12 0. BACKGROUND
For each λ k let be the simple k[T ]-module k[T ]/(T λ). If all the images
FM (Sλ) are indecomposable and pairwise non-isomorphic, then {FM (Sλ)}λ∈k is
called an affine one-parameter family (of indecomposable modules).
0.2.2 (Tame algebras). Let A be a finite dimensional algebra over an alge-
braically closed field k. Then A is called tame, if for each natural number d almost
all indecomposable A-modules of dimension d lie in a finite number of affine one-
parameter families, that is, given d there are finitely many k[T ]-A-bimodules Mi,
free of finite rank over k[T ], such that all but finitely many indecomposable A-
modules of dimension d are isomorphic to FMi (Sλ) for some i and some λ k.
0.2.3 (Generic modules). In the study of one-parameter families the concept of a
generic module is important ([19], also [50]). An A-module M is called generic [19],
if it is indecomposable, of infinite length over A, but of finite length over its endo-
morphism ring. Note that for each affine one-parameter family, given by a functor
FM , a generic A-module is given by FM (k(T )), where k(T ) is the field of rational
functions in one variable.
Crawley-Boevey [19] has shown that, over an algebraically closed field, A is
tame if and only if for any natural number d there is only a finite number of generic
modules of endolength d. (In the latter case one also says that A is generically
tame. This notion makes sense over any field.) He showed that in this case the
generic modules correspond to the one-parameter families.
0.2.4 (The Kronecker algebra). The Kronecker algebra Λ over an algebraically
closed field k provides the prototype of a tame algebra as well as of a one-parameter
family. It is defined to be the path algebra of the quiver

and is isomorphic to Λ =
k 0
k2
k
, where k
2
= k⊕k is considered as k-k-bimodule.
The module category mod(Λ), as well as its Auslander-Reiten quiver, has a partic-
ular simple shape, it is trisected
mod(Λ) = P R Q,
where P is the preprojective component, consisting of the Auslander-Reiten orbits
of two projective indecomposables, Q is the preinjective component, consisting of
the Auslander-Reiten orbits of two injective indecomposables, and R consists of the
regular indecomposable modules, all lying in homogeneous tubes. One can say that
P and Q form the discrete part of mod(Λ) and R forms the continuous part, since
the tubes are parametrized by the projective line
P1(k).
Moreover, if one forms the
category
H
def
= Q[−1] P R
inside the bounded derived category of mod(Λ), then H is equivalent to
coh(P1(k)),
the category of coherent sheaves over
P1(k).
The regular indecomposable modules of a fixed dimension form the one-param-
eter families for Λ (leave out one tube for an affine family). The regular part R
itself forms a separating tubular family.
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