Noetherian Rings and Modules - Main Theorem

Hilbert's Basis Theorem establishes that the Noetherian property extends to polynomial rings, the cornerstone of algebraic geometry.

TheoremHilbert's Basis Theorem

If $R$ is a Noetherian ring, then the polynomial ring $R[x]$ is Noetherian.

By induction, $R[x_1, \ldots, x_n]$ is Noetherian for all $n \geq 1$ .

This fundamental result implies every finitely generated algebra over a Noetherian ring is Noetherian, since such algebras are quotients of polynomial rings.

Remark

Hilbert proved this theorem in 1890 for polynomial rings over fields, revolutionizing invariant theory by showing finitely many generators suffice for invariant rings. The general version for arbitrary Noetherian rings appeared later in the work of Emmy Noether.

TheoremConsequences for Algebraic Geometry

The Hilbert Basis Theorem implies:

Every ideal in $k[x_1, \ldots, x_n]$ is finitely generated
Every algebraic variety is defined by finitely many polynomial equations
Every ascending chain of subvarieties stabilizes
The Zariski topology is Noetherian (every open set is quasi-compact)

Without this theorem, much of classical algebraic geometry would fail to have finite presentations.

ExampleFinitely Many Equations

An affine variety $V \subseteq \mathbb{A}^n$ is defined by ideal $I(V) = \{f \in k[x_1, \ldots, x_n] : f|_V = 0\}$ .

By Hilbert's Basis Theorem, $I(V) = (f_1, \ldots, f_m)$ for some finite set. Thus $V$ is defined by finitely many equations $f_1 = \cdots = f_m = 0$ , even though $I(V)$ may be conceptually infinite.

TheoremKrull's Intersection Theorem

Let $R$ be a Noetherian local ring with maximal ideal $\mathfrak{m}$ and $M$ a finitely generated $R$ -module. Then: $\bigcap_{n=1}^\infty \mathfrak{m}^n M = 0$

provided $M$ is not trivial. This describes the "separation property" of the $\mathfrak{m}$ -adic topology.

Remark

Krull's Intersection Theorem fails without the Noetherian hypothesis. It ensures the $\mathfrak{m}$ -adic filtration is "Hausdorff," crucial for completions and formal geometry.

TheoremArtin-Rees Lemma

Let $R$ be Noetherian, $I$ an ideal, and $M \supseteq N$ finitely generated $R$ -modules. Then there exists $k$ such that for all $n \geq k$ : $I^n M \cap N = I^{n-k}(I^k M \cap N)$

This controls how powers of ideals interact with submodules, essential for studying filtrations and completions.

ExampleApplication of Artin-Rees

The Artin-Rees Lemma implies that if $M$ is finitely generated and $N \subseteq M$ , the $I$ -adic topology on $N$ coincides with the subspace topology from $M$ . This makes completion well-behaved: $(M/N)^\wedge \cong \hat{M}/\hat{N}$ .

TheoremKrull's Principal Ideal Theorem

Let $R$ be a Noetherian ring and $x \in R$ neither a unit nor zero divisor. Then every minimal prime over $(x)$ has height $\leq 1$ .

More generally, for an ideal $I = (x_1, \ldots, x_n)$ , every minimal prime over $I$ has height $\leq n$ . This constrains the dimension of quotients.

Remark

Krull's Principal Ideal Theorem is also called the height theorem or Hauptidealsatz. It provides upper bounds on dimensions of varieties defined by few equations, central to dimension theory.

TheoremNoetherian Normalization

For a finitely generated algebra $R$ over a field $k$ that is an integral domain, there exist algebraically independent elements $y_1, \ldots, y_d \in R$ such that $R$ is integral over $k[y_1, \ldots, y_d]$ and $d = \dim(R)$ .

This combines Noether normalization with the Noetherian property to establish finite covers of polynomial algebras.

These theorems establish fundamental finiteness and dimension properties, making Noetherian rings the natural setting for geometric investigations.