Homotopy invariant presheaves with framed transfers

Garkusha, Grigory; Panin, Ivan

doi:10.48550/arxiv.1504.00884

Cited by 20 publications

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“…We claim that the restriction map F (U ) → F (k(U )) is injective. This was shown for local schemes in [8]. The same proof works over semi-local schemes if we use [12, 2.2, 2.3, 4.3].…”

Section: Effective Framed Bispectramentioning

confidence: 53%

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The triangulated categories of framed bispectra and framed motives

Garkusha¹,

Panin²

2018

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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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“…We claim that the restriction map F (U ) → F (k(U )) is injective. This was shown for local schemes in [8]. The same proof works over semi-local schemes if we use [12, 2.2, 2.3, 4.3].…”

Section: Effective Framed Bispectramentioning

confidence: 53%

“…with α : F → F nis the canonical sheafification map. Since F nis is a framed radditive A 1invariant quasi-stable presheaf of Abelian groups by [8], then so are the presheaves Ker α, Coker α.…”

Section: Effective Framed Bispectramentioning

confidence: 99%

The triangulated categories of framed bispectra and framed motives

Garkusha¹,

Panin²

2018

Preprint

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“…As an application, explicit computations of motivic infinite loop spaces are given as follows: Section 10]. Based on [1,17,18,19], a motivic recognition principle for motivic infinite loop spaces is given in [14] in the language of infinity categories.…”

Section: Introductionmentioning

confidence: 99%

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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“…Definition 2.8. ( [GP2,Definitions 2.4,2.5,2.11,Remark 2.12]) The exterior composition of framed correspondences (or linear framed correspondences) between objects of Sm/k defines categories:…”

Section: Statement Of the Main Theorem And Its Reduction To The Other...mentioning

confidence: 99%

“…We also need the following notion, due to Voevodsky, given in [GP2] Definition 2.9. (=[GP2, Definition 2.7]) A F r + -presheaf F of Abelian groups is stable if for any k-smooth variety the pull-back map σ * X : F(X) → F(X) equals the identity map, where σ X = (X × 0, X × A 1 , t; pr X ) ∈ F r 1 (X, X).…”

Section: Statement Of the Main Theorem And Its Reduction To The Other...mentioning

confidence: 99%

A motivic Segal theorem for open pairs of smooth schemes over an infinite perfect field

Tsybyshev

2020

Preprint

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V. Voevodsyky laid the groundwork of delooping motivic spaces in order to provide a new, more computation-friendly, construction of the stable motivic category SH(k), G. Garkusha and I. Panin made that project a reality, while collaborating with A. Ananievsky, A. Neshitov and A. Druzhinin. In particular, G. Garkusha and I. Panin proved that for an infinite perfect field k and any k-smooth scheme X the canonical morphism of motivic spacesis Nisnevich-locally a group-completion. In the present work, a generalisation of that theorem to the case of smooth open pairs (X, U ), where X is a k-smooth scheme, U is its open subscheme intersecting each component of X in a nonempty subscheme. We claim that in this case the motivic space C * F r((X, U )) is Nisnevich-locally connected, and the motivic space morphism C * F r((X, U )) → Ω ∞ P 1 Σ ∞ P 1 (X/U ) is Nisnevichlocally a weak equivalence. Moreover, we show that if the codimension of S = X − U in each component of X is greater than r 0, the simplicial sheaf C * F r((X, U )) is locally r-connected.

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Homotopy invariant presheaves with framed transfers

Cited by 20 publications

References 0 publications

The triangulated categories of framed bispectra and framed motives

The triangulated categories of framed bispectra and framed motives

Framed motivic $Γ$-spaces

A motivic Segal theorem for open pairs of smooth schemes over an infinite perfect field

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