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
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