In 1911 Steinitz extended the structure theorem for finitely generated abelian
groups by proving his classification of isomorphism classes in fingen(^), the cate-
gory of finitely generated modules over (what are now called) Dedekind domains Q
[S]. One of the most interesting parts of Steinitz's results is that he was also able to
describe the direct-sum relations in fingen(O). Steinitz's theorem has been partic-
ularly resistant to generalization. In fact, the only noetherian integral domains Q
(other than Dedekind domains) for which such a module classification for fingen(^)
was known prior to completion of this project seem to be the Dedekind-like rings
studied in [L2] (a subclass of the rings called "Dedekind-like" in the present mem-
oir). However, for commutative noetherian rings that are not domains, some other
results exist [NR, NRSB].
On the other hand, during the past 30 years or so, the concept of wild repsen-
tation type has arisen as an insurmountable obstacle to getting a classification
theorem for the isomorphism classes in fingen(^4) when A is a finite dimensional
algebra over a field. For an early result that provided part of the motivation for
this series of papers, see [Ri]. The results there and after it show that the vast
majority of finite dimensional algebras over algebraically closed fields have wild
representation type. Moreover, no wild algebra A has a complete classification the-
orem for the isomorphism classes in fingen(A). (However, we know of no theorem
in mathematical logic saying that such a classification cannot be proved.)
The objective of this memoir is to describe the category fingen(fJ) where ft
is any commutative noetherian ring such that fingen(fJ) does not have wild rep-
resentation type. This completes the project begun in [KL1, KL2] where Q was
restrictred to be a complete local ring. We say that Q is fingen-tame — more
completely, fingen(Q) has tame representation type — if we can describe the iso-
morphism classes, direct-sum relations, and local-global relations in this category,
and give some information about the homomorphisms in this category.
Let m be a maximal ideal of a commutative noetherian ring Q. Informally,
we say that Q is finlen-wild (wrt m) — more completely, the category finlen(fJ)
of ^-modules of finite length has wild representation type (with respect to m) —
if any description of all isomorphism classes in fmlen(fi) would have to contain a
description of all isomorphism classes in finlen(A) for all finite-dimensional (non-
commutative!) algebras A over the field k = Q/m. See Definition 14.5 for our
formal definition of finlen-wild, and [KL1, Remarks 2.3] for a brief introduction to
the notion of wildness from the standpoint of commutative noetherian rings.
We define a class of commutative noetherian rings A called "Dedekind-like"
[Definition 10.1], and a (very small) class of commutative artinian local rings of
length 4 called "Klein rings" [Definitions 14.2]. The reason for these definitions
is the following result (whose "indecomposability" hypothesis involves no loss of
generality, since ft is noetherian).
2.1 (see Theorem 14.5). If an indecomposable a commutative noe-
therian ring Q is such that finlen(fi) fails to be wild (wrt m) for all maximal ideals
m, then ft is a homomorphic image of a Dedekind-like ring, or else is a Klein ring.
Received by the editor February 20, 2003, and in revised form June 30, 2004. Corrections
February 2, 2005.