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.
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