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

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.