---
tag: DMYXA8HZ2P
kind: statement
role: theorem
name: The local ring of holomorphic germs is Noetherian and factorial
msc: [32B05, 13J07, 13F15]
uses: [EVPZ7ZG1HB, 7PJ8KVZZCQ, MZ02BKPHJA, 0670T59HGE, 0W4X9G6263]
about: [EVPZ7ZG1HB]
sources:
  - url: https://en.wikipedia.org/wiki/Function_of_several_complex_variables
---

## Statement

Let $n \ge 0$ and let $\mathcal{O}_n = \C\{z_1, \dots, z_n\}$ be the [[ring of convergent power series|EVPZ7ZG1HB]], which is isomorphic to the ring of [[germs of holomorphic functions|7PJ8KVZZCQ]] at $0 \in \C^n$.
Then:

1. $\mathcal{O}_n$ is a [[local ring|MZ02BKPHJA]] with maximal ideal $\mathfrak{m} = (z_1, \dots, z_n)$.
2. $\mathcal{O}_n$ is a [[Noetherian ring|0670T59HGE]].
3. $\mathcal{O}_n$ is a [[unique factorization domain|0W4X9G6263]].

## Remarks

Part 1 holds because a series with nonzero constant term has a convergent inverse.
Parts 2 and 3 are proved together by induction on $n$, with the Weierstrass preparation and division theorems as the main tools.
For a nonzero ideal $I$, after a linear change of coordinates $I$ contains an element $f$ regular of order $d$ in $z_n$.
Division by $f$ shows that $\mathcal{O}_n/(f)$ is a finitely generated module over $\mathcal{O}_{n-1}$, which is Noetherian by induction, so $I/(f)$ and hence $I$ is finitely generated.
For factoriality, preparation reduces factorization in $\mathcal{O}_n$ to factorization of Weierstrass polynomials in $\mathcal{O}_{n-1}[z_n]$, which is a unique factorization domain by Gauss's lemma, since $\mathcal{O}_{n-1}$ is one by induction.
Example: in $\mathcal{O}_2$, the germ $z_1^2 - z_2^3$ is irreducible, while $z_2^2 - z_1^2(1 + z_1)$ factors as $\big(z_2 - z_1\sqrt{1 + z_1}\big)\big(z_2 + z_1\sqrt{1 + z_1}\big)$, where $\sqrt{1 + z_1}$ is the convergent binomial series.
The ring of holomorphic functions on an open set, rather than germs, is not Noetherian.
On $\C$, the entire functions vanishing at all but finitely many of the points $1, 2, 3, \dots$ form an ideal that is not finitely generated.

## Explanations

### L1 human

Near a point, holomorphic functions behave algebraically like polynomials.
Every ideal of germs is generated by finitely many germs, and every germ factors into irreducible germs in an essentially unique way.
This is what makes it possible to study the local shape of zero sets of holomorphic functions with algebra.

### L2 human

The theorem is the starting point of local analytic geometry.
Noetherianity gives that every germ of an analytic set is a finite union of irreducible germs, and that descending chains of analytic germs stabilize.
Factoriality shows that the germ of a hypersurface $\set{f = 0}$ decomposes according to the irreducible factors of $f$.
The ring $\mathcal{O}_n$ is moreover regular of dimension $n$, and Oka's coherence theorem extends its Noetherian property to the sheaf of holomorphic functions.