K-theory, Chow groups and Riemann-Roch
نویسنده
چکیده
1 K and K0 of schemes Let X be a quasi-projective variety. We define K(X) as follows. Let Fvb(X) be the free abelian group on all isomorphism classes of vector bundles (of finite rank) over X and consider the equivalence relation Rvb(X) generated by the following. If 0 → E1 → E2 → E3 → 0 is an exact sequence of vector bundles, then [E2] = [E1] + [E3]. The quotient Fvb(X)/Rvb(X) is called K(X). Similarly we define K0(X) by taking the free abelian group Fcoh(X) generated by all isomorphism classes of coherent sheaves on X modulo the relation Rcoh(X) generated by [F2] = [F1] + [F3] whenever we have an exact sequence of coherent sheaves, 0 → F1 → F2 → F3 → 0. K(X) is a commutative ring where multiplication is given by tensor products and K0(X) is a module over K(X), the action given by tensor product. We also have a natural map K(X) → K0(X) given by [E] going to itself. We may also consider the free abelian group on all isomorphism classes of coherent sheaves of finite projective dimension and going modulo a relation exactly as above and let me temporarily call this group L(X). We have natural homomorphisms, K(X) → L(X) → K0(X). Easy to check that the first map is an isomorphism. We will see that all are isomorphisms if X is smooth. K is a ∗Lectures given at the Algebraic Geometry Seminar at Washington University.
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