8 INTRODUCTION

• A tubular algebra admits generic modules with up to three different (non-

isomorphic) endomorphism rings.

• The endomorphism ring of the generic module over a tame hereditary

algebra is commutative if and only if all multiplicities are equal to one,

a condition automatically satisfied over algebraically closed base fields.

It is surprising that this condition, which essentially says that the mor-

phisms between preprojective and regular representations behave “well”,

yields the commutativity of the generic module’s endomorphism ring, and

conversely.

The results on the function field also provide an explanation of the strange fact

(pointed out in [90]) that a bimodule like

R

HH, given by noncommutative data,

leads to a commutative function field

Quot

(

R[U, V ]/(U

2

+ V

2

+ 1)

)

,

whereas a bimodule like

Q

Q(

√

2,

√

3)Q(√2,√3), given by commutative data, leads to

a noncommutative function field, the quotient division ring of

Q U, V /(UV + V U, V

2

+ 2U

2

− 3).

There are a number of inspiring papers dealing with tame hereditary alge-

bras. For example, those by Dlab and Ringel on bimodules and hereditary alge-

bras [24, 89, 27, 26, 29] (see additionally [28, 22, 23]), in particular Ringel’s

Rome proceedings paper [90], as well as those by Lenzing [64], Baer, Geigle and

Lenzing [7], and by Crawley-Boevey [18], dealing with the structure of the param-

eter curves for tame hereditary algebras over arbitrary fields.

By perpendicular calculus and insertion of weights many problems for concealed

canonical algebras (and in particular for tame hereditary algebras) can be reduced

to the special class of tame bimodule algebras. This means that we often may

restrict our attention to a tame hereditary k-algebra of the form Λ =

G 0

M F

,

where M =

F

MG is a tame bimodule over k, that is, the product of the dimensions

of M over the skew fields F and G, respectively, equals 4. These are the analogues

of the Kronecker algebra

k 0

k2 k

, which is isomorphic to the path algebra of the

following quiver.

• •

In this homogeneous case X parametrizes the simple regular representations of

Λ. This situation was studied in the cited papers by Dlab and Ringel, by Baer,

Geigle and Lenzing, and by Crawley-Boevey. Over the real numbers the structure

of X as topological space is described explicitly in [24, 25, 26]. In [89, 29] and

more generally in [18] an aﬃne part of X is described by the simple modules over a

(not necessarily commutative) principal ideal domain. In [18] additionally a (com-

mutative) projective curve is constructed, which parametrizes the points of X and

is the centre of the noncommutative projective curves considered in [64] and [7].

A model-theoretic approach using the Ziegler spectrum is described by Prest [86]

and Krause [51, Chapter 14]. One advantage provided by the present notes is that

the geometry of X is described in terms of graded factorial coordinate algebras.

This is useful in particular for studying the properties of the sheaf category H

by forming natural localizations (Chapter 2) and for analyzing the automorphism

group of

Db(H)

(Chapter 3). It is also exploited in our proof of the characterization