On the Riemann-Roch formula without projective hypothesis

TL;DR

This paper establishes a Riemann-Roch formula connecting higher algebraic K-theory and motivic cohomology for proper morphisms between smooth schemes over a noetherian base without requiring projective assumptions, extending classical results.

Contribution

It proves a projective-hypothesis-free Riemann-Roch formula and an arithmetic version involving Arakelov's higher K-theory, broadening the scope of classical theorems.

Findings

Riemann-Roch formula without projective assumptions

Arithmetic Riemann-Roch theorem involving Arakelov's K-theory

Results derived from a motivic statement in the stable homotopy category

Abstract

Let $S$ be a finite dimensional noetherian scheme. For any proper morphism between smooth $S$-schemes, we prove a Riemann-Roch formula relating higher algebraic $K$-theory and motivic cohomology, thus with no projective hypothesis neither on the schemes nor on the morphism. We also prove, without projective assumptions, an arithmetic Riemann-Roch theorem involving Arakelov's higher $K$-theory and motivic cohomology as well as an analogue result for the relative cohomology of a morphism. These results are obtained as corollaries of a motivic statement that is valid for morphisms between oriented absolute spectra in the stable homotopy category of $S$.