INTRODUCTION 7

It is important to study representation theory over arbitrary base fields since

many applications deal with algebras defined over fields which are not algebraically

closed. For example, the base field of real numbers is of interest for applications

in analysis, the field of rational numbers for number theory, finite fields for the

relationships to quantum groups (like Ringel’s Hall algebra approach), etc.

When attempting to generalize statements first proven over algebraically closed

fields to arbitrary base fields, three typical scenarios of different nature can be

observed. Frequently statements and proofs carry over to the more general situation

without essential change. Also often the statements remain true but require new

proofs, frequently leading to better insights and streamlined arguments even for

the algebraically closed

case2.

On the other hand, in a significant number of cases

completely new and unexpected effects occur, causing the statements to fail in the

general case. The present notes focus in particular on these kinds of new effects.

The representation theoretical analogues of the exceptional curves X and their

hereditary categories H are given by the concealed canonical algebras [70] and

their module categories mod(Λ). The link between the two concepts is given by an

equivalence

Db(H) Db(mod(Λ))

of derived categories which leads to a translation

between geometric and representation theoretic notions. We illustrate this in the

typical case where Λ is a tame hereditary algebra: the subcategory H0 of objects

of finite length corresponds to the full subcategory R of mod(Λ) formed by the

regular representations. Simple objects Sx in H correspond to simple regular rep-

resentations. Vector bundles correspond to preprojective (or preinjective) modules,

line bundles L to preprojective modules P (or preinjective modules) of defect −1

(or 1, respectively). In particular, the multiplicities e(x) are also definable in terms

of preprojective modules of defect −1 and simple regular representations. The

function field of X agrees with the endomorphism ring of the unique generic [19]

Λ-module. The importance of the generic module for the representation theory of

tame hereditary algebras is demonstrated in [90]. Our results on exceptional curves

all have direct applications to representation theory. In particular:

• Let Λ be a tame hereditary algebra. The (small) preprojective algebra

n≥0

HomΛ(P, τ

−nP

),

where P is a projective module of defect −1 and τ

−

is the (inverse)

Auslander-Reiten translation on mod(Λ), is a graded factorial domain

if the underlying tame bimodule is of dimension type (1, 4) (or (4,

1))3.

Note that the (small) preprojective algebra contains the full information

on Λ and its representation theory.

• In general there are automorphisms of

Db(mod(Λ))

fixing all objects but

acting non-trivially on morphisms, contrary to the algebraically closed

case.

• A tubular algebra requires up to three different projective curves of genus

zero to parametrize the indecomposable modules.

2Some

examples for this can be seen in the results of Happel and Reiten about the charac-

terization of hereditary abelian categories with tilting object ([39], generalizing [38]) and in the

proof of the transitivity of the braid group action on complete exceptional sequences for hereditary

Artin algebras by Ringel ([94], generalizing [20]) and by Meltzer and the author for exceptional

curves ([60], summarized in Section 7.1), generalizing [78].

3This

is also true for many tame bimodules of dimension type (2, 2).