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Noncommutative Curves of Genus Zero: Related to Finite Dimensional Algebras
Dirk Kussin, Universität Paderborn, Germany

Memoirs of the American Mathematical Society
2009; 128 pp; softcover
Volume: 201
ISBN-10: 0-8218-4400-8
ISBN-13: 978-0-8218-4400-7
List Price: US$67
Individual Members: US$40.20
Institutional Members: US$53.60
Order Code: MEMO/201/942
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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

  • Introduction
  • Background
Part 1. The homogeneous case
  • Graded factoriality
  • Global and local structure of the sheaf category
  • Tubular shifts and prime elements
  • Commutativity and multiplicity freeness
  • Automorphism groups
Part 2. The weighted case
  • Insertion of weights
  • Exceptional objects
  • Tubular exceptional curves
  • Appendix A. Automorphism groups over the real numbers
  • Appendix B. The tubular symbols
  • Bibliography
  • Index
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