The first stable homotopy groups of motivic spheres

Abstract

We compute the 1-line of stable homotopy groups of motivic spheres over fields of characteristic not two in terms of hermitian and Milnor K-groups. This is achieved by solving questions about convergence and differentials in the slice spectral sequence.Definition 2.3: The Λ-local stable model structure is the left Bousfield localization of the stable model structure on Spt Σ P 1 MS with respect to the set of naturally induced mapsall integers s, t, and X ∈ Sm S . Denote the corresponding homotopy category by SH Λ . Remark 2.4: A map α : E → F is a weak equivalence in the Λ-local stable model structure if and only if α ∧ 1 Λ : E Λ → F Λ is a stable motivic weak equivalence. When Λ = Q, this defines the rational stable motivic homotopy category. By [20, Theorem 3.3.19(1)] there exists a left Quillen functor from the stable to the Λ-local stable model structure on Spt Σ P 1 MS. We shall refer to its derived functor as the Λ-localization functor. Proof. This follows from Corollaries 2.16, 2.17, Remark 2.18 applied to every connected component of the base scheme, and the computation of s q (KQ) over fields of characteristic unequal to two in [63, Theorem 4.18].Recall that a motivic spectrum E ∈ SH Λ is called slice-wise cellular if s q (E) is contained in the full localizing triangulated subcategory of SH Λ generated by the qth suspension Σ 2q,q MΛ [73, Definition 4.1]. Let D MΛ denote the homotopy category of MΛ − mod. Replacing SH Λ by D MΛ gives an equivalent definition of slice-wise cellular spectra.Corollary 2.16: Every cellular spectrum in SH Λ is slice-wise cellular.

An explicit KO‐degree map and applications

“…Ormsby and Østvaer [35] have established Morel's conjecture after taking sections over fields k having cohomological dimension 2. Recently, Röndigs, Spitzweck and Østvaer have established Morel's conjecture for fields having characteristic 0 [37]; their results are compatible with ours. The validity of our conjecture on the structure of π A 1 n (A n \ 0) would immediately imply Morel's conjecture at the level of sheaves.…”

Witt sheaves and the η-inverted sphere spectrum

“…Explicit computations of π 1 (S k ) * (k) for fields of cohomological dimension at most 2 have been made by Ormsby-Østvaer in [16]; these refine our general statement in that the torsion has bounded exponent, although the groups are still not in general finite. These computations have been extended in a recent work of Röndigs-Spitzweck-Østvaer [19], in which they calculate π 1 (S k ) * (k) for an arbitrary characteristic zero field k and for * −4; for * < −4, they have a partial computation that gives an upper bound for π 1 (S k ) * (k). In particular, their results show that π 1 (S k ) * (k) has exponent dividing 48 for * = 0, 1, has exponent dividing 24 for * = 2, and is zero for * > 2.…”

Remarks on motivic Moore spectra

“…The unit map 1 → kq induces a map C nh → kq nh , which induces the identity map on the slice summands MZ/2n, Σ q,q MZ/2{α q 1 }, Σ q+1,q MZ/2{α q 1 }, Σ q+2,q MZ/2{α q−3 1 α 3 }, and Σ q+3,q MZ/2{α q−3 1 α 3 }, and the map Proof. The form of the slices follows from [10,Section 8] or [21,Theorem 2.12], since h induces multiplication by 2 on slices, and the slice functors are triangulated. The map C nh → kq nh is determined by the unit map 1 → kq, whose behaviour on slices can be read off from [21, Lemmas 2.28, 2.29].…”

“…Moreover, the ε-graded commutativity of π * +(⋆) 1 implies that ν 2 = −ν 2 , whence a map K MW 4+⋆ {ν 2 }/(ην 2 , 2ν 2 ) → π 2−(⋆) 1 of K MW -modules exists. The slice spectral sequence for π 2+(⋆) 1 shows its injectivity, as well as the isomorphism statement, using results from [21] and [17,Theorem 8.3]; details are to be given in [20].…”