An extension in the Adams spectral sequence in dimension 54

Burklund, Robert

doi:10.1112/blms.12428

Cited by 5 publications

(5 citation statements)

References 6 publications

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“…Below, we depict some of the E ∞ -page of the -Bockstein spectral sequence for π * , * , * kq 2 . We follow the convention introduced in [BHS19, §A.2] and [Bur20] of ing blue symbols to denote -torsion classes, while black symbols denote -torsion free classes. In particular, black symbols contribute not only to the homotopy group corresponding to the box in which they appear, but also to the groups corresponding to boxes directly below where they appear.…”

Section: Odd Primesmentioning

confidence: 99%

Galois reconstruction of Artin-Tate $\mathbb{R}$-motivic spectra

Burklund¹,

Hahn²,

Andrew³

2020

Preprint

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We explain how to reconstruct the category of Artin-Tate R-motivic spectra as a deformation of the purely topological C 2 -equivariant stable category. The special fiber of this deformation is algebraic, and equivalent to an appropriate category of C 2 -equivariant sheaves on the moduli stack of formal groups. As such, our results directly generalize the cofiber of τ philosophy that has revolutionized classical stable homotopy theory.A key observation is that the Artin-Tate subcategory of R-motivic spectra is easier to understand than the previously studied cellular subcategory. In particular, the Artin-Tate category contains a variant of the τ map, which is a feature conspicuously absent from the cellular category. Contents 1. Introduction 1 2. Constructing the element ta 17 3. The ta-local category 19 4. Galois reconstruction 24 5. Modules over the cofiber of ta 30 6. The Chow t-structure and ν R 44 7. The a-local category 53 8. Completions 55 9. Odd primes 60 10. Examples and computations 61 Appendix A. Recollections on compact rigid generation 65 Appendix B. Recollections on filtered objects 68 Appendix C. A machine for deforming homotopy theories 70 References 75

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Section: Odd Primesmentioning

confidence: 99%

Galois reconstruction of Artin-Tate $\mathbb{R}$-motivic spectra

Burklund¹,

Hahn²,

Andrew³

2020

Preprint

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“…See [15] and [16] for more details. We are grateful to John Rognes for pointing out a mistake in [30,Lemma 4.56 and Table 27] concerning the hidden 2 extension on h 0 h 5 i.…”

Section: Hidden 2 Extensionsmentioning

confidence: 99%

“…Now shuffle to obtain (ησ + ) η, ηκ, τ θ 4.5 + ν 2 ν, ηκ, τ θ 4.5= ησ + ν 2 , η ν , ηκ τ θ 4.The matric Toda bracket ησ+ ν 2 , η ν , ηκ must equal {0, ν 2 σ }, since ν 2 σ = {h 2 1 h 4 c 0 } is the only non-zero element of π 25,15 , and that element belongs to the indeterminacy because it is a multiple of ν 2 .…”

mentioning

confidence: 99%

Stable homotopy groups of spheres: from dimension 0 to 90

Isaksen

Wang

2023

Publ.math.IHES

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Using techniques in motivic homotopy theory, especially the theorem of Gheorghe, the second and the third author on the isomorphism between motivic Adams spectral sequence for $C\tau $ C τ and the algebraic Novikov spectral sequence for $BP_{*}$ B P ∗ , we compute the classical and motivic stable homotopy groups of spheres from dimension 0 to 90, except for some carefully enumerated uncertainties.

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“…This ability to "tilt" an Adams spectral sequence, first obtained for the Adams-Novikov spectral sequence using motivic methods by Gheorghe, Isaksen, Wang and Xu [41], [49], [50], has important computational consequences. For example, Burklund, Hahn and Senger use Cτ methods to prove strong results about the classical Adams spectral sequence, solving -among other things -several conjectures in geometric topology [26], [28], [29], [25].…”

Section: Derived 8-categories and Goerss-hopkins Theorymentioning

confidence: 99%

Adams spectral sequences and Franke's algebraicity conjecture

Patchkoria¹,

Pstrągowski²

2021

Preprint

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To any well-behaved homology theory we associate a derived 8-category which encodes its Adams spectral sequence. As applications, we prove a conjecture of Franke on algebraicity of certain homotopy categories and establish homotopy-coherent monoidality of the Adams filtration. 7.1. The algebraic model 87 7.2. Splittings of abelian categories and the Bousfield functor 89 7.3. Bousfield adjunction 91 7.4. The truncated thread monad 94 7.5. Proof of Franke's conjecture 99 8. Applications of the conjecture 101 8.1. Modules over ring spectra 102 8.2. Chromatic algebraicity 105 8.3. Diagram 8-categories 105 Appendix A. Comodules and quotients of abelian categories 110 References 112

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An extension in the Adams spectral sequence in dimension 54

Cited by 5 publications

References 6 publications

Galois reconstruction of Artin-Tate $\mathbb{R}$-motivic spectra

Galois reconstruction of Artin-Tate $\mathbb{R}$-motivic spectra

Stable homotopy groups of spheres: from dimension 0 to 90

Adams spectral sequences and Franke's algebraicity conjecture

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