Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Gap Sheaves and Vogel Cycles   Throughout our work on the L\^e cycles of an affine hypersurface singularity,
our primary algebraic tool consisted of a method for taking the Jacobian ideal
of a complex analytic function and decomposing it into pure-dimensional
"pieces". These pieces were obtained by considering the relative polar
varieties of L\^e and Teissier as gap sheaves. A gap sheaf is a formal device
which gives a scheme-theoretic meaning to the analytic closure of the
difference of an initial scheme and an analytic set. We would like to extend
our results on L\^e cycles to functions on an arbitrary complex analytic space,
and so we generalize this algebraic approach.
  We begin with an ordered set of generators for an ideal, and produce a
collection of pure-dimensional analytic cycles, the Vogel cycles, which seem to
contain a great deal of ``geometric'' data related to the original ideal. We
prove a number of useful results, including some extremely general
L\^e-Iomdine-Vogel formulas; these formulas generalize the L\^e Iomdine
formulas that we used so profitably in previous work.

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The categories of this paper are: alg-geom
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The categories of this paper are: hep-lat
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The categories of this paper are: chem-ph
1
The categories of this paper are: cs
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The categories of this paper are: mtrl-th
12
The categories of this paper are: nucl-th
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The categories of this paper are: atom-ph
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The categories of this paper are: math-ph
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The categories of this paper are: funct-an
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The categories of this paper are: nlin
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The categories of this paper are: bayes-an
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The categories of this paper are: econ
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The categories of this paper are: ao-sci
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The categories of this paper are: hep-ex
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The categories of this paper are: nucl-ex
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The categories of this paper are: patt-sol