viii CONTENT S
6.5. Birationality 80
6.6. The exceptional curve 82
6.7. The structure of the exceptional curve 84
6.8. The strict transform 88
6.9. A result on KQ of some categories 91
Chapter 7. Derived categories 93
7.1. Generalities 93
7.2. Admissible compositions of morphisms between quasi-schemes 93
Chapter 8. The derived category of a non-commutative blowup 96
8.1. The formalism of semi-orthogonal decompositions 96
8.2. Generalities 97
8.3. Computation of some derived functors 98
8.4. The main theorem 101
Chapter 9. Some results on graded algebras and their sections 105
9.1. Generalities 105
9.2. The case of a blowing up 111
Chapter 10. Quantum plane geometry 113
10.1. Multiplicities of some objects 113
10.2. Classification of lines and conies 118
Chapter 11. Blowing up n points in an elliptic quantum plane 121
11.1. Derived categories 121
11.2. Exceptional simple objects 126
Chapter 12. Non-commutative cubic surfaces 129
Appendix A. Two-categories 132
Appendix B. Summary of notations 135
Appendix C. Index of terminology 138
Bibliography 139
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