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.
Categories: math.AG math.AT math.KT