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.
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
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.