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

Abstract

Let ${\mathcal{H}}_d^{(t)}$ ($t\ge -d$, $t>-3$) 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 $I◃\mathbb{C}[z_1,\dots ,z_d]$ (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 $p>2$.

Some mistakes fixed and details added to the proof of the main result. 17 pages

References in corpus (5)

• p-Summable Commutators in Dimension d
• Essentially Reductive Hilbert Modules II
• Stable polynomial division and essential normality of graded Hilbert modules
• Essentially Reductive Weighted Shift Hilbert Modules
• An Analytic Grothendieck Riemann Roch Theorem