---
tag: 3VRZ80X45P
kind: statement
role: theorem
name: Rückert Nullstellensatz
msc: [32B10, 32B05]
uses: [Q9B74BYZ8C, 7PJ8KVZZCQ, 437YRCNYRJ]
about: [Q9B74BYZ8C]
sources:
  - url: https://en.wikipedia.org/wiki/Complex_analytic_variety
---

## Statement

Let $a \in \C^n$, let $\mathcal{O}_{n,a}$ be the ring of [[germs of holomorphic functions|7PJ8KVZZCQ]] at $a$, and let $I \subseteq \mathcal{O}_{n,a}$ be an ideal.
Let $V(I)$ be the [[germ of the analytic set|Q9B74BYZ8C]] at $a$ defined by $I$, and let $\mathcal{I}(V(I))$ be the ideal of germs vanishing on $V(I)$.
Then
$$\mathcal{I}(V(I)) = \sqrt{I},$$
the [[radical|437YRCNYRJ]] of $I$.
In other words, if a germ $f \in \mathcal{O}_{n,a}$ vanishes on $V(I)$, then $f^m \in I$ for some integer $m \ge 1$.

## Remarks

The inclusion $\sqrt{I} \subseteq \mathcal{I}(V(I))$ is clear: if $f^m \in I$, then $f^m$, and hence $f$, vanishes on $V(I)$.
For the other inclusion, write $\sqrt{I}$ as a finite intersection of prime ideals, which is possible because $\mathcal{O}_{n,a}$ is Noetherian, and reduce to the case of a prime ideal $\mathfrak{p}$.
For $\mathfrak{p}$ prime, the local parametrization of the irreducible germ $V(\mathfrak{p})$ shows that a germ vanishing on $V(\mathfrak{p})$ lies in $\mathfrak{p}$.
Example: for $I = (z_1^2) \subseteq \mathcal{O}_{2,0}$, the germ $V(I)$ is the line $\set{z_1 = 0}$, and $\mathcal{I}(V(I)) = (z_1) = \sqrt{I}$; here $z_1^2 \in I$.
As a consequence, $I \mapsto V(I)$ and $A_a \mapsto \mathcal{I}(A_a)$ are mutually inverse bijections between radical ideals of $\mathcal{O}_{n,a}$ and germs of analytic sets at $a$, and prime ideals correspond to irreducible germs.
As for Hilbert's Nullstellensatz, the base field matters: the statement fails for real analytic germs, since $x^2 + y^2$ vanishes only at $0 \in \R^2$ but $x \notin \sqrt{(x^2 + y^2)}$.

## Explanations

### L1 human

The theorem says that the germs vanishing on the common zeros of some germs are exactly those of which some power can be built from them.
It is the local, holomorphic version of Hilbert's Nullstellensatz for polynomials.
It means that the shape of an analytic set near a point and the algebra of the ideal of germs determine each other.

### L2 human

The Nullstellensatz is the dictionary of local analytic geometry.
With it, the decomposition of an analytic germ into irreducible components corresponds to the primary decomposition of its ideal.
The dimension of a germ $A_a$ equals the Krull dimension of $\mathcal{O}_{n,a}/\mathcal{I}(A_a)$.
It is also an ingredient in Cartan's coherence theorem for the ideal sheaf of an analytic set.