A comparison of motivic and classical stable homotopy theories

Abstract

Let k be an algebraically closed field of characteristic zero. 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 consequence, c induces an isomorphism c * : πn(E) −→ Πn,0(c(E))(k) for all spectra E and all n ∈ Z.Fix an embedding σ : k → C and let ReB : SH(k) → SH be the associated Betti realization. We show that the slice tower for the motivic sphere spectrum over k, S k , has Betti realization which is strongly convergent. This gives a spectral sequence 'of motivic origin' converging to the homotopy groups of the sphere spectrum S ∈ SH; this spectral sequence at E 2 agrees with the E2 terms in the Adams-Novikov spectral sequence after a reindexing. Finally, we show that, for E a torsion object in SH(k) eff , the Betti realization induces an isomorphism Πn,0(E)(k) → πn(ReBE) for all n, generalizing the Suslin-Voevodsky theorem comparing mod N Suslin homology and mod N singular homology.As a special case, Theorem 1 implies the following corollary:Corollary 2. Let k be an algebraically closed field of characteristic zero. Let S k be the motivic sphere spectrum in SH(k) and S the classical sphere spectrum in SH. Then the constant presheaf functor induces an isomorphismIn fact, the corollary implies the theorem, by a density argument (see Lemma 6.5). Remark.(1) As pointed out by the referee, the functor c is induced by a (left) Quillen functor between model categories (see the proof of Lemma 6.5), so we do achieve a comparison of 'homotopy theories', as stated in the title, rather than just the underlying homotopy categories.(2) The functor c is not full in general. In fact, for a perfect field k, Morel [22, Lemma 3.10, Corollary 6.43] has constructed an isomorphism of Π 0,0 S k (k) with the Grothendieck-Witt group GW(k) of symmetric bilinear forms over k. As long as not every element of k is a square, the augmentation ideal in GW(k) is non-zero, hence c : π 0 (S) → Π 0,0 S k (k) is not surjective. Of course, if k is algebraically closed, then GW(k) = Z by rank, and thus c : π 0 (S) → Π 0,0 S k (k) is an isomorphism. This observation can be viewed as the starting point for our main result.We also have a homotopy analog of the theorem of Suslin-Voevodsky [33, Theorem 8.3] comparing Suslin homology and singular homology with mod N coefficients:Theorem 3. Let k be an algebraically closed field of characteristic zero with an embedding σ : k → C. Then, for all X ∈ Sm/k, all N > 1 and n ∈ Z, the Betti realization associated to σ induces an isomorphism

The first stable homotopy groups of motivic spheres

“…An explicit answer to (1) is out of reach because the slices of 1 involve the E 2 -page of the topological Adams-Novikov spectral sequence as conjectured by Voevodsky [73] and verified by Levine [33] for fields. Our solution applies to Dedekind domains of mixed characteristic.…”

Witt sheaves and the η-inverted sphere spectrum

“…By [9,Lemma 4.5], f s n X is in SH(k) n for all X ∈ SH(k) and thus HoSpt T (k)(K n ) ⊂ SH(k) n . Thus, by the universal property of τ n , we have for X ∈ SH(k) a commutative triangle By [9,Lemma 4.3], the map f s n X → X induces an isomorphism on π A 1 a,b for all a b + n, and thus f s n X → τ n X is an isomorphism. Thus, SH(k) n ⊂ HoSpt T (k)(K n ).…”

The stable Galois correspondence for real closed fields

“…The map in 4.1(ii) is identical to the map [S n , S C ] C → [S n , S] induced by complex Betti realization. This is an isomorphism by Levine's theorem [8].…”