Equivariant Algebraic K-Theory and Derived completions III: Applications

TL;DR

This paper applies derived completion theorems to establish general Riemann-Roch and Lefschetz-Riemann-Roch theorems for equivariant K-theory, covering a broad class of algebraic varieties and group actions.

Contribution

It extends derived completion applications to Riemann-Roch problems for higher equivariant K-theory and constructs spectral sequences for homotopy groups.

Findings

Riemann-Roch theorems for equivariant G-theory and homotopy K-theory on various varieties

Lefschetz-Riemann-Roch theorems for fixed point schemes

Spectral sequences computing homotopy groups of derived completions

Abstract

In the present paper, we discuss applications of the derived completion theorems proven in our previous two papers. One of the main applications is to Riemann-Roch problems for forms of higher equivariant K-theory, which we are able to establish in great generality both for equivariant G-theory and equivariant homotopy K-theory with respect to actions of linear algebraic groups on normal quasi-projective schemes over a given field. We show such Riemann-Roch theorems apply to all toric and spherical varieties. We also obtain Lefschetz-Riemann-Roch theorems involving the fixed point schemes with respect to actions of diagonalizable group schemes. We also show the existence of certain spectral sequences that compute the homotopy groups of the derived completions of equivariant G-theory starting with equivariant Borel-Moore motivic cohomology.