Noncommutative curves of genus zero: related to finite dimensional algebras

146 Pages·2009·1.931 MB·English

by  Dirk Kussin

Noncommutative Curves of
Genus Zero
Related to Finite Dimensional Algebras
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M
EMOIRS
of the
American Mathematical Society
Number 942
Noncommutative Curves of
Genus Zero
Related to Finite Dimensional Algebras
Dirk Kussin
(cid:1)(cid:2)(cid:3)(cid:4)(cid:2)(cid:5)(cid:6)(cid:2)(cid:7)(cid:8)(cid:9)(cid:10)(cid:10)(cid:11)(cid:8)(cid:8)(cid:12)(cid:8)(cid:8)(cid:13)(cid:14)(cid:15)(cid:16)(cid:5)(cid:2)(cid:8)(cid:9)(cid:10)(cid:17)(cid:8)(cid:8)(cid:12)(cid:8)(cid:8)(cid:18)(cid:16)(cid:5)(cid:6)(cid:2)(cid:7)(cid:8)(cid:11)(cid:19)(cid:9)(cid:8)(cid:20)(cid:21)(cid:22)(cid:7)(cid:23)(cid:4)(cid:8)(cid:14)(cid:21)(cid:8)(cid:24)(cid:8)(cid:25)(cid:16)(cid:5)(cid:6)(cid:2)(cid:7)(cid:23)(cid:26)(cid:8)(cid:8)(cid:12)(cid:8)(cid:8)(cid:27)(cid:1)(cid:1)(cid:18)(cid:8)(cid:10)(cid:10)(cid:28)(cid:24)(cid:29)(cid:11)(cid:9)(cid:28)(cid:28)
American Mathematical Society
Providence, Rhode Island
2000MathematicsSubjectClassification.
Primary14H45,16G10;Secondary14H60,14A22.
Library of Congress Cataloging-in-Publication Data
Kussin,Dirk,1967–
Noncommutativecurvesofgenuszero: relatedtofinitedimensionalalgebras/DirkKussin.
p.cm. —(MemoirsoftheAmericanMathematicalSociety,ISSN0065-9266;no. 942)
“Volume201,number942(firstof5numbers).”
Includesbibliographicalreferencesandindex.
ISBN978-0-8218-4400-7(alk.paper)
1. Curves, Algebraic. 2. Representations of rings (Algebra) 3. Noncommutative algebras.
I.Title.
QA565.K87 2009
516.3(cid:1)52—dc22 2009019382
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Dedicated to the memory of my beloved partner
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Contents
Introduction 1
Chapter 0. Background 11
0.1. Notation 11
0.2. One-parameter families, generic modules and tameness 11
0.3. Canonical algebras and exceptional curves 14
0.4. Tubular shifts 19
0.5. Tame bimodules and homogeneous exceptional curves 22
0.6. Rational points 24
Part 1. The homogeneous case 27
Chapter 1. Graded factoriality 29
1.1. Efficient automorphisms 30
1.2. Prime ideals and universal extensions 33
1.3. Prime ideals as annihilators 35
1.4. Noetherianness 38
1.5. Prime ideals of height one are principal 39
1.6. Unique factorization 41
1.7. Examples of graded factorial domains 43
The non-simple bimodule case 44
The quaternion case 46
The square roots case 46
Chapter 2. Global and local structure of the sheaf category 49
2.1. Serre’s theorem 49
2.2. Localization at prime ideals 51
2.3. Noncommutativity and the multiplicities 56
2.4. Localizing with respect to the powers of a prime element 58
2.5. Zariski topology and sheafification 59
Chapter 3. Tubular shifts and prime elements 61
3.1. Central prime elements 61
3.2. Non-central prime elements and ghosts 62
Chapter 4. Commutativity and multiplicity freeness 69
4.1. Finiteness over the centre 69
4.2. Commutativity of the coordinate algebra 70
4.3. Commutativity of the function field 71
Chapter 5. Automorphism groups 75
vii
viii CONTENTS
5.1. The automorphism group of a homogeneous curve 76
5.2. The structure of Aut(H) 77
5.3. The twisted polynomial case 78
5.4. On the Auslander-Reiten translation as functor 80
5.5. The quaternion case 82
5.6. The homogeneous curves over the real numbers 82
5.7. Homogeneous curves with finite automorphism group 85
Part 2. The weighted case 87
Chapter 6. Insertion of weights 89
6.1. p-cycles 89
6.2. Insertion of weights into central primes 91
6.3. Automorphism groups for weighted curves 95
Chapter 7. Exceptional objects 97
7.1. Transitivity of the braid group action 97
7.2. Exceptional objects and graded factoriality 99
Chapter 8. Tubular exceptional curves 101
8.1. Slope categories and the rational helix 102
8.2. The index of a tubular exceptional curve 105
8.3. A tubular curve of index three 107
8.4. A related tubular curve of index two 109
8.5. Line bundles which are not exceptional 110
Appendix A. Automorphism groups over the real numbers 113
A.1. Tables for the domestic and tubular cases 113
Appendix B. The tubular symbols 119
Bibliography 121
Index 127
Abstract
In these notes we investigate noncommutative smooth projective curves of
genuszero,alsocalledexceptionalcurves. Asamainresultweshowthateachsuch
curve 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
andJordan. Moreover, thereisa naturalbijectionbetweenthepoints ofXand the
homogeneous prime ideals of height one in R, and these prime ideals are principal
in a strong sense.
Curves of genus zero have strong applications in the representation theory of
finite dimensional algebras being natural index sets for one-parameter families of
indecomposable modules. They play a key role for an understanding of the notion
oftamenessandconjecturallyforanextensionofDrozd’sTameandWildTheorem
to arbitrary base fields. The function field of X agrees with the endomorphism
ring of the unique generic module over the associated tame hereditary algebra.
This skew field is of finite dimension over its centre which is an algebraic function
field in one variable. As another main result we show that the function field is
commutative if and only if the multiplicities determined by the homomorphism
spaces from line bundles to simples sheaves (originally defined by Ringel for tame
hereditary algebras) are equal to one for every point.
The study provides major insights into the nature of arithmetic complications
intherepresentationtheoryoffinitedimensionalalgebrasthatariseifthebasefield
is not algebraically closed.
ReceivedbytheeditorMarch19,2006andinrevisedformMarch18,2007.
2000MathematicsSubjectClassification. Primary14H45,16G10;Secondary14H60,14A22,
16S38.
Keywordsandphrases. noncommutativecurve,genuszero,exceptionalcurve,one-parameter
family,separatingtubularfamily,tamebimodule,canonicalalgebra,tubularalgebra,orbitalgebra,
gradedfactorialdomain,efficientautomorphism,ghostautomorphism.
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