INTRODUCTION 5
ideal Py associated to the point y) induces a graded algebra automorphism γy on
R, given by the formula rπy = πyγy(r). This in turn induces an automorphism
γy

Aut(X) whose action on the set of all points of X is invisible, but it is a non-
trivial element of Aut(X) if (under an additional assumption, see 3.2.4) for all units
u the element πyu is not central. This means that the functor γy

fixes all objects
but acts non-trivially on morphisms. Such a functor we call a ghost automorphism.
The simplest example in which this effect arises is given by the curve X with
underlying bimodule M = C(C C)C over k = R, where C acts from the right on
the second component via conjugation. A projective coordinate algebra is given by
the graded twisted polynomial ring R = C[X; Y, · ], graded by total degree, where
X is a central variable and for the variable Y we have Y a = aY for all a C. We
write R = C[X, Y ]. Then Y is a prime element which is not central (up to units).
It follows that complex conjugation induces a ghost automorphism of X. Moreover,
denote by σx and σy the (efficient) tubular shifts corresponding to the points x
and y associated with the prime ideals generated by X and Y , respectively. Then
C[X, Y ] = Π(L, σx) holds.
The following theorem expresses the interrelation between various automor-
phisms in more detail.
Theorem. Let R = Π(L, σ), where σ is efficient. Let πy be a prime element of
degree d in R, associated to the point y and γy the induced graded algebra automor-
phism. Let σy be the tubular shift associated to y. Then there is an isomorphism
of functors σy
σd
γy
∗.
The theorem contains important information about the structure of the Picard
group Pic(X), defined as the subgroup of Aut(H) generated by all tubular shifts
σx (x X). In particular, in contrast to the algebraically closed case, the Picard
group may not be isomorphic to Z.
In Chapter 5 we develop a technique which allows explicit calculation of the
automorphism group Aut(X) in many cases. We illustrate this for the preceding
example, where R = C[X, Y ]. The ghost group is the subgroup of Aut(X) consisting
of all ghost automorphisms.
Proposition. Let X be the homogeneous curve with projective coordinate al-
gebra R = C[X, Y ]. Then R = Π(L, σx), and Aut(X) is generated by
the automorphism γy

of order two, induced by complex conjugation, gen-
erating the ghost group;
transformations of the form Y aY for a R+;
the automorphism induced by exchanging X and Y .
Moreover, the Picard group Pic(X) is isomorphic to Z × Z2, and for the Auslander-
Reiten translation the following formula holds true:
τ = σx
−1
σy
−1
= σx
−2
γy
∗.
See Sections 5.3 and 5.4 for more general statements. In general the functo-
rial properties of the Auslander-Reiten translation have not been extensively stud-
ied. The preceding result shows that interesting effects appear. On objects the
Auslander-Reiten translation τ acts like σx
−2,
which agrees with the degree shift by
−2. But on morphisms the ghost automorphism induced by complex conjugation
enters the game.
Previous Page Next Page