Twisted K-theory in motivic homotopy theory

TL;DR

This paper develops a motivic approach to twisted algebraic K-theory, constructing spectral sequences and proving descent properties, thereby extending previous results and connecting to birational geometry.

Contribution

It introduces a motivic spectral sequence for twisted K-theory and proves new representability and descent results using slice filtration and birational techniques.

Findings

Constructed the $ ext{A}$-twisted motivic spectral sequence.

Proved cdh descent and Milnor excision for twisted homotopy K-theory.

Expressed twisted K-theory as an extension of twisted Grassmannian.

Abstract

In this paper, we study twisted algebraic $K$-theory from a motivic viewpoint. For a smooth variety $X$ over a field of characteristic zero and an Azumaya algebra $\mathcal{A}$ over $X$, we construct the $\mathcal{A}$-twisted motivic spectral sequence, by computing the slices of the motivic twisted algebraic $K$-theory spectrum as a twisted form of motivic cohomology. This generalizes previous results due to Kahn-Levine where $\mathcal{A}$ is assumed to be pulled back from a base field. Our methods use interaction between the slice filtration and birational geometry. Along the way, we prove a representability result, expressing the motivic space of twisted $K$-theory as an extension of the twisted Grassmannian by the sheaf of "twisted integers". This leads to a proof of cdh descent and Milnor excision for twisted homotopy $K$-theory.