## 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 {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_{i}f_{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 ℓ^{1} 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.

Original language | English |
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Pages (from-to) | 273-289 |

Number of pages | 17 |

Journal | Journal of the London Mathematical Society |

Volume | 83 |

Issue number | 2 |

DOIs | |

State | Published - 1 Jan 2011 |

Externally published | Yes |

## ASJC Scopus subject areas

- General Mathematics