On the Conservativity of the Functor Assigning to a Motivic Spectrum its Motive

Given a 0-connective motivic spectrum $E \in SH(k)$ over a perfect field k, we determine $h_0$ of the associated motive $M E \in DM(k)$ in terms of $\pi_0 (E)$. Using this we show that if k has finite 2-\'etale cohomological dimension, then the functor M 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-\'etale cohomological dimension by considering what we call "real motives". As a by-product we reprove a variant of a rigidity Theorem of R\"ondings-{\O}stv{\ae}r.