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

TL;DR

This work advances the Arvesonouglas program by showing that closures of ideals with stable/approximate stable division in the Hilbert spaces al H_d^{(t)} are p-ssentially normal for all p>d (for t>-3). It proves that quasi homogeneous ideals in two variables have the stable division property, and combines these results to yield a new proof that closures of quasi homogeneous ideals in C[x,y] are p-ssentially normal for p>2 on al H_2^{(t)} (t  -2). The approach centers on expressing ideals as sums of principal (or stable) submodules via stable division, leveraging principal-ideal essential normality results and Beurling-type reductions to extend to general ideals. This stable-division framework provides a versatile method for establishing essential normality beyond homogeneous cases and offers new lines of evidence toward the two-variable Arvesonouglas conjecture with potential extensions to broader settings.

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 $p>2$.

Theorems & Definitions (25)

• Definition 1.1
• Lemma 2.1
• proof
• Remark 2.2
• Theorem 2.3
• proof
• Theorem 2.4
• proof
• Proposition 2.5: Fang-Xia
• Remark 2.6