Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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 $H^2$
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.

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The categories of this paper are: q-bio
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The categories of this paper are: plasm-ph
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The categories of this paper are: astro-ph
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The categories of this paper are: patt-sol
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The categories of this paper are: quant-ph
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The categories of this paper are: chao-dyn
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The categories of this paper are: acc-phys
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The categories of this paper are: nucl-ex
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The categories of this paper are: stat
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The categories of this paper are: hep-ex
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The categories of this paper are: q-alg
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The categories of this paper are: eess
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The categories of this paper are: comp-gas
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The categories of this paper are: gr-qc
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The categories of this paper are: ao-sci
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The categories of this paper are: q-fin