Stable division and essential normality: the non-homogeneous and quasi homogeneous cases

TL;DR

This paper proves that certain ideals in specific reproducing kernel Hilbert spaces have closures that are p-essentially normal, extending previous results to non-homogeneous and quasi-homogeneous cases with new proof techniques.

Contribution

It introduces the approximate stable division property for ideals and proves p-essential normality in broader non-homogeneous and quasi-homogeneous contexts.

Findings

Ideals with approximate stable division are p-essentially normal in these spaces.

All quasi homogeneous ideals in two variables have the stable division property.

New proof that closures of quasi homogeneous ideals in two variables are p-essentially normal for p>2.

Abstract

Let $\mathcal{H}_d^{(t)}$ ($t \geq -d$, $t>-3$) be the reproducing kernel Hilbert space on the unit ball $\mathbb{B}_d$ with kernel \[ k(z,w) = \frac{1}{(1-\langle z, w \rangle)^{d+t+1}} . \] We prove that if an ideal $I \triangleleft \mathbb{C}[z_1, \ldots, z_d]$ (not necessarily homogeneous) has what we call the "approximate stable division property", then the closure of $I$ in $\mathcal{H}_d^{(t)}$ is $p$-essentially normal for all $p>d$. We then show that all quasi homogeneous ideals in two variables have the stable division property, and combine these two results to obtain a new proof of the fact that the closure of any quasi homogeneous ideal in $\mathbb{C}[x,y]$ is $p$-essentially normal for $p>2$.