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.