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The vanishing ideal of a reduced hypersurface germ is principal

Statement

Let $n \geq 1$ , let $p \in C^{n}$ and let $f \in O_{C^{n}, p}$ be a reduced nonzero nonunit. Write $Z (f)$ for the zero set germ of $f$ at $p$ and set

$I_{p} (Z (f)) : = {h \in O_{C^{n}, p} : h vanishes on the zero set of f near p},$

where $h$ vanishes on the zero set of $f$ near $p$ when some representative of $h$ vanishes at every point of the zero set of some representative of $f$ on a common neighbourhood of $p$ . Then $I_{p} (Z (f))$ is an ideal of the germ ring and

$I_{p} (Z (f)) = (f) .$

More generally, for an arbitrary nonzero nonunit germ $g$ with square-free reduction $g_{r e d}$ ,

$I_{p} (Z (g)) = (g_{r e d}) .$

Facts & Assumptions

Given: A reduced nonzero nonunit germ $f$ at $p$ , and the ideal $I_{p} (Z (f))$ of germs vanishing on its zero set near $p$ .

[F1]

The square-free reduction $g_{r e d} = q_{1} \dots q_{r}$ of a nonzero nonunit $g$ is reduced, its associate class depends only on $g$ , and $Z (g_{r e d}) = Z (g)$ on a common neighbourhood of $p$ (Square-free reduction of a holomorphic equation).

[F2]

Center at $p$ and choose the invertible complex-linear map $T$ and product representative of the finite-projection theorem. With $Φ (z) = p + T z$ , the germ $f \circ Φ$ is regular of order $d \geq 1$ and equals $u W$ for a unit $u$ and a degree- $d$ Weierstrass polynomial $W$ ; their zero sets coincide on that representative (After a linear coordinate change, every nonzero germ is regular in the last variable, Weierstrass preparation theorem, Finite local projection of a reduced hypersurface germ).

[F3]

Units have nonzero value at the base point, so $(f \circ Φ) = (W)$ and their zero germs agree (A germ is a unit exactly when its value at $0$ is nonzero, so $O_{m, 0}$ is local).

[F4]

Weierstrass division: every $h \in O_{n, 0}$ is uniquely $q W + r_{0} + r_{1} T + \dots + r_{d - 1} T^{d - 1}$ with $q \in O_{n, 0}$ and coefficients $r_{j} \in O_{n - 1, 0}$ (Weierstrass division theorem).

[F5]

The prepared $W$ of the reduced germ $f$ is square-free over $K = Frac (O_{n - 1, 0})$ and its discriminant $D_{W} = {Disc}_{T} (W)$ is a nonzero base germ; $D_{W} (z^{'}) = 0$ exactly when the slice $W (z^{'}, \cdot)$ has a repeated root (Reduced preparation has nonzero discriminant, Discriminant and branch set of a fixed Weierstrass projection, The discriminant is $\prod_{i < j} (α_{i} - α_{j})^{2}$ and vanishes exactly when a monic polynomial has a repeated root).

[F6]

A nonzero holomorphic function on a connected open set does not vanish on a nonempty open subset (A holomorphic function vanishing on a nonempty open subset of a domain vanishes identically).

[F7]

A monic complex polynomial of degree $d$ has $d$ roots counted with multiplicity (A complex polynomial of degree $n$ has exactly $n$ roots counted with multiplicity). A nonzero polynomial of degree at most $d - 1$ over a field has fewer than $d$ distinct roots (A nonzero polynomial of degree $n$ over an integral domain has at most $n$ distinct roots).

Proof technique: direct — prepare in generic coordinates, divide by $W$ , and force the remainder to vanish on a dense base set.

Proof

1.1givenF2F3

Choose $T$ and $Φ (z) = p + T z$ with $f \circ Φ = u W$ as in [F2]. The pullback $Φ^{*} : h \mapsto h \circ Φ$ is a ring isomorphism $O_{C^{n}, p} \to O_{n, 0}$ , with inverse pullback by $Φ^{- 1}$ . It sends $(f)$ to $(W)$ by [F3] and sends $I_{p} (Z (f))$ to $I_{0} (Z (W))$ , since $Φ$ carries the corresponding zero germs onto each other. Hence it suffices to prove $I_{0} (Z (W)) = (W)$ .

1.2givenF4

The reverse inclusion is immediate: if $h = c W$ then every representative of $h$ vanishes at every point where $W$ vanishes, so $(W) \subseteq I_{0} (Z (W))$ .

2.1step 1.1F4

Let $h \in I_{0} (Z (W))$ . By [F4] write $h = q W + r$ with $r = \sum_{j < d} r_{j} T^{j}$ , $r_{j} \in O_{n - 1, 0}$ . Since $q W$ vanishes on $Z (W)$ and $h$ does too, the remainder $r = h - q W$ vanishes on $Z (W)$ near the origin.

3.1step 2.1F2F5F7

Choose a common product $V \times {∣ T ∣ < ρ}$ on which the division identity and vanishing of $r$ on $Z (W)$ hold. Write $W = T^{d} + \sum_{j < d} a_{j} (z^{'}) T^{j}$ . Since $a_{j} (0) = 0$ , shrink the connected base polydisc $V$ until $\sum_{j < d} ∣ a_{j} (z^{'}) ∣ ρ^{j - d} < 1$ . For $∣ T ∣ \geq ρ$ the lower terms have sum of absolute values strictly less than $∣ T ∣^{d}$ , so no slice root lies there. For every $z^{'} \in V$ with $D_{W} (z^{'}) \neq 0$ , [F5] and [F7] therefore give $d$ distinct roots, all within the common product. The polynomial $r (z^{'}, \cdot)$ has degree less than $d$ and vanishes at all these roots, so [F7] makes it the zero polynomial. Thus $r_{j} (z^{'}) = 0$ for every $j < d$ .

4.1step 3.1F5F6

If $n = 1$ , the base is the single point $V \subset C^{0}$ and $D_{W}$ is a nonzero constant, so $V ∖ {D_{W} = 0} = V$ . Each $r_{j}$ is also a constant; step 3.1 says it vanishes at this sole point, hence $r_{j} = 0$ . If $n \geq 2$ , then $V ∖ {D_{W} = 0}$ is nonempty and open: $D_{W}$ is a nonzero holomorphic germ, so it cannot vanish on a nonempty open subset of $V$ , and its zero set is closed. Since each $r_{j} \in O_{n - 1, 0}$ vanishes on this nonempty open set, [F6] gives $r_{j} = 0$ for every $j < d$ . In either case $r = 0$ and $h = q W \in (W)$ ; combined with step 1.2 this gives $I_{0} (Z (W)) = (W)$ .

5.1step 1.1step 4.1F1∎

Undoing the coordinate change of step 1.1 gives $I_{p} (Z (f)) = (f)$ for reduced $f$ . For an arbitrary nonzero nonunit $g$ with square-free reduction $g_{r e d}$ , the zero germs agree on a neighbourhood and $g_{r e d}$ is reduced by [F1], so $I_{p} (Z (g)) = I_{p} (Z (g_{r e d})) = (g_{r e d})$ .

Depends on

Used by

Puiseux discs normalise a reduced plane curve germ Corollary
Complex-analytic hypersurface germ and its reduced equation Definition
Local Krull dimension of a hypersurface germ Definition
Regular and singular points of an analytic hypersurface Definition
Total quotient ring and normalisation of a reduced plane curve germ Definition
An ordinary node has two smooth branches Example
The coordinate axes form a reduced crossing Example
The cusp y²=x³ has Puiseux parameter (t²,t³) Example
The plane branch y²=x⁵ has Puiseux parameter (t²,t⁵) Example
Convergent Puiseux parametrisation of an irreducible plane branch Theorem
Finite unique irreducible components of a hypersurface germ Theorem
Singular locus of a reduced analytic hypersurface Theorem

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Sources

Jiří Lebl, Tasty Bits of Several Complex Variables, Chapter 6 §§6.1–6.7 (standard reference, not scraped)
Jean-Pierre Demailly, Complex Analytic and Differential Geometry, Chapter II §§2, 4 and 6 (standard reference, not scraped)