The Riemann-Roch theorem for the Adams operations

TL;DR

This paper extends the Riemann-Roch theorem to Adams operations in algebraic K-theory, providing new formulas valid for various morphisms and analyzing their relation to Chern characters.

Contribution

It proves the Riemann-Roch theorems for Adams operations with coefficients and for different types of morphisms, and clarifies their relation to Chern characters using oriented cohomology theories.

Findings

Proved Riemann-Roch formulas for Adams operations with coefficients.

Established relations between Adams operations and Chern characters.

Unified approach using oriented cohomology theories.

Abstract

We prove the classical Riemann-Roch theorems for the Adams operations $\,\psi^j\,$ on $K$-theory: a statement with coefficients on $\mathbb{Z}[j^{-1}]$, that holds for arbitrary projective morphisms, as well as another one 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 work. Their notion of oriented cohomology theory of algebraic varieties allows to use classical arguments to prove general and neat statements, which imply all the aforementioned results as particular cases.