Comparison of stable homotopy categories and a generalized Suslin-Voevodsky theorem

Abstract

Let k be an algebraically closed field of exponential characteristic p. Given any prime ℓ≠p, we construct a stable étale realization functorÉt_ℓ: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 the definition). This is induced by the étale topological realization functor á la Friedlander. The constant presheaf functor naturally induces the functorSH[1/p]→SH(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.