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
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.