---
tag: ES08X651A3
kind: definition
name: Irreducible analytic germ
msc: [32B10, 32B05]
defines: 7VPJWF0AZ7
uses: [A2WHCVP6JN, 00MHHZ9SNE, 2TGPZ8VW2D]
sources:
  - url: https://en.wikipedia.org/wiki/Complex_analytic_variety
---

## Definition

Let $a \in \C^n$ and let $A_a$ be a [[germ of an analytic set|A2WHCVP6JN]] at $a$.
The germ $A_a$ is *irreducible* if:

1. $A_a$ is not the empty germ, and
2. whenever $A_a = B_a \cup C_a$ for germs of analytic sets $B_a, C_a$ at $a$, one has $A_a = B_a$ or $A_a = C_a$.

## Remarks

Equivalently, the ideal $\mathcal{I}(A_a)$ of [[holomorphic germs|2TGPZ8VW2D]] vanishing on $A_a$ is a [[prime ideal|00MHHZ9SNE]] of $\mathcal{O}_{n,a}$.
Example: the germ at $0$ of the cusp $A = \set{z_1^2 = z_2^3}$ in $\C^2$ is irreducible.
The germ $f = z_1^2 - z_2^3$ is irreducible in $\mathcal{O}_{2,0}$, since it is a Weierstrass polynomial of degree $2$ in $z_1$.
A factorization would give a germ $h$ with $h(z_2)^2 = z_2^3$, which is impossible by comparing orders of vanishing.
Since $\mathcal{O}_{2,0}$ is a unique factorization domain, the ideal $(f)$ is prime, hence radical, and the [[Rückert Nullstellensatz|3VRZ80X45P]] gives $\mathcal{I}(A_0) = (f)$, a prime ideal.
Non-example: the germ at $0$ of $\set{z_2^2 = z_1^2(1 + z_1)}$ is reducible.
With $s = \sqrt{1 + z_1}$, the convergent binomial series, it is the union of the germs of the two smooth curves $\set{z_2 = z_1 s}$ and $\set{z_2 = -z_1 s}$, and neither contains the other.
Every germ of an analytic set has a decomposition $A_a = A_{1,a} \cup \cdots \cup A_{k,a}$ into irreducible germs, unique up to order when no $A_{i,a}$ is contained in another.