0.2. ONE-PARAMETER FAMILIES, GENERIC MODULES AND TAMENESS 13
There is (up to isomorphism) precisely one generic Λ-module, given by the
representation
k(T )
1
·T
k(T ) ,
where k(T ) is the field of rational functions in one variable, which is the function
field of
P1(k);
the endomorphism ring of this generic module is k(T ).
Let A be a tame k-algebra. By Drozd’s theorem all one-parameter families for
A are rational. In all known examples these parametrizations can be realized by a
functor mod(Λ) −→ mod(A).
0.2.5. Over an arbitrary field k there is still no convenient definition of tameness.
The definition of generically tameness makes sense over any field and has many
advantages, but it does not capture the geometric flavour of one-parameter families.
One should expect that an extension of Drozd’s Tame and Wild Theorem over
arbitrary field k holds in the sense that, roughly speaking, the indecomposable finite
dimensional modules over any non-wild finite dimensional k-algebra lie essentially in
one-parameter families which are indexed by (affine parts of) the noncommutative
curves of genus zero. The projective line is related to the Kronecker algebra, just
as the noncommutative curves of genus zero are related (up to weights) to the tame
bimodules M =
F
MG and their associated hereditary algebras
Λ =
G 0
M F
,
which were studied by Dlab and Ringel in several papers (for example [24, 89, 29],
to name a few). Therefore the tame bimodules are of fundamental importance in
the study of one-parameter families. Note that in general different one-parameter
families of genus zero for a fixed finite dimensional k-algebra may be induced by
different tame bimodules over k, as the discussion in Chapter 8 shows.
0.2.6 (The weighted case). In general one has do deal with the so-called weighted
case which leads to the study of the canonical algebras and to the weighted pro-
jective lines (as Ringel pointed out in his survey [93]). Over algebraically closed
fields, the canonical algebras were defined by Ringel [91] and the weighted pro-
jective lines by Geigle and Lenzing [34]. Both definitions were later extended to
arbitrary fields. In the case of the canonical algebras this was done by Ringel and
Crawley-Boevey [92], in the case of the weighted projective lines by Lenzing [68]
who called the more general objects exceptional curves. The canonical algebras
can be characterized (essentially up to tilting equivalence) as the class of finite
dimensional algebras admitting a separating tubular family [70]. These tubular
families are parametrized by the exceptional curves. The tame bimodule algebras
correspond to the subclass of finite dimensional algebras whose tubes are all homo-
geneous. So we call the tame bimodule case also the homogeneous or unweighted
case, the general case also the weighted case.
By some general techniques (perpendicular calculus [35]), insertion of weights
[68]) the general, weighted case can be reduced essentially to the homogeneous case.
Therefore, main parts of this article are concerned with the homogeneous case.
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