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.