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

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Let k be an algebraically closed field of exponential characteristic p. Given any prime ℓ≠ p, we construct a stable étale realization functor Étℓ:Spt(k)→ Pro(Spt)HZ/ℓ from the stable ∞-category of motivic P¹-spectra over k to the stable ∞-category of (HZ/ℓ)*-local pro-spectra (see section 3 for definition). This is induced by the étale topological realization functor á la Friedlander. The constant presheaf functor naturally induces the functor SH[1/p](k)[1/p], where k and p are as above and SH and SH(k) are the classical and motivic stable homotopy categories, respectively. We use the stable étale realization functor to show that this functor is fully faithful. Furthermore, we conclude with a homotopy theoretic generalization of the étale version of the Suslin-Voevodsky theorem.