f \in k [x_{1}, \dots, x_{n}]$ vanishes at every point in the zero-locus of $I$ (i.e., $f (p) = 0$ for all $p$ such that $g (p) = 0$ for all $g \in I$ ), then some power of $f$ is in the ideal $I$ . That is, $f^{r} \in I$ for some integer $r \geq 1$ .

Geometric Interpretation

The Nullstellensatz (German for "theorem of zeros") establishes a fundamental correspondence:

Geometric objects (algebraic sets, which are the solution sets to systems of polynomial equations) correspond to...
Algebraic objects (radical ideals in a polynomial ring)

Weak Form

The weak form of the theorem states that a system of polynomial equations has a solution in an algebraically closed field if and only if the ideal generated by the polynomials is not the entire ring. This can be seen as a vast generalization of the Fundamental Theorem of Algebra.

Applications

Application 1: Algebraic Geometry

These theorems are fundamental to algebraic geometry, providing the bridge between algebra and geometry.

Application 2: Commutative Algebra

The Nullstellensatz is a cornerstone of commutative algebra, providing tools for understanding the structure of polynomial rings.

Application 3: Computer Algebra

These results are essential for computational algebraic geometry and computer algebra systems.

Examples

Example 1: The Circle

Consider the ideal $I = (x^{2} + y^{2} - 1)$ in $C [x, y]$ . The zero-locus of $I$ is the unit circle. The Nullstellensatz tells us that any polynomial that vanishes on the circle must be in the radical of $I$ .

Example 2: The Parabola

For the ideal $I = (y - x^{2})$ in $C [x, y]$ , the zero-locus is the parabola $y = x^{2}$ . Any polynomial vanishing on this parabola must be in the radical of $I$ .

Example 3: Noether Normalization

For the ring $A = C [x, y] / (x^{2} + y^{2} - 1)$ , Noether normalization tells us that $A$ is a finite module over $C [x]$ .

Advanced Topics

The Strong Nullstellensatz

The strong form of the Nullstellensatz provides a precise correspondence between radical ideals and algebraic sets.

The Weak Nullstellensatz

The weak form provides a criterion for when a system of equations has a solution.

Applications to Dimension Theory

Noether normalization is fundamental to the theory of dimension in algebraic geometry.

Summary

Noether normalization and Hilbert's Nullstellensatz establish a profound dictionary between the algebra of polynomial rings and the geometry of shapes defined by polynomial equations. These theorems are fundamental to modern algebraic geometry and have applications throughout mathematics.

The correspondence between geometric objects and algebraic objects provided by these theorems is the foundation of algebraic geometry and continues to be an active area of research with connections to many other areas of mathematics.