Contents

Chapter I: Introduction 1

1. Elementar y matrice s 4

2. Stabilit y 7

3. K X(A [t]) an d unipotents 12

4. Grothendiec k and Whitehead groups of categories with exact sequences 14

5. A Grothendieck style definition o f K

X

(A) 18

6. Homology , mapping cones, and Euler characteristics 2 0

7. Resolutio n theorems 2 3

8. Projectiv e resolutions and regular rings 2 7

9. Hilbert s syzygy theorem 3 2

Appendix 1 3 6

Chapter II: Localisation, Divissage and Applications 4 1

10. Th e localisation sequence 4 2

11. Categorie s of nilpotent endomorphisms 4 6

12. K

x(A[t,rl\)

4 9

13. Devissag e 5 3

14. Som e calculations with the localisation sequence and K

Y

-Laurent theorem .... 5 6

15. Computation s with the Mayer-Vietoris sequence 6 3

References 6 7

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