INTRODUCTION 3

(although not under this name) and is frequently used in representation theory,

see [7, 65, 49]. Note that Π(L, σ) typically is noncommutative. M. Artin and J. J.

Zhang used orbit algebras to define noncommutative projective schemes [2] and to

prove an analogue of Serre’s theorem [102].

Let for example σ be the inverse Auslander-Reiten translation τ −. Then it is

easy to see that the pair (L, τ −) is a so-called ample pair ([2, 105]), and thus by

the theorem of Artin-Zhang [2, Thm. 4.5]

H

modZ(Π(L,

τ

−))

mod0

Z

(Π(L, τ −))

,

the quotient category modulo the Serre subcategory of Z-graded modules of finite

length. Hence Π(L, τ −) is a projective coordinate algebra for X, and it coincides

with the (small) preprojective algebra defined in [7]. However the graded algebras

constructed in this way are often not practical for studying the geometry of X ex-

plicitly. For example, in the case of the projective line X =

P1(k)

over k (understood

in the scheme sense) we have

Π(L, τ

−)

=

k[X2,

XY, Y

2],

which consists of the polynomials in X and Y of even degree. This algebra is a

projective coordinate algebras for

P1(k),

as is the full polynomial algebra k[X, Y ],

graded by total degree. This example illustrates the well-known fact that projec-

tive coordinate algebras are not uniquely determined, and also that some projective

coordinate algebras are more useful than others. Of the two, only k[X, Y ] is graded

factorial.

Main results. We show that there exists a graded factorial coordinate algebra

in general, given as orbit algebra Π(L, σ) for a suitable autoequivalence σ on H. Of

course, one has to replace the usual factoriality by a noncommutative version.

The geometry of X is given by the hereditary category H. For this an un-

derstanding of the interplay between vector bundles and objects of finite length is

important. In particular, with the structure sheaf L, for each point x ∈ X and the

corresponding simple object Sx ∈ Ux the bimodule

End(Sx)

Hom(L, Sx)End(L)

is of interest. By Serre duality this is equivalent to studying the bimodule

End(L)

Ext1(Sx,

L)End(Sx),

and this leads directly to the universal extension

0 −→ L

πx

−→ L(x) −→

Sx(x) e

−→ 0

with the multiplicity (originally defined by Ringel in [90])

e(x) =

[Ext1(Sx,

L) : End(Sx)].

The above universal extension (for L) is a special case of a more general construction

which leads to the tubular shift automorphism σx of H, sending an object A to A(x).

We realize the kernels πx (for each x ∈ X) as homogeneous elements in a

suitable orbit algebra. This is accomplished by an automorphism σ on H which we

call eﬃcient (in 1.1.3). We show that such an automorphism always exists and has

the property that for any x the middle term L(x) in the universal extension is of

the form L(x)

σd(L)

for some positive integer d, depending on x.