Let k be an algebraically closed field of characteristic zero. Let SH(k) be the motivic stable homotopy category of T-spectra over k, SH the classical stable homotopy category and let c:SH → SH(k) be the functor induced by sending a space to the constant presheaf of spaces on Sm/k. We show that c is fully faithful. In particular, c induces an isomorphism πn(E) → πn,0(c(E)) for all spectra E.
Fix an embedding of k into the complex numbers and let Re:SH(k) → SH be the associated Betti realization. Let Sk be the motivic sphere spectrum. We show that the Tate-Postnikov tower for Sk has Betti realization which is strongly convergent. This gives a spectral sequence of algebro-geometric origin converging to the homotopy groups of the classical sphere spectrum; this spectral sequence at E2 agrees with the E2 terms in the Adams-Novikov spectral sequence.
Marc Levine <firstname.lastname@example.org>