The Riemann–Roch theorem for the Adams operations

Abstract

We prove the classical Riemann–Roch theorems for the Adams operations ${\psi }^j$ on $K$-theory: a statement with coefficients on $Z\left[j^{-1}\right]$, that holds for arbitrary projective morphisms, as well as another statement with integral coefficients, that is valid for closed immersions. In presence of rational coefficients, we also analyze the relation between the corresponding Riemann–Roch formula for one Adams operation and the analogous formula for the Chern character. To do so, we complete the elementary exposition of the work of Panin–Smirnov that was initiated by the first author in a previous paper. Their notion of oriented cohomology theory on algebraic varieties allows to use classical arguments to prove general and neat statements, which imply all the aforementioned results as particular cases.