(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  and to
prove an analogue of Serre’s theorem .
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]
(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 . 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 =
over k (understood
in the scheme sense) we have
which consists of the polynomials in X and Y of even degree. This algebra is a
projective coordinate algebras for
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
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
is of interest. By Serre duality this is equivalent to studying the bimodule
and this leads directly to the universal extension
0 −→ L
−→ L(x) −→
with the multiplicity (originally defined by Ringel in )
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)
for some positive integer d, depending on x.