Cancellation theorem for framed motives of algebraic varieties

Ananyevskiy, Alexey; Garkusha, Grigory; Panin, Ivan

doi:10.48550/arxiv.1601.06642

Cited by 14 publications

(39 citation statements)

References 2 publications

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“…By assumption, υ is a stable motivic equivalence. Therefore Z → F [1] is a level equivalence. Regarding the entries of Z and F [1] as injective cofibrant motivic spaces, we conclude that…”

Section: Theorem the Category Of Spectral Framed Functors Shmentioning

confidence: 93%

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The triangulated categories of framed bispectra and framed motives

Garkusha¹,

Panin²

2018

Preprint

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An alternative approach to the classical Morel-Voevodsky stable motivic homotopy theory SH(k) is suggested. The triangulated category of framed bispectra SH f r nis (k) and effective framed bispectra SH f r,e f f nis (k) are introduced in the paper. Both triangulated categories only use Nisnevich local equivalences and have nothing to do with any kind of motivic equivalences. It is shown that SH f r nis (k) and SH f r,e f f nis (k) recover the classical Morel-Voevodsky triangulated categories of bispectra SH(k) and effective bispectra SH e f f (k) respectively.We also recover SH(k) and SH e f f (k) as the triangulated category of framed motivic spectral functors SH f r S 1 [F r 0 (k)] and the triangulated category of framed motives S H f r (k) respectively constructed in the paper.

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Section: Theorem the Category Of Spectral Framed Functors Shmentioning

confidence: 93%

“…Proof. (1). In each weight j, the S 1 -spectrum A * , j has a natural filtration A * , j = colim n L n (A * , j ), where L n (A * , j ) is the spectrum A 0, j , A 1, j , .…”

Section: Theorem Let a Be A Bispectrum Such That Every Entry A I J Of...mentioning

confidence: 99%

The triangulated categories of framed bispectra and framed motives

Garkusha¹,

Panin²

2018

Preprint

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“…In our quest to carry over Segal's programme for Γ-spaces to A 1 -homotopy theory we begin by formulating some homotopical axioms for framed motivic Γ-spaces. These axioms concern both of the "variables" Γ op and Sm/k + in (1). Informally speaking, the pointed finite sets accounts for the S 1 -suspension whereas the framed correspondences accounts for the G m -suspension in stable motivic homotopy theory.…”

Section: Introductionmentioning

confidence: 99%

“…A Γ-space gives rise to a simplicial functor and hence an associated S 1 -spectrum; for details, see [10,Chapter 2]. Similarly in the motivic setting, see (7), every U ∈ Sm/k + and X as in (1) give rise to a presheaf of S 1 -spectra X (S,U ). We refer to [24] for a comprehensive introduction to the homotopical algebra of such presheaves.…”

Section: Introductionmentioning

confidence: 99%

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Framed motivic $Γ$-spaces

Garkusha¹,

Panin²,

Østvær³

2019

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We combine several mini miracles to achieve an elementary understanding of infinite loop spaces and very effective spectra in the algebro-geometric setting of motivic homotopy theory. Our approach combines Γ-spaces and framed correspondences into the concept of framed motivic Γspaces; these are continuous or enriched functors of two variables that take values in motivic spaces and are equipped with a framing. We craft proofs of our main results by imposing further axioms on framed motivic Γ-spaces such as a Segal condition for simplicial Nisnevich sheaves, cancellation, A 1 -and σ -invariance, Nisnevich excision, Suslin contractibility, and grouplikeness. This adds to the discussion in the literature on coexisting points of view on the A 1 -homotopy theory of algebraic varieties.

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The Unit Map of the Algebraic Special Linear Cobordism Spectrum

Yakerson¹

2019

J. Inst. Math. Jussieu

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In joint work with Elmanto, Hoyois, Khan and Sosnilo [EHK + 19a], we computed infinite P 1 -loop spaces of motivic Thom spectra, using the technique of framed correspondences. This result allows us to express non-negative G m -homotopy groups of motivic Thom spectra in terms of geometric generators and relations. Using this explicit description, we show that the unit map of the algebraic special linear cobordism spectrum induces an isomorphism on G m -homotopy sheaves.

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Cancellation theorem for framed motives of algebraic varieties

Cited by 14 publications

References 2 publications

The triangulated categories of framed bispectra and framed motives

The triangulated categories of framed bispectra and framed motives

Framed motivic $Γ$-spaces

The Unit Map of the Algebraic Special Linear Cobordism Spectrum

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