Stable motivic<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"><mml:msub><mml:mrow><mml:mi>π</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:math>of low-dimensional fields

Ormsby, Kyle; Østvær, Paul Arne

doi:10.1016/j.aim.2014.07.024

Cited by 19 publications

(19 citation statements)

References 39 publications

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“…Morel has given a complete description of the 0-line π n,n (k) in terms of Milnor-Witt K-theory [34]. The 1 line π n+1,n (k) is determined by Hermitian and Milnor K-theory groups by the work of Röndigs, Spitzweck and Østvaer [41], which generalizes the partial results obtained by Ormsby and Østvaer in [39]. Ormsby has investigated the case of related invariants over p-adic fields [37] and the rationals [38], and Dugger and Isaksen have analyzed the case over the real numbers [13].…”

Section: Introductionmentioning

confidence: 55%

Two-complete stable motivic stems over finite fields

Wilson

Østvær

2017

Algebr. Geom. Topol.

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Let $\ell$ be a prime and $q = p^{\nu}$ where $p$ is a prime different from $\ell$. We show that the $\ell$-completion of the $n$th stable homotopy group of spheres is a summand of the $\ell$-completion of the $(n, 0)$ motivic stable homotopy group of spheres over the finite field with $q$ elements $F_q$. With this, and assisted by computer calculations, we are able to explicitly compute the two-complete stable motivic stems $\pi_{n, 0}(F_q)^{\wedge}_2$ for $0\leq n\leq 18$. Additionally, we compute $\pi_{19, 0}(F_q)^{\wedge}_2$ and $\pi_{20, 0}(F_q)^{\wedge}_2$ when $q \equiv 1 \bmod 4$ assuming Morel's connectivity theorem for $F_q$ holds.Comment: 4 figures, 2 tables. Published version now availabl

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Section: Introductionmentioning

confidence: 55%

Two-complete stable motivic stems over finite fields

Wilson

Østvær

2017

Algebr. Geom. Topol.

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“…The rightmost map in (1.2) is surjective for n ≥ −4 (compare with [4,Corollary 6]). The exact sequence (1.2) vastly generalizes computations in [51] for fields of cohomological dimension at most two, and in [13] and [19] for the real numbers. Our computations of motivic stable homotopy groups are carried out on F -points.…”

Section: Main Results and Outline Of The Papermentioning

confidence: 88%

The first stable homotopy groups of motivic spheres

2019

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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.

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“…The integral version of this conjecture says that, for a general base field F , there is a short exact sequence 0 → K M 2 (F )/24 → π 1 S F → F × /(F × ) 2 ⊕ Z/2 → 0. The second-named author and P. Østvaer have previously verified the integral version of Morel's conjecture for fields of cohomological dimension less than three [42].…”

Section: Introductionmentioning

confidence: 89%

“…(Recall that (2, η)-completion is the same as 2-completion when the 2primary cohomological dimension of k[i] is finite.) By [42,Lemma 5.12], the map π 1 (S k ) ∧ 2 → K M 1 (k)/2 ⊕ Z/2 is surjective, taking u η s to ([u], 1) (where u represents the quadratic form uX 2 in GW (k)). It follows that η s and −1 η s are linearly independent.…”

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confidence: 99%

Galois equivariance and stable motivic homotopy theory

Heller

Ormsby

2016

Trans. Amer. Math. Soc.

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Abstract. For a finite Galois extension of fields L/k with Galois group G, we study a functor from the G-equivariant stable homotopy category to the stable motivic homotopy category over k induced by the classical Galois correspondence. We show that after completing at a prime and η (the motivic Hopf map) this results in a full and faithful embedding whenever k is real closed and L = k[i]. It is a full and faithful embedding after η-completion if a motivic version of Serre's finiteness theorem is valid. We produce strong necessary conditions on the field extension L/k for this functor to be full and faithful. Along the way, we produce several results on the stable C 2 -equivariant Betti realization functor and prove convergence theorems for the p-primary C 2 -equivariant Adams spectral sequence.

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Stable motivicπ1of low-dimensional fields

Cited by 19 publications

References 39 publications

Two-complete stable motivic stems over finite fields

Two-complete stable motivic stems over finite fields

The first stable homotopy groups of motivic spheres

Galois equivariance and stable motivic homotopy theory

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