Stable polynomial division and essential normality of graded Hilbert modules

TL;DR

This paper introduces the stable division property for modules, linking algebraic techniques to operator theory, and proves that modules with this property are essentially normal, advancing the understanding of Arveson's conjecture.

Contribution

It establishes the stable division property for modules and shows that modules with this property are p-essentially normal, providing a new approach to Arveson's conjecture.

Findings

Certain classes of modules have the stable division property.

Modules with stable division are p-essentially normal for p > dimension.

Reducing the problem to ideals generated by quadratic polynomials.

Abstract

The purpose of this paper is to initiate a new attack on Arveson's resistant conjecture, that all graded submodules of the $d$-shift Hilbert module $H^2$ are essentially normal. We introduce the stable division property for modules (and ideals): a normed module $M$ over the ring of polynomials in $d$ variables has the stable division property if it has a generating set $\{f_1, ..., f_k\}$ such that every $h \in M$ can be written as $h = \sum_i a_i f_i$ for some polynomials $a_i$ such that $\sum \|a_i f_i\| \leq C\|h\|$. We show that certain classes of modules have this property, and that the stable decomposition $h = \sum a_i f_i$ may be obtained by carefully applying techniques from computational algebra. We show that when the algebra of polynomials in $d$ variables is given the natural $\ell^1$ norm, then every ideal is linearly equivalent to an ideal that has the stable division property. We then show that a module $M$ that has the stable division property (with respect to the appropriate norm) is $p$-essentially normal for $p > \dim(M)$, as conjectured by Douglas. This result is used to give a new, unified proof that certain classes of graded submodules are essentially normal. Finally, we reduce the problem of determining whether all graded submodules of the $d$-shift Hilbert module are essentially normal, to the problem of determining whether all ideals generated by quadratic scalar valued polynomials are essentially normal.