The Krull Intersection Theorem

The Krull intersection theorem determines the intersection of all powers of an ideal in a Noetherian ring, providing a fundamental tool used throughout commutative algebra.

The Theorem

Theorem7.7Krull Intersection Theorem

Let $R$ be a Noetherian ring, $I \subseteq R$ an ideal, and $M$ a finitely generated $R$ -module. Then $\bigcap_{n=1}^\infty I^n M = \{m \in M : (1 - a)m = 0 \text{ for some } a \in I\}$ In particular:

If $(R, \mathfrak{m})$ is a Noetherian local ring, then $\bigcap_{n=1}^\infty \mathfrak{m}^n = 0$ .
If $R$ is a Noetherian domain and $I$ is a proper ideal, then $\bigcap_{n=1}^\infty I^n = 0$ .

The element $a \in I$ in the theorem is obtained from the Artin-Rees lemma applied to the submodule $N = \bigcap I^n M$ .

Key Consequence

Theorem7.8Separation of the I-adic Topology

For a Noetherian local ring $(R, \mathfrak{m})$ , the $\mathfrak{m}$ -adic topology is Hausdorff: the only element contained in every neighborhood of $0$ is $0$ itself. Consequently, the natural map $R \to \hat{R}$ is injective.

ExampleNon-Noetherian failure

The Krull intersection theorem can fail without the Noetherian hypothesis. In the ring $R = k[x_1, x_2, x_3, \ldots]$ with $\mathfrak{m} = (x_1, x_2, \ldots)$ , the element $x_1 x_2 x_3 \cdots$ (which makes sense in the completion) witnesses the failure. More concretely, in certain non-Noetherian valuation rings, the maximal ideal can equal its own square.

The Artin-Rees Lemma

Theorem7.9Artin-Rees Lemma

Let $R$ be a Noetherian ring, $I \subseteq R$ an ideal, and $N \subseteq M$ finitely generated $R$ -modules. Then there exists an integer $c \geq 0$ such that for all $n \geq c$ , $I^n M \cap N = I^{n-c}(I^c M \cap N)$

The Artin-Rees lemma says that the $I$ -adic filtration on $M$ induces essentially the $I$ -adic filtration on the submodule $N$ (up to a shift). This is the technical engine behind both the Krull intersection theorem and the exactness of completion.

RemarkProof strategy

The Artin-Rees lemma is proved by introducing the Rees algebra $\mathcal{R}(I) = \bigoplus_{n \geq 0} I^n t^n \subseteq R[t]$ and the Rees module $\bigoplus_{n \geq 0} I^n M \cdot t^n$ . Since $R$ is Noetherian, $\mathcal{R}(I)$ is Noetherian (it is generated over $R$ by $I \cdot t$ ), and the Rees module is finitely generated, from which the stabilization follows.