14 0. BACKGROUND
0.3. Canonical algebras and exceptional curves
In this section we describe briefly the general class of finite dimensional alge-
bras admitting a separating tubular family. This is the class of concealed canoni-
cal algebras, which contains the class of canonical algebras and the class of tame
hereditary algebras, in particular tame bimodule algebras. These algebras have as
geometric counterpart the exceptional curves. These curves correspond to the con-
cealed canonical algebras via tilting theory, and accordingly are derived equivalent
to the corresponding algebra. Thus the study of (concealed) canonical algebras is
essentially equivalent to the study of exceptional curves. Since we are interested
in the geometrical aspects of algebras, we prefer in this paper the usage of the
language and theory of the exceptional curves.
0.3.1 (Concealed canonical algebras). Let k be a field and Σ a finite dimensional
k-algebra, which is assumed to be connected. Denote by mod(Σ) the category
of finitely generated right Σ-modules. Then Σ is concealed canonical ([70], see
also [104]) if and only if mod(Σ) contains a sincere separating exact subcategory
mod0(Σ). This means
Exactness. mod0(Σ) is an exact abelian subcategory of mod(Σ), which is
stable under Auslander-Reiten translation τ = D Tr and τ = Tr D
Separation. Each indecomposable from mod(Σ) belongs either to
mod0(Σ) or to mod+(Σ), which consists of all M mod(Σ) such that
Hom(mod0(Σ), M ) = 0, or to mod−(Σ), which consists of all N mod(Σ)
such that Hom(N, mod0(Σ)) = 0.
Sincerity. For each non-zero M mod+(Σ) there is a non-zero morphism
from M to mod0(Σ) and for each non-zero N mod−(Σ) there is non-zero
morphism from mod0(Σ) to N.
Stability. Each projective module belongs to mod+(Σ) and each injective
module to mod−(Σ).
0.3.2. The most prominent classes of examples are the following:
(1) The canonical algebras, as defined by Ringel and Crawley-Boevey in [92].
Actually, every concealed canonical algebra is tilting equivalent to a canonical al-
gebra. A canonical algebra is defined to be the tensor algebra of a species
D1
D1
D1 ··· D1
D1
D1
D2
D2
D2
···
D2
D2
D2
F
F
MG
G
.
.
.
.
.
.
.
.
.
.
.
.
Dt
Dt
Dt
···
Dt
Dt
Dt
modulo certain relations (for details we refer to [92]). Here,
F
MG is a tame bi-
module (see 0.3.16 below), and there are t arms, the i-th arm of length pi 1,
and the Di are finite dimensional skew fields over k, with k lying in their centres.
Moreover, there are F -Di-bimodule Ui and Di-G-bimodules Vi (k acting centrally)
on the arrows starting in the source and ending in the sink, respectively.
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