**Memoirs of the American Mathematical Society**

2009;
128 pp;
Softcover

MSC: Primary 14; 16;

Print ISBN: 978-0-8218-4400-7

Product Code: MEMO/201/942

List Price: $71.00

AMS Member Price: $42.60

MAA member Price: $63.90

**Electronic ISBN: 978-1-4704-0556-4
Product Code: MEMO/201/942.E**

List Price: $71.00

AMS Member Price: $42.60

MAA member Price: $63.90

# Noncommutative Curves of Genus Zero: Related to Finite Dimensional Algebras

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*Dirk Kussin*

In these notes the author investigates noncommutative smooth projective curves of genus zero, also called exceptional curves. As a main result he shows that each such curve \(\mathbb{X}\) admits, up to some weighting, a projective coordinate algebra which is a not necessarily commutative graded factorial domain \(R\) in the sense of Chatters and Jordan. Moreover, there is a natural bijection between the points of \(\mathbb{X}\) and the homogeneous prime ideals of height one in \(R\), and these prime ideals are principal in a strong sense.

#### Table of Contents

# Table of Contents

## Noncommutative Curves of Genus Zero: Related to Finite Dimensional Algebras

- Contents vii8 free
- Introduction 112 free
- Chapter 0. Background 1122 free
- Part 1. The homogeneous case 2738
- Chapter 1. Graded factoriality 2940
- Chapter 2. Global and local structure of the sheaf category 4960
- Chapter 3. Tubular shifts and prime elements 6172
- Chapter 4. Commutativity and multiplicity freeness 6980
- Chapter 5. Automorphism groups 7586
- 5.1. The automorphism group of a homogeneous curve 7687
- 5.2. The structure of Aut(H) 7788
- 5.3. The twisted polynomial case 7889
- 5.4. On the Auslander-Reiten translation as functor 8091
- 5.5. The quaternion case 8293
- 5.6. The homogeneous curves over the real numbers 8293
- 5.7. Homogeneous curves with finite automorphism group 8596

- Part 2. The weighted case 8798
- Appendix A. Automorphism groups over the real numbers 113124
- Appendix B. The tubular symbols 119130
- Bibliography 121132
- Index 127138