The single-equation proof does not cover arbitrary analytic sets

The hypersurface theory on this page applies to set germs cut out by one nonzero nonunit holomorphic equation (Complex-analytic hypersurface germ and its reduced equation). It does not extend to arbitrary analytic set germs, and the standard example shows why.

Consider the germ of the origin in ${\mathbb{C}}^2$,

$X=(\{0\},0)=(Z(x)\cap Z(y),0)\mathrm{.}$

It is an analytic set germ, the common zero set of the two coordinate functions, and its vanishing ideal is the maximal ideal ${\mathfrak{m}}_0=(x,y)$ of ${\mathcal{O}}_{{\mathbb{C}}^2,0}$: a germ vanishes on the set germ $\{0\}$ exactly when its value at $0$ is zero. This ideal is not principal. Indeed, suppose $(x,y)=(g)$ for a germ $g$. Then $Z(g)=Z(x)\cap Z(y)=\{0\}$ as set germs, while $g$ is a nonzero holomorphic germ; but by the zero-set theorem a nonzero holomorphic function on a domain in ${\mathbb{C}}^2$ has no isolated zeros, so every point of $Z(g)$ is a limit point of $Z(g)\setminus \{0\}$ and $Z(g)$ cannot equal the singleton germ $\{0\}$ near the origin (A nonzero holomorphic hypersurface in complex dimension at least two has no isolated points). Hence $\{0\}$ is not a hypersurface germ: there is no nonzero nonunit $f$ with $(Z(f),0)=(\{0\},0)$, and the hypersurface definition, the preparation theorem argument, the discriminant and the gradient criterion $\mathrm{\nabla }f$ all have no single equation to act on here.

This example shows the limit of the single-equation setup: general analytic set germs are described by ideals, which need not be principal. The germ $\{0\}\subset {\mathbb{C}}^2$ is itself a smooth zero-dimensional submanifold, even though its vanishing ideal ${\mathfrak{m}}_0=(x,y)$ is not principal. General singular-locus, resolution, and parametrisation questions for analytic set germs require their own arguments; they do not follow from the one-equation hypersurface proofs on this page. For a reduced hypersurface, the singular locus is cut out locally by $f,{\mathrm{\partial }}_{1}f,\dots ,{\mathrm{\partial }}_{n}f$ and is treated by the gradient criterion and the results of this page.