Comparison of Stable Homotopy Categories and a Generalized Suslin-Voevodsky Theorem

Let $k$ be an algebraically closed field of exponential characteristic $p$. Given any prime $\ell\neq p$, we construct a stable \'etale realization functor $$\underline{\text{\'Et}}_{\ell}:\text{Spt}(k)\rightarrow \text{Pro}(\text{Spt})^{H\mathbb{Z}/\ell}$$ from the stable $\infty$-category of motivic $\mathbb{P}^1$-spectra over $k$ to the stable $\infty$-category of $(H\mathbb{Z}/\ell)^*$-local pro-spectra (see section 3 for definition). This is induced by the \'etale topological realization functor \'a la Friedlander. The constant presheaf functor naturally induces the functor $$\text{SH}[1/p]\rightarrow\text{SH}(k)[1/p],$$ where $k$ and $p$ are as above and $\text{SH}$ and $\text{SH}(k)$ are the classical and motivic stable homotopy categories, respectively. We use the stable \'etale realization functor to show that this functor is fully faithful. Furthermore, we conclude with a homotopy theoretic generalization of the \'etale version of the Suslin-Voevodsky theorem.