Essentially Reductive Hilbert Modules II

TL;DR

This paper investigates the property of essential reductivity in Hilbert modules over polynomial rings, extending known results to specific domains and ideal types, and analyzing the conditions under which this property is preserved.

Contribution

The paper extends the class of Hilbert modules known to be essentially reductive to include certain Reinhardt domains and quasi-homogeneous ideals in multiple variables.

Findings

Essential reductivity is preserved in Hilbert modules over ellipsoids in two variables.

Extension of results to quasi-homogeneous ideals with one-dimensional zero variety.

Analysis of conditions for essential reductivity in higher-dimensional settings.

Abstract

Many Hilbert modules over the polynomial ring in m variables are essentially reductive, that is, have commutators which are compact. Arveson has raised the question of whether the closure of homogeneous ideals inherit this property and provided motivation to seek an affirmative answer. Positive results have been obtained by Arveson, Guo, Wang and the author. More recently, Guo and Wang extended the results to quasi-homogeneous ideals in two variables. Building on their techniques, in this note the author extends this result to Hilbert modules over certain Reinhardt domains such as ellipsoids in two variables and analyzes extending the result to the closure of quasi-homogeneous ideals in m variables when the zero variety has dimension one.