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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 ∈ 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.

Stable polynomial division and essential normality of graded Hilbert modules