The ring of holomorphic germs at $0$ and its maximal ideal

Fix $m\ge 1$. Two holomorphic functions on neighbourhoods of $0\in {\mathbb{C}}^m$ (Holomorphic functions on an open subset of ${\mathbb{C}}^m$) are equivalent at $0$ when they agree on some smaller neighbourhood of $0$. An equivalence class is a holomorphic germ at $0$, and the set of all such germs is denoted ${\mathcal{O}}_{m,0}$.

For the $m=1$ boundary case used later, also set ${\mathcal{O}}_{0,0}:=\mathbb{C},{\mathfrak{m}}_{0,0}:=\{0\}\mathrm{.}$ This is the ring of constant germs at the unique point of ${\mathbb{C}}^0$.

If $[f]$ and $[g]$ are germs, choose representatives defined on a common neighbourhood of $0$ and set

$[f]+[g]:=[f+g],[f][g]:=[fg]\mathrm{.}$

These are well defined because agreement on a smaller neighbourhood is preserved by pointwise addition and multiplication. Thus ${\mathcal{O}}_{m,0}$ is a commutative ring with identity $[1]$.

Its distinguished ideal is

${\mathfrak{m}}_{m,0}:=\{[f]\in {\mathcal{O}}_{m,0}:f(0)=0\}\mathrm{.}$

This is well defined because equivalent representatives have the same value at $0$. The local-ring terminology used later is that of A local ring is a nonzero commutative ring with a unique maximal ideal.