Isotropic stable motivic homotopy groups of spheres

Tanania, Fabio

doi:10.1016/j.aim.2021.107696

Cited by 4 publications

(13 citation statements)

References 21 publications

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“…For n = ∞, these are realisations closely related to ψ p,E of DM (k) c from the previous section. For not formally-real fields, these coincide with the categories studied by Tanania in [21] and [22].…”

supporting

confidence: 72%

“…For example, it was computed for p = 2- [11,Theorem 3.7] that the isotropic motivic cohomology of a point form an external algebra with generators -duals of Milnor's operations. This, in turn, leads to the computation of the stable isotropic homotopy groups of spheres by F.Tanania, who identified these groups with the E 2 -term of the classical Adams spectral sequence- [9].…”

Section: Introductionmentioning

confidence: 99%

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On isotropic and numerical equivalence of cycles

Vishik

2022

Sel. Math. New Ser.

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We study the conjecture claiming that, over a flexible field, isotropic Chow groups coincide with numerical Chow groups (with $$\mathbb {F}_p$$ F p -coefficients). This conjecture is essential for understanding the structure of the isotropic motivic category and that of the tensor triangulated spectrum of Voevodsky category of motives. We prove the conjecture for the new range of cases. In particular, we show that, for a given variety X, it holds for sufficiently large primes p. We also prove the p-adic analogue. This permits to interpret integral numerically trivial classes in $${\text {CH}}(X)$$ CH ( X ) as $$p^{\infty }$$ p ∞ -anisotropic ones.

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confidence: 72%

Section: Introductionmentioning

confidence: 99%

On isotropic and numerical equivalence of cycles

Vishik

2022

Sel. Math. New Ser.

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“…More precisely, in [9] the stable motivic homotopy groups of C , the cofiber of , are identified with the E 2 -page of the classical Adams-Novikov spectral sequence, while in [5] the motivic spectrum C is provided with an E 1 -ring structure inducing an isomorphism of rings with higher products between .C / and the classical Adams-Novikov E 2 -page. A parallel result for isotropic categories was obtained in [19], where the isotropic sphere spectrum X was equipped with an E 1 -ring structure inducing an isomorphism of rings with higher products between .X/ and the classical Adams E 2 -page. Moreover, in [6] the category of C -cellular spectra is described, and is proved to be equivalent as a stable 1-category equipped with a t-structure (see Lurie [13]) to the derived category of left BP BP-comodules concentrated in even degrees, where BP is the Brown-Peterson spectrum and BP BP its BP-homology.…”

Section: Introductionmentioning

confidence: 80%

“…In [21], Vishik introduced the isotropic triangulated category of motives and computed the isotropic motivic cohomology of the point, which is strongly related to the Milnor subalgebra. By following this lead, we studied in [19] the isotropic stable motivic homotopy category. In particular, we identified the isotropic motivic homotopy groups of the sphere spectrum with the cohomology of the topological Steenrod algebra, ie the E 2 -page of the classical Adams spectral sequence.…”

Section: Introductionmentioning

confidence: 99%

Cellular objects in isotropic motivic categories

Tanania

2023

Geom. Topol.

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Our main purpose is to describe the category of isotropic cellular spectra over flexible fields. Guided by Gheorghe, Wang and Xu (Acta Math. 226 (2021) 319-407), we show that it is equivalent, as a stable 1-category equipped with a t-structure, to the derived category of left comodules over the dual of the classical topological Steenrod algebra. In order to obtain this result, the category of isotropic cellular modules over the motivic Brown-Peterson spectrum is also studied, and isotropic Adams and Adams-Novikov spectral sequences are developed. As a consequence, we also compute hom sets in the category of isotropic Tate motives between motives of isotropic cellular spectra.

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“…For example, it was computed for p = 2 -[7, Theorem 3.7] that the isotropic motivic cohomology of a point form an external algebra with generators -duals of Milnor's operations. This, in turn, leads to the computation of the stable isotropic homotopy groups of spheres by F.Tanania, who identified these groups with the E 2 -term of the classical Adams spectral sequence - [5].…”

Section: Introductionmentioning

confidence: 99%

On isotropic and numerical equivalence of cycles

Vishik¹

2022

Preprint

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We study the conjecture claiming that, over a flexible field, isotropic Chow groups coincide with numerical Chow groups (with F p -coefficients). This conjecture is essential for understanding the structure of the isotropic motivic category and that of the tensor triangulated spectrum of Voevodsky category of motives. We prove the conjecture for the new range of cases. In particular, we show that, for a given variety X, it holds for sufficiently large primes p. We also prove the p-adic analogue. This permits to interpret integral numerically trivial classes in CH(X) as p ∞ -anisotropic ones. Isotropic Chow groupsEverywhere below k will be a field of characteristic zero.Let n ∈ N and Ch * = CH * /n be Chow groups modulo n.Definition 2.1 Let n ∈ N and X be a scheme of finite type over k. We say that X is n-anisotropic, if degrees of all closed points on X are divisible by n.Definition 2.2 Let n ∈ N, X be a scheme over k and x ∈ CH r (X). Then x is n-anisotropic, if there exists a proper map f : Y → X from an n-anisotropic scheme Y and y ∈ CH r (Y ) such that f * (y) = x. Now we can introduce isotropic Chow groups:Ch k/k := Ch /( n-anisotropic classes ).

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Isotropic stable motivic homotopy groups of spheres

Cited by 4 publications

References 21 publications

On isotropic and numerical equivalence of cycles

On isotropic and numerical equivalence of cycles

Cellular objects in isotropic motivic categories

On isotropic and numerical equivalence of cycles

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