A motivic analogue of the K(1)-local sphere spectrum

TL;DR

This paper provides a motivic analogue of the classical K(1)-local sphere resolution, identifying the $KGL/2$-local sphere via Hermitian K-theory and introducing a new conservativity technique in motivic homotopy theory.

Contribution

It introduces a motivic analogue of the classical K(1)-local sphere resolution and develops a novel conservativity argument applicable to stable motivic homotopy theory.

Findings

Identification of the motivic $KGL/2$-local sphere as a fiber of $ ext{ extpsi}^3 - 1$

Extension of the localization description to arbitrary motivic spectra

Development of a new conservativity argument in motivic homotopy theory

Abstract

We identify the motivic $KGL/2$-local sphere as the fiber of $\psi^3-1$ on $(2,\eta)$-completed Hermitian $K$-theory, over any base scheme containing $1/2$. This is a motivic analogue of the classical resolution of the $K(1)$-local sphere, and extends to a description of the $KGL/2$-localization of an arbitrary motivic spectrum. Our proof relies on a novel conservativity argument that should be of broad utility in stable motivic homotopy theory.