Local Krull dimension of a hypersurface germ

Fix $n\ge 1$, a point $p\in {\mathbb{C}}^n$ and a nonempty complex-analytic hypersurface germ $X$ at $p$ (Complex-analytic hypersurface germ and its reduced equation). Write $I_p(X)$ for the vanishing ideal of $X$, the ideal of all holomorphic germs at $p$ vanishing on a representative of $X$ near $p$, and define the local ring of the germ $X$ at $p$ as the quotient

$A_p(X):={\mathcal{O}}_{{\mathbb{C}}^n,p}\mathrm{/}{I}_p(X)\mathrm{.}$

Since $X$ is a hypersurface germ, $I_p(X)$ is a principal ideal: choosing a defining equation $f$ of $X$ with square-free reduction $f_{\mathrm{r}\mathrm{e}\mathrm{d}}$, the principal vanishing-ideal lemma gives $I_p(X)=(f_{\mathrm{r}\mathrm{e}\mathrm{d}})$, so that

$A_p(X)={\mathcal{O}}_{{\mathbb{C}}^n,p}\mathrm{/}(f_{\mathrm{r}\mathrm{e}\mathrm{d}})$

is the quotient of the holomorphic germ ring by a principal ideal generated by a reduced germ (The vanishing ideal of a reduced hypersurface germ is principal, Reduced holomorphic germ for a hypersurface). The ideal $(f_{\mathrm{r}\mathrm{e}\mathrm{d}})$ is proper because $f_{\mathrm{r}\mathrm{e}\mathrm{d}}$ is a nonunit, so the quotient is a nonzero commutative ring and its Krull dimension is defined (Krull dimension of a nonzero ring).

The local dimension of $X$ at $p$ is

${\dim }_{p}X:=\dim A_p(X),$

the Krull dimension of the local ring $A_p(X)$, that is, the supremum of the lengths of strict chains of prime ideals of $A_p(X)$; the value is allowed to be infinite, and it is a numerical invariant of the germ.

Well-definedness. The definition does not depend on the defining equation or on the chosen representative. The vanishing ideal $I_p(X)$ is attached to the set germ $X$ alone: two representatives of $X$ agree on a neighbourhood of $p$, so they have the same vanishing ideal, and a germ vanishes on one representative near $p$ exactly when it vanishes on the other. If $f^{\mathrm{\prime }}$ is any other defining equation of $X$, then $f_{\mathrm{r}\mathrm{e}\mathrm{d}}^{\mathrm{\prime }}$ generates the same principal ideal by the principal vanishing-ideal lemma, applied to $f^{\mathrm{\prime }}$ and to $f$; hence ${\mathcal{O}}_{{\mathbb{C}}^n,p}\mathrm{/}(f_{\mathrm{r}\mathrm{e}\mathrm{d}})={\mathcal{O}}_{{\mathbb{C}}^n,p}\mathrm{/}(f_{\mathrm{r}\mathrm{e}\mathrm{d}}^{\mathrm{\prime }})$ as quotients of the same ring, with the same prime ideals and therefore the same Krull dimension. Thus ${\dim }_{p}X$ depends only on the set germ $X$, and it is computed by any reduced local equation. The translation convention of Reduced holomorphic germ for a hypersurface identifies ${\mathcal{O}}_{{\mathbb{C}}^n,p}$ with the germ ring at the origin, and the definition is transported along it.

Degenerate cases. The definition applies only to nonempty hypersurface germs, i.e. to nonzero nonunit equations, so the empty set germ and the whole space germ are excluded; this is why $A_p(X)$ is not the zero ring. For $n=1$, the square-free reduction of a nonzero nonunit germ of one variable is $\zeta \mapsto \zeta -p$ up to a unit, so $X=\{p\}$ as a set germ and $A_p(X)$ is the quotient of ${\mathcal{O}}_{\mathbb{C},p}$ by its maximal ideal, a field; hence ${\dim }_{p}X=0$ for $n=1$. The general computation of ${\dim }_{p}X$ for hypersurface germs is the pure-codimension statement proved later on this page.