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).
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