On the conservativity of the functor assigning to a motivic spectrum its motive

Abstract

Given a $0$-connective motivic spectrum $E\in \mathbf{SH}\left(k\right)$ over a perfect field $k$, we determine ${\underline{h}}_0$ of the associated motive $ME\in \mathbf{DM}\left(k\right)$ in terms of ${\underline{\pi }}_0\left(E\right)$. Using this, we show that if $k$ has finite $2$-étale cohomological dimension, then the functor $M:\mathbf{SH}\left(k\right)\to \mathbf{DM}\left(k\right)$ is conservative when restricted to the subcategory of compact spectra and induces an injection on Picard groups. We extend the conservativity result to fields of finite virtual $2$-étale cohomological dimension by considering what we call real motives.