0.3. CANONICAL ALGEBRAS AND EXCEPTIONAL CURVES 15

(2) The tame hereditary algebras. In particular, the tame bimodule algebras

(see 0.3.16 and 0.5.1). Actually, by the so-called insertion of weights [68], and

by the perpendicular calculus [35], two processes which are inverse to each other,

many problems for concealed canonical algebras can be reduced to the special class

of tame bimodule algebras. (We will explain this in 0.3.16.)

0.3.3 (Separating tubular family). A sincere separating exact subcategory

mod0(Σ) defines a separating tubular family of stable tubes [92]: there is the

coproduct of categories

mod0(Σ) =

x∈X

Ux,

where Ux are connected, uniserial length categories, containing neither non-zero

projective nor non-zero injective modules. The full subcategory Tx = ind(Ux) of

indecomposable objects in Ux is called a stable tube. Moreover, each non-zero

morphism from an object in mod+(Σ) to an object in mod−(Σ) factorizes through

any prescribed tube Ux.

0.3.4 (Associated hereditary category). In the preceding coproduct, X is an

index set, which is equipped with geometric structure. In [70] there is defined an

associated hereditary abelian k-category H. Hereditary means that ExtH(−,

i

−) = 0

for all i ≥ 2. Roughly speaking, to construct H one takes the union of mod0(Σ)

and mod+(Σ) and forms inside the bounded derived category

Db(mod(Σ))

(see [37])

the closure of this union under all inverse shift automorphisms defined to tubes in

mod0(Σ). By the construction it is immediate that the categories H and mod(Σ)

are derived equivalent.

In the most important special case when Σ is a tame bimodule algebra we

describe the category H more explicitly in 0.5.1.

0.3.5 (Bundles/objects of finite length). Denote by H0 (H+, respectively) the

full subcategory of H of objects of finite length (of objects, not containing objects

= 0 of finite length, respectively). Then each indecomposable object in H is either

in H0 or in H+, and HomH(H0, H+) = 0. The objects of H (H+, respectively) are

also called sheaves (vector bundles or torsionfree, respectively). By construction of

H we have H0 = mod0(Σ) =

x∈X

Ux.

0.3.6 (Exceptional curves). X, together with the category H, is called an excep-

tional curve [68], and one sometimes writes H = coh(X). This class of categories H

is characterized independently of the construction above by the following properties:

• H is a connected small abelian k-category with finite dimensional mor-

phism and extension spaces.

• H is hereditary and noetherian and contains no non-zero projective object.

• H admits a tilting object (see the following number).

0.3.7 (Tilting object). T ∈ H is called a tilting object, if

• ExtH(T,

1

T ) = 0, and

• If X ∈ H, then HomH(T, X) = 0 = ExtH(T,

1

X) implies X = 0.

A tilting object lying in H+ is called a tilting bundle.

There exists even a tilting bundle T such that EndH(T ) is a canonical algebra

([70, Prop. 5.5]).