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 $\ell \ne p$, we construct a stable étale realization functor $${\underline{\text{Eˊt}}}_{\ell }:\text{Spt}(k)\to \text{Pro}(\text{Spt}{)}^{H\mathbb{Z}/\ell }$$ from the stable $\infty$-category of motivic ${\mathbb{P}}^1$-spectra over $k$ to the stable $\infty$-category of $(H\mathbb{Z}/\ell {)}^{\ast }$-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 \[\text{SH}[1/p]\rightarrow\text{SH}(k)[1/p],\] 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 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.

arXiv comment21 pages. Changed title, improved the introduction, and replaced proposition 4.4 of previous version with lemmas 6.11 and 7.2. Comments Welcome!