Part I. ALGEBRAIC PRELIMINARIES:

GAP SHEAVES AND VOGEL CYCLES

Chapter 0. INTRODUCTION

Throughout this book, our primary algebraic tool consists of a method for taking a coher-

ent sheaf of ideals and decomposing it into pure-dimensional "pieces". Actually, we begin

with an ordered set of generators for the 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. Part I of this book contains the construction of the Vogel

cycles; it is, regrettably, very technical in nature. The Vogel cycles are defined using gap

sheaves, together with the associated analytic cycles which they define, the gap cycles. 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.

If the underlying space is not Cohen-Macaulay, the main technical problem is that there

are, at least, three different reasonable definitions of the gap sheaves and cycles; we select

as "the" definition the one that works most nicely in inductive proofs. We show, however,

that if one re-chooses the functions defining the ideal in a suitably "generic" way, then all

competing definitions for the gap cycles and Vogel cycles agree.

In Chapter 3 of this part, we prove some extremely general Le-Iomdine-Vogel formulas;

as we shall see in later chapters, these formulas are an amazingly effective tool for trans-

forming problems about a given singularity into problems involving a singularity of smaller

dimension.

The reader who wishes to bypass this technical portion of the book can jump to the

Summary of Part I, which begins on page 31.

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