Construction of the motivic cellular spectrum $\mathbf{KO}^{geo}$ over $Spec(\mathbb{Z})$

Abstract

We construct a periodic motivic spectrum over $Spec(\mathbb{Z})$ which when pulled back to any scheme $S$ with $\frac{1}{2}\in\Gamma(S,\mathcal{O}_S)$ is the $HP^1-$spectrum constructed by Panin and Walter. This spectrum $\mathbf{KO}^{geo}$ is constructed using closed subschemes of the Grassmannians $Gr(r,n)$. Using this we show that $\mathbf{KO}^{geo}$ is cellular.