The motivic Hopf map solves the homotopy limit problem for $K$-theory

TL;DR

This paper proves that the homotopy limit problem for algebraic K-theory over certain fields can be solved using motivic tools, specifically the motivic slice filtration and the first motivic Hopf map.

Contribution

It provides an affirmative solution to the homotopy limit problem for K-theory over fields of finite virtual cohomological dimension using motivic methods.

Findings

Homotopy limit problem for K-theory is solved over specified fields.

Motivic slice filtration is effective in addressing K-theory problems.

First motivic Hopf map plays a key role in the solution.

Abstract

We solve affirmatively the homotopy limit problem for $K$-theory over fields of finite virtual cohomological dimension. Our solution employs the motivic slice filtration and the first motivic Hopf map.