Remarks on motivic Moore spectra

Röndigs, Oliver

doi:10.1090/conm/745/15026

Cited by 4 publications

(5 citation statements)

References 12 publications

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“…Similar to the treatment of very effective hermitian K-theory kq, the homotopy groups of the successive slice filtrations f q 1 will be discussed as K MW -modules, starting with π 1 f 4 1 whose image in π 1 1 will be zero. The treatment for π 1 1 given here can also be found in [Rön20].…”

Section: Lemma 33mentioning

confidence: 99%

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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Section: Lemma 33mentioning

confidence: 99%

The first stable homotopy groups of motivic spheres

2019

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“…This justifies the relevance of the following statement. Proof This is a consequence of [23,Theorem 5.5] in the formulation given in [22,Theorem 2.5]; see also [24,Theorem 1.1]. First of all, the unit map π 1+( ) 1 → π 1+( ) kq is surjective, whence the same is true for the induced map on the quotients.…”

Section: Over a Fieldmentioning

confidence: 79%

“…Proof The Toda bracket η, h, η = {6ν, −6ν} from [22,Proposition 4.1] implies by Proposition B.1 that there exists an element h ∈ π 2+(2) P 2 which on the one hand maps to h ∈ η π 0+(0) 1, and on the other hand is such that h •η is the image of 6ν ∈ π 1+(2) 1. Inspecting the short exact sequence (3.3) in weight 3 gives an isomorphism π 1+(2) 1/ηπ 1+(1) 1 ∼ = π 2+(3) P 2 by Theorem 3.2.…”

Section: Lemma 34mentioning

confidence: 99%

“…It is as non-commutative as the endomorphism ring it belongs to. Along the way, the homotopy modules π 1 P 2 and π 2 P 2 will be determined, based on computations in [23] and [22]. These computations provide an ingredient to complete the program Aravind Asok, Jean Fasel and Ben Williams developed in [3] to prove Suslin's conjecture on the Suslin-Hurewicz homomorphism from Quillen to Milnor K -theory in degree four.…”

Section: Introductionmentioning

confidence: 99%

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Endomorphisms of the projective plane and the image of the Suslin–Hurewicz map

Röndigs

2023

Invent. math.

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The endomorphism ring of the projective plane over a field F of characteristic neither two nor three is slightly more complicated in the Morel–Voevodsky motivic stable homotopy category than in Voevodsky’s derived category of motives. In particular, it is not commutative precisely if there exists a square in F which does not admit a sixth root. A byproduct of these computations is a proof of Suslin’s conjecture on the Suslin–Hurewicz homomorphism from Quillen to Milnor K-theory in degree four, based on work of Asok et al. (Invent Math 219:39-73, 2020).

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“…It is as non-commutative as the endomorphism ring it belongs to. Along the way, the homotopy modules π 1 P 2 and π 2 P 2 will be determined, based on computations in [RSØ19] and [Rön20].…”

Section: Introductionmentioning

confidence: 99%