The local ring of holomorphic germs is Noetherian and factorial

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.

Going deeper

For a student

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.