12 0. BACKGROUND

For each λ ∈ k let Sλ 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.