(ICTP, Trieste, 2002) Lecture 1

In order to formalize his work on the Riemann-Roch theorem (in the spirit of Hirzebruch), Grothendieck introduced a new contravariant functor [BS] defined on the category of non singular algebraic varieties X. He named this functor K(X), the “K-theory ” of X. It seems the terminology “K ” came out from the German word “Klassen”, since K(X) may be thought of as a group of “classes ” of vector bundles on X. Grothendieck could not use the terminology C(X) since his thesis (in functional analysis) made an heavy use of the ring C(X) of continuous functions on the space X. In order to define K(X), one considers first the free abelian group L(X) generated by the isomorphism classes [E] of vector bundles E on X. The group K(X) is then the quotient of L(X) by the subgroup generated by the following relations [E’] + [E”] = [E] when we have an exact sequence of vector bundles 0 zzc E ’ zzc E zzc E ” zzc 0 Exercise: compute K(X) when X is a Riemann surface in complex Algebraic Geometry. This group K(X) has nice algebraic properties, very much related to the usual ones in cohomology. As a matter of fact, the theory of characteristic classes, e.g. the Chern character denoted by Ch- which we shall review later on- might be thought of as defining a functor from K(X) to suitable cohomology theories. For instance, if f: X zzc Y is a morphism of projective varieties, we have a “Gysin map ” f