Lectures on the K-Functor in Algebraic Geometry

CONTENTS Introduction Guide to the literature § 1. The Grothendieck groups K_{\displaystyle\boldsymbol\cdot}(X) and K^{\displaystyle\boldsymbol\cdot}(X) § 2. K(X) and cycles § 3. Self-intersection and exterior powers § 4. Projectivized bundles § 5. Computation of K(\mathbf{P}(\mathscr{E})) and the splitting principle § 6. Computation of K(\mathbf{P}(\mathscr{E})) (conclusion) § 7. K(X) as a covariant functor § 8. \gamma-filtration of the ring K^{\displaystyle\boldsymbol\cdot}(X) § 9. Filtration and dimension § 10. The connection between K(X) and \operatorname{Pic} X § 11. Chern classes and the Adams operation § 12. The structure of monoidal transformations § 13. The behaviour of K(X) under a monoidal transformation § 14. The behaviour of K(X) under a monoidal transformation (continuation) § 15. The behaviour of K(X) under a monoidal transformation (conclusion) § 16. The Adams operations and the homomorphism of the direct image § 17. The sheaf of differentials § 18. The Riemann-Roch theorem for embeddings § 19. The Riemann-Roch theorem for projections References