# Equivariant Grothendieck-Riemann-Roch and localization in operational K-theory

## TL;DR

This paper develops a generalized Riemann-Roch transformation in operational K-theory, extending classical theorems and applying to toric and spherical varieties, with implications for equivariant and derived schemes.

## Contribution

It introduces a new Grothendieck transformation from bivariant operational K-theory to Chow, generalizing classical Riemann-Roch and localization theorems.

## Key findings

- Constructed a Riemann-Roch formula generalizing classical results
- Identified a toric variety with non-surjective equivariant K-theory
- Described operational K-theory of spherical varieties via fixed points

## Abstract

We produce a Grothendieck transformation from bivariant operational $K$-theory to Chow, with a Riemann-Roch formula that generalizes classical Grothendieck-Verdier-Riemann-Roch. We also produce Grothendieck transformations and Riemann-Roch formulas that generalize the classical Adams-Riemann-Roch and equivariant localization theorems. As applications, we exhibit a projective toric variety $X$ whose equivariant $K$-theory of vector bundles does not surject onto its ordinary $K$-theory, and describe the operational $K$-theory of spherical varieties in terms of fixed-point data. In an appendix, Vezzosi studies operational $K$-theory of derived schemes and constructs a Grothendieck transformation from bivariant algebraic $K$-theory of relatively perfect complexes to bivariant operational $K$-theory.