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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