16 0. BACKGROUND

0.3.8 (Exceptional object). An object E in H is called exceptional if it is

indecomposable and ExtH(E,

1

E) = 0. It follows then by an argument by Happel

and Ringel [41] that EndH(E) is a skew field.

0.3.9 (Serre duality). For an exceptional curve there is an autoequivalence τ

on H such that Serre duality

ExtH(X,

1

Y ) D HomH(Y, τX)

holds functorially in X, Y ∈ H, where D is the duality Homk(−, k).

Since the category H is hereditary, the (bounded) derived category

Db(X)

def

=

Db(H)

=

Db(Σ)

is just the repetitive category of coh(X). Moreover, H has almost

split sequences and the Serre functor τ : H −→ H serves as Auslander-Reiten

translation. Denote by τ

−

the inverse Auslander-Reiten translation.

0.3.10 (Grothendieck group). Denote by K0(X) the Grothendieck group of H.

Since H and mod(Σ) have the same bounded derived category, we have K0(X) =

K0(Σ), and this is a free abelian group of finite rank. We denote by [X] the class

in K0(X) of an object X ∈ H.

K0(X) is equipped with the (normalized) Euler form −,−. This bilinear form

is defined on classes of objects X, Y in H by

[X], [Y ] =

1

m

(

dimk HomH(X, Y ) − dimk ExtH(X,

1

Y )

)

,

where m is a positive integer such that the image of the resulting bilinear form

generates Z.

The Auslander-Reiten translation τ induces the Coxeter transformation, which

we also denote by τ (by a slight abuse of notation), and which is an automorphism

on K0(X) = K0(Σ) preserving the Euler form. The radical of K0(X) is defined by

Rad(K0(X)) = {x ∈ K0(X) | τ x = x}.

0.3.11 (Weights). For each x ∈ X let p(x) be the rank of the tube Tx. That is,

p(x) is the number of isomorphism classes of simple objects in Ux. The tube Tx, or

the point x, is called homogeneous ([91]) , if p(x) = 1, exceptional otherwise. X is

called homogeneous if all p(x) = 1. Clearly, a point x is exceptional if and only if

a simple object Sx in Ux is exceptional.

Each exceptional curve admits only a finite number of exceptional points. De-

note by x1, . . . , xt ∈ X the exceptional points. We call the numbers pi = p(xi) 1

weights, accordingly (p1, . . . , pt) the weight sequence.

0.3.12 (Rank). We define the rank of sheaves: Let x0 ∈ X, and let S0 be

a simple sheaf in the tube Ux0 of rank p0. Let w :=

∑p0−1[τ

j=0

j

S0], which is an

element of Rad K0(X). By [70] we can assume that x0 is a so-called rational point

(see 0.4.4), that is, Zw is a direct summand of Rad K0(X). After normalizing the

linear form −, w on K0(X) by the factor c := [Z : K0(X), w ], we get a surjective

linear form, compatible with the Coxeter transformation: For each x ∈ K0(X)

define rk x :=

1

c

x, w , and moreover rk(X) = rk([X]) for each X ∈ H. Let X ∈ H

be indecomposable. Then rk(X) = 0 if and only if X ∈ H0; if X ∈ H+, then

rk(X) 0.

0.3.13 (Function field). The quotient category of H modulo the Serre subcat-

egory H0, formed by the objects of finite length, is equivalent to the category of

finite dimensional vector spaces over some skew field which is (up to isomorphism)