文档介绍：LECTURES ON THE ^-FUNCTOR IN
ALGEBRAIC GEOMETRY
Yu. I. Manin
Contents
Introduction 1
Guide to the literature 3
§1. The Grothendieck groups K. (I) and K' (X) 4
§2. K(X)nnd cycles 9
§3. Self-intersection and exterior powers 12
§4. Projectivized bundles 17
§5. Computation of K(¥ (%) and the splitting principle ... 21
§6. Computation of K(P (£)) (conclusion) 25
§7. K(X) as a covariant functor 29
§8. Ύ-filtration of the ring K'(X) 34
§9. Filtration and dimension 39
§10. The connection between K(X) and Pic X 43
§11. Chern classes and the Adams operation 47
§12. The structure of monoidal transformations 51
§13. The behaviour of K(X) under a raonoidal transformation . 55
§14. The behaviour of K(X) under a monoidal transformation
(continuation) 61
§15. The behaviour of K(X) under a monoidal transformation
(conclusion) 64
§16. The Adams operations and the horaomorphism of the direct
image 69
§17. The sheaf of differentials 76
§18. The Riemann-Roch theorem for embeddings 82
§19. The Riemann-Roch theorem for projections 84
References 88
Introduction
FVom 1966 to 1968 I gave a two-year course of lectures in the Faculty
of Mathematics and Mechanics at the Moscow state University.
The course was planned as an introduction to algebraic geometry; notes
of the first part [2] were published last year. I wanted not only to give
some of the basic principles of the theory of schemes, but also to show
how they work in more significant situations. I presented Grothendieck's
theory of the rings K, which leads ultimately to the proof of the
Riemann-Roch theorem, as a essful example of such a significant
mathematical application. On the one hand it shows the technique of
2 Yu. I. Manin
computation with coherent sheaves without requiring too detailed a study
of local properties of morphisms or the problem of the representability of
functors (I had no time for this). On the other hand it is very close to
the classical pr