Riemann-Roch for homotopy invariant K-theory and Gysin morphisms

TL;DR

This paper establishes a Riemann-Roch theorem for homotopy invariant K-theory and various cohomologies, extending classical results to broader contexts without smoothness constraints, and introduces Gysin morphisms for spectra-based cohomologies.

Contribution

It develops a new Riemann-Roch theorem for homotopy invariant K-theory and constructs Gysin morphisms for spectra-based cohomologies, broadening the scope of classical theorems.

Findings

Proved Riemann-Roch for homotopy invariant K-theory without smoothness assumptions.

Constructed Gysin morphisms for spectra-based cohomologies.

Established a motivic version of the Riemann-Roch theorem.

Abstract

We prove the Riemann-Roch theorem for homotopy invariant $K$-theory and projective local complete intersection morphisms between finite dimensional noetherian schemes, without smoothness assumptions. We also prove a new Riemann-Roch theorem for the relative cohomology of a morphism. In order to do so, we construct and characterize Gysin morphisms for regular immersions between cohomologies represented by spectra (examples include homotopy invariant $K$-theory, motivic cohomology, their arithmetic counterparts, real absolute Hodge and Deligne-Beilinson cohomology, rigid syntomic cohomology, mixed Weil cohomologies) and use this construction to prove a motivic version of the Riemann-Roch.