Stable polynomial division and essential normality of graded Hilbert modules

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 H² 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 {f1, ..., fk} such that every h ∈ M can be written as h = ∑_i a _i f_i for some polynomials a_i such that ∑ ∥a_if_i∥ ≤ C∥h∥. We show that certain classes of modules have this property, and that the stable decomposition h = ∑ a_i f_i may be obtained by carefully applying algorithms from computational algebra. We show that when the algebra of polynomials in d variables is given the natural ℓ¹ norm, then every ideal is linearly equivalent to an ideal that has the stable division property. We then show that, for a submodule M that has the stable division property (with respect to the appropriate norm), the quotient module H ²/M 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.