Contents
Chapter 1. Introduction 1
1.1. Motivation 1
1.2. Construction 3
1.3. General properties 5
1.4. Non-commutative Del-Pezzo surfaces 6
1.5. Exceptional simple objects 6
1.6. Non-commutative cubic surfaces 7
1.7. Acknowledgement 7
Chapter 2. Preliminaries on category theory 8
Chapter 3. Non-commutative geometry 10
3.1. Bimodules 10
3.2. Graded modules, bimodules and algebras 18
3.3. Quotients of the identity functor 19
3.4. Ideals in the identity functor 22
3.5. Quasi-schemes 26
3.6. Divisors 28
3.7. Proj 29
3.8. Condition "x" a n d cohomological dimension 32
3.9. Higher inverse images 39
3.10. Algebras which are strongly graded modulo a Serre subcategory 40
3.11. The positive part of certain graded algebras. 41
3.12. Veronese subalgebras 42
Chapter 4. Pseudo-compact rings 44
Chapter 5. Cohen-Macaulay curves embedded in quasi-schemes 53
5.1. Preliminaries 53
5.2. Some computations 57
5.3. Completion of objects in mod(X) 60
5.4. Completion of bimodules 61
5.5. The category C/,p 64
5.6. Completion of algebras 68
5.7. Multiplicities in the case that r has infinite order 69
Chapter 6. Blowing up a point on a commutative divisor 72
6.1. Some ideals 72
6.2. Some Rees algebras 76
6.3. Definition of blowing up 77
6.4. The normal bundle 79
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