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A quotient by a Weierstrass polynomial is a finite module over the smaller germ ring

Statement

Let $W$ be a Weierstrass polynomial of degree $d$ in $z_{m}$ . Then the quotient $O_{m, 0} / (W)$ is a finitely generated $O_{m - 1, 0}$ -module, where $O_{0, 0} = C$ when $m = 1$ , generated by the residue classes of

$1, z_{m}, \dots, z_{m}^{d - 1} .$

Facts & Assumptions

Given: A degree- $d$ Weierstrass polynomial $W$ .

[L1]

A Weierstrass polynomial is the monic degree- $d$ polynomial in the last variable from Weierstrass polynomials in the last variable.

[L2]

Weierstrass division gives unique quotient and remainder of degree $< d$ upon division by $W$ (Weierstrass division theorem).

[L3]

Noetherian-module language is that of Noetherian commutative rings and modules.

Proof

technique · direct

1.1

By [L1] and [L2], every germ $f \in O_{m, 0}$ can be written uniquely as $f = q W + r_{0} + r_{1} z_{m} + \dots + r_{d - 1} z_{m}^{d - 1}$ with $r_{j} \in O_{m - 1, 0}$ . Modulo $(W)$ this becomes $[f] = r_{0} [1] + r_{1} [z_{m}] + \dots + r_{d - 1} [z_{m}^{d - 1}],$ so the listed residue classes generate the quotient as an $O_{m - 1, 0}$ -module.

L1L2L3

2.1

The same division theorem [L2] makes the remainder unique, so those generators give a canonical normal form for every class in the quotient. Since there are only $d$ generators, the quotient is a finite $O_{m - 1, 0}$ -module.

step 1.1L2∎

Depends on

Used by

The ring of holomorphic germs is Noetherian Theorem

Dependency tree · two levels

14 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Jiří Lebl, Tasty Bits of Several Complex Variables, Section 6.4 (standard reference, not scraped)
Jaap Korevaar and Jan Wiegerinck, Several Complex Variables, Section 4.5 (standard reference, not scraped)