CHAPTER 0

Preliminaries

This chapter assembles some facts that apply to rings more general than the

Dedekind-like rings that occupy the bulk of this memoir. Many of these facts are

known, but are included so that readers not familiar with them can find clear

statements in the form in which we use them. We include short proofs of known

results when these proofs are no longer than a precise series of references, or when

we do not know good references

3. Standard Notation and Terminology

This section collects some terminology and notation that occurs often and re-

tains its significance throught this memoir. The more complete Terminology Index

and Notation Index at the end of the memoir make it easy to find other definitions

when they are encountered outside of the section in which they are defined.

NOTATION

3.1. Throughout this memoir, ring means "commutative ring" and

local ring means "noetherian local ring" unless otherwise specified. A ring is reduced

if it has no nonzero nilpotent elements.

A always denotes a Dedekind-like ring [defined in §10] unless otherwise specified.

Essentially the only "otherwise specified" situation is that we regularly use it for

rings that we are proving to be Dedekind-like.

A regular element of a ring is a non-zero-divisor in that ring. The total quotient

ring of a ring is the localization that inverts all regular elements of the ring. Maps

denoted by

L-

and — » denote injections and surjections, respectively.

T always denotes the normalization of a reduced noetherian ring (integral clo-

sure in its total quotient ring), and denotes the normalization of A if no other ring

is explicitly specified. There is a unique decomposition

(3.i.i) r = e ^ i \

where each Th is an integral domain. If T is the normalization of A, then each Th

is a Dedekind domain [Definition 10.1]. (Note that we do not consider fields to be

Dedekind domains.) We call the rings I \ the coordinate rings of I\

(... )q = (... )Q(Q) denotes localization at the complement of the finite set of

minimal prime ideals of a noetherian ring Q. When O is reduced, this is the same

as the localization that inverts all regular elements of A, that is, OQ is the total

quotient ring of Q. See Section 4 for the many equivalent interpretations of (... )Q,

which we make extensive use of. In particular, if T is the normalization of a reduced

noetherian ring Q and Q — Q(ft), then TQ = QQ and — in the notation of (3.1.1)

— each ( I \ ) Q is the field of quotients of I \ [Corollary 4.4].

As in the earlier papers of this series, we write functions on the left, with one

exception: functions that will be represented by matrix multiplication are written

on the right.

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