Witt sheaves and the η-inverted sphere spectrum

Ananyevskiy, Alexey; Levine, Marc; Panin, Ivan

doi:10.1112/topo.12015

Cited by 23 publications

(51 citation statements)

References 22 publications

(43 reference statements)

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“…Proof. Ananyevsky, Levine and Panin show that the groups π s,w (F ) are torsion for s > w ≥ 0 in [3]. It follows that the group π s,w (F ) is the sum of its ℓ-primary subgroups π s,w (F ) (ℓ) .…”

Section: 2mentioning

confidence: 98%

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: 2mentioning

confidence: 98%

Two-complete stable motivic stems over finite fields

Wilson

Østvær

2017

Algebr. Geom. Topol.

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“…A different but related question is to determine rational motivic stable homotopy theory. By a recent result of Ananyevskiy-Levine-Panin [ALP17] we have SH(k) − Q ≃ DM W (k, Q), where the right hand side denotes a category of rational Witt-motives. Our results show easily that…”

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confidence: 92%

Motivic and real étale stable homotopy theory

Bachmann¹

2018

Compositio Math.

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Let S be a Noetherian scheme of finite dimension and denote by ρ ∈ [½, Gm] SH(S) the (additive inverse of the) morphism corresponding to −1 ∈ O × (S). Here SH(S) denotes the motivic stable homotopy category. We show that the category obtained by inverting ρ in SH(S) is canonically equivalent to the (simplicial) local stable homotopy category of the site S rét , by which we mean the small realétale site of S, comprised ofétale schemes over S with the realétale topology.One immediate application is that SH(R)[ρ −1 ] is equivalent to the classical stable homotopy category. In particular this computes all the stable homotopy sheaves of the ρ-local sphere (over R). As further applications we show that D A 1 (k, Z[1/2]) − ≃ DM W (k)[1/2] (improving a result of Ananyevskiy-Levine-Panin), reprove Röndigs' result that π i (½[1/η, 1/2]) = 0 for i = 1, 2 and establish some new rigidity results.

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“…There has been some progress in this direction. Now that we know the validity of motivic Serre finiteness [1], Heller and Ormsby have proved that, after η-completion, there is a fully faithful functor from the C 2 -equivariant stable homotopy category to the stable motivic homotopy category of real closed fields [9]. The ideas behind the construction of the stablé etale realization functor constructed in this paper may lead to Galois equivariant generalizations.…”

mentioning

confidence: 78%

“…Remark 6.4. It is now a theorem of Ananyevskiy, Levine, and Panin that motivic Serre finiteness is valid for all fields k [1].…”

Section: Proof Of the Main Theoremmentioning

confidence: 99%

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Comparison of stable homotopy categories and a generalized Suslin-Voevodsky theorem

Zargar

2019

Advances in Mathematics

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Let k be an algebraically closed field of exponential characteristic p. Given any prime ℓ = p, we construct a stableétale realization functoŕ Et ℓ : Spt(k) → Pro(Spt) HZ/ℓ from the stable ∞-category of motivic P 1 -spectra over k to the stable ∞-category of (HZ/ℓ) * -local prospectra (see section 3 for definition). This is induced by theétale topological realization functorá la Friedlander. The constant presheaf functor naturally induces the functor

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Witt sheaves and the η-inverted sphere spectrum

Cited by 23 publications

References 22 publications

Two-complete stable motivic stems over finite fields

Two-complete stable motivic stems over finite fields

Motivic and real étale stable homotopy theory

Comparison of stable homotopy categories and a generalized Suslin-Voevodsky theorem

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