Reduced holomorphic germ for a hypersurface

Fix $n\ge 1$ and a point $p\in {\mathbb{C}}^n$, and write ${\mathcal{O}}_{{\mathbb{C}}^n,p}$ for the ring of holomorphic germs at $p$, with addition and multiplication of germs defined by representatives on a common neighbourhood (The ring of holomorphic germs at $0$ and its maximal ideal). Recall that a germ is a unit exactly when its value at $p$ is nonzero (A germ is a unit exactly when its value at $0$ is nonzero, so ${\mathcal{O}}_{m,0}$ is local).

A germ $f\in {\mathcal{O}}_{{\mathbb{C}}^n,p}$ is a hypersurface equation germ at $p$ when $f$ is nonzero and not a unit.

Such an equation germ $f$ is reduced when no irreducible element of ${\mathcal{O}}_{{\mathbb{C}}^n,p}$ divides $f$ twice: there is no irreducible $q\in {\mathcal{O}}_{{\mathbb{C}}^n,p}$ with $q^2\mid f$ in ${\mathcal{O}}_{{\mathbb{C}}^n,p}$. In other words, a reduced equation germ is a nonzero nonunit that is not divisible by the square of an irreducible germ.

The zero germ and the unit germs are excluded from hypersurface equations, so reducedness is only defined for nonzero nonunits.

Convention for a general centre. The published germ ring is defined at the origin, and on this page ${\mathcal{O}}_{{\mathbb{C}}^n,p}$ is read through the translation convention: the biholomorphism $z\mapsto z+p$ of ${\mathbb{C}}^n$ pulls germs at $p$ back to germs at $0$, so ${\mathcal{O}}_{{\mathbb{C}}^n,p}$ denotes the isomorphic ring of holomorphic germs at $p$, a germ at $p$ corresponds to its translate $z\mapsto f(z+p)$ at $0$, and the maximal ideal is the translate of ${\mathfrak{m}}_{n,0}$; the dimension lemma proved below identifies it with the ideal generated by the coordinate differences $z_i-p_i$. Every definition and result on this page is transported along this identification, which only relabels the base point. At $p=0$ the convention is the identity.

By the unique factorisation property of the holomorphic germ ring (The ring of holomorphic germs is a UFD) a nonzero nonunit $f$ has a factorisation

$f=u{q}_1^{e_1}\cdots q_r^{e_r}$

with $u$ a unit, $r\ge 1$, the $q_i$ pairwise nonassociate irreducible germs, and exponents $e_i\ge 1$; the $r$-tuple of associate classes of the $q_i$ and the exponents $e_i$ are determined by $f$. Comparing two such factorisations shows that $f$ is reduced exactly when every $e_i=1$: if some $e_i\ge 2$ then $q_i^2\mid f$, and conversely a divisor $q^2\mid f$ with $q$ irreducible makes $q$ associate to one of the $q_i$ with $e_i\ge 2$, by uniqueness of the factorisation applied to a factorisation of the quotient.

Reducedness is a property of the equation germ, not of its zero set. The germs $z_1$ and $z_1^2$ at the origin of ${\mathbb{C}}^2$ are both nonzero nonunits and have the same zero set near the origin, but only $z_1$ is reduced. The geometric identification of equations that cut out the same zero set germ is the subject of the later definition of a hypersurface germ on this page.