On the Riemann--Roch theorem without denominators

Abstract

A proof of the Riemann-Roch theorem without denominators is given. It is also proved that Grothendieck's ring functor CHmult is not an oriented cohomology pretheory. The Riemann-Roch formula without denominators for a closed embedding i : Y� → X of codimension d expresses the Chern class cd(i∗OY )i n terms of the class (Y ) ∈ CH d (X). In the present paper, we give a proof of this formula in the spirit of Verdier (Ve) but without using local Chern classes. The proof consists of several reductions. First, we use excision of singularities and deformation to the normal cone to reduce the proof to the case of an embedding of a smooth variety Y in a projective bundle P(E )o verY .A t the next step, we reduce the proof to the simplest case of an embedding of a rational point in a projective space. Finally, we present two simple arguments that prove the formula in this case: one involves the Koszul complex, and the other involves the ring functor CHmult, which arises naturally under the study of the behavior of the Chern classes under direct images. In the second part of the paper, we present some properties of the functor CHmult and prove that this functor is not an oriented cohomology pretheory. The authors are thankful to I. A. Panin and K. I. Pimenov for generous guidance and support and to J. L. Colliot-Thelene (Orsay University, Paris) and B. Erez (University of Bordeaux I) for inspiring discussions and hospitality. In the present paper, we use the following notation. 1) X is a quasiprojective variety over a field k. 2) CH ∗ (X )= ⊕CH i (X) is the Chow ring of classes of rationally equivalent cycles on X. 3) K � 0 (X) is the Grothendieck K-group of the category of coherent sheaves on X. If F is a coherent sheaf on X, then: 4) ci(F )i s theith Chern class of the sheaf F (see (Fu)); 5) si(F )i s theith Segre class of the sheaf F (see (Fu)). If f : Y → X is a flat morphism of varieties, then: 6) f CH : CH ∗ (X) → CH ∗ (Y ) is the inverse image homomorphism for the Chow