Riemann-Roch theorem for oriented cohomology, by Ivan Panin

Dear Readers,

A. Smirnov doesn't acknowledge his authorship for preprint 0542, and it was a mistake for me to post it without his approval, for which I apologize to you and to him. Please make no references to preprint 0542, and please discard any copies of it. This new preprint is essentially the same as 0542, aside from the change in authorship.

The present article in particular contains proofs of Riemann-Roch type theorems stated in preprint Push-forwards in oriented cohomology theories of algebraic varieties, by I.Panin and A.Smirnov . Since the body of that preprint can be naturally subdivided into two parts (a construction of integrations and a use of integrations) one can present each part separately. Here we present the part devoted to the proof of Riemann-Roch type theorems stated in that preprint. More precisely, the notion of an oriented cohomology pretheory on algebraic varieties is introduced, and a Riemann-Roch theorem for ring morphisms between oriented pretheories is proved. An explicit formula for the Todd genus related to a ring morphism is given. The results are illustrated by classical and other examples.


Ivan Panin <panin@pdmi.ras.ru>