Gersten conjecture for K-theory on Henselian schemes and $\phi$-motivic localisation

TL;DR

This paper proves the Gersten Conjecture for K-theory on Henselian schemes by introducing a new ta-motivic localization, extending known results in motivic homotopy theory and Cousin complexes.

Contribution

It introduces a ta-motivic localization framework that generalizes support triviality results for motivic ta-homotopies, leading to a proof of the Gersten Conjecture for K-theory on Henselian schemes.

Findings

Proves Gersten Conjecture for K-theory on Henselian schemes.

Develops ta-motivic localization for cellular motivic spaces.

Extends results to Cousin complexes and motivic ta-homotopies.

Abstract

A key triviality result for support extension maps for motivic $\mathbb{A}^1$-homotopies of cellular motivic spaces $S$ over a DVR spectrum $B$ is proven. Combining with earlier known results on Gersten complex and the K-theory motivic spectrum we achieve a proof of the Gersten Conjecture for K-theory on essentially smooth local Henselian $B$-schemes. Additionally, we outline generalisations for Cousin complexes associated to motivic $\mathbb{A}^1$- and $\square$-homotopies of cellular $B$-spectra. The proof is based on two ingredients: (1) A new ``motivic localisation'' over $B$, called \emph{$\phi$-motivic}, % localisation giving rise to the $\phi$-motivic homotopy category such that the triviality of the support extension maps and the acyclicity of Cousin complexes hold for all objects $S$, not necessarily cellular. (2) An interpretation of some classes in the motivic $\mathbb{A}^1$-homotopies with support defined with respect to the Morel-Voevodsky motivic homotopy category of smooth $B$-schemes in terms of the construction of $\phi$-motivic homotopy category mentioned in Point (1).