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.