Framed motives of algebraic varieties (after V. Voevodsky)

Garkusha, Grigory; Panin, Ivan

doi:10.48550/arxiv.1409.4372

Cited by 25 publications

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“…Remark. In the language of framed motives [10] if A ∈ M • is (ℓ − 1)-biconnected and the base field is (infinite) perfect then the framed motive M f r (A c ) (respectively the space C * Fr(A c )), where A c is a cofibrant resolution of A in the projective model structure of spaces, is locally (ℓ − 1)-connected as an S 1 -spectrum (respectively as a motivic space).…”

Section: Theorem (Comparison) the Following Natural Adjunctions Betwe...mentioning

confidence: 99%

“…It is of independent interest to investigate the problem when the quotient Nisnevich sheaves of G-orbits Y /G, Y ∈ Sm/S, are of the form X /U with X ∈ Sm/k and U open in X . Since framed correspondences with coeffitients in X /U have an explicit geometric description due to Voevodsky (see the "Voevodsky Lemma" in [10] as well), a possible application of the problem is computation of the framed motive M f r (X /U ) in terms of framed motives of associated linear algebraic groups.…”

Section: 1] As Well)mentioning

confidence: 99%

“…that first takes a bispectrum E to its naive projective cofibrant resolution E c and then one sets in each bidegree C * F r(E) i, j := C * Fr(E c i, j ) (we refer the reader to [10] for the definition of C * Fr(X ), X ∈ M • ). We should note that the localization functor C * F r is isomorphic to the big framed motives localization functor M b f r of [10] (see [12] as well). We then define SH new (k) as the category of…”

Section: On the Localization Functor C * F Rmentioning

confidence: 99%

“…that first takes a bispectrum E to its naive projective cofibrant resolution E c and then one sets in each bidegree C * F r(E) i, j := C * Fr(E c i, j ). The localization functor C * F r is isomorphic to the big framed motives localization functor M b f r of [10] (see [12] as well). Then SH new (k) is defined as the category of C * F r-local objects in SH nis (k).…”

Section: Introductionmentioning

confidence: 99%

“…In Theorem 6.3 we prove that C * F r and C * F r G ,n are naturally isomorphic. As a result, one can incorporate linear algebraic groups into the theory of motivic infinite loop spaces and framed motives developed in [10].…”

Section: Introductionmentioning

confidence: 99%

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Group schemes and motivic spectra

Garkusha¹

2018

Preprint

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By a theorem of Mandell-May-Schwede-Shipley [18] the stable homotopy theory of classical S 1 -spectra is recovered from orthogonal spectra. In this paper general linear, special linear, symplectic, orthogonal and special orthogonal motivic spectra are introduced and studied. It is shown that the stable homotopy theory of motivic spectra is recovered from one of these types of spectra. An application is given for the localization functorin the sense of [12] that converts the Morel-Voevodsky stable motivic homotopy theory SH(k) into the equivalent local theory of framed bispectra [12]. CONTENTS 1. Introduction 1 2. Diagram motivic spaces and diagram motivic spectra 3 3. Motivic spectra associated with group schemes 4 4. Model structures for C -spectra 10 5. The comparison theorem 11 6. On the localization functor C * F r 18 References 21

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Section: Theorem (Comparison) the Following Natural Adjunctions Betwe...mentioning

confidence: 99%

Section: 1] As Well)mentioning

confidence: 99%

Section: On the Localization Functor C * F Rmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Group schemes and motivic spectra

Garkusha¹

2018

Preprint

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Framed and MW-transfers for homotopy modules

Ananyevskiy

Neshitov

2019

Sel. Math. New Ser.

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In the paper we use the theory of framed correspondences to construct Milnor-Witt transfers on homotopy modules. As a consequence we identify the zeroth stable A 1homotopy sheaf of a smooth variety with the zeroth homology of the corresponding MWmotivic complex and prove that the hearts of the homotopy t-structures on the stable A 1derived category and the category of Milnor-Witt motives are equivalent. We also show that a homotopy invariant stable linear framed Nisnevich sheaf admits a unique structure of a presheaf with MW-transfers compatible with the framed structure.2010 Mathematics Subject Classification. 14F42.

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Rigidity for linear framed presheaves and generalized motivic cohomology theories

Ananyevskiy

Druzhinin

2018

Advances in Mathematics

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A rigidity property for the homotopy invariant stable linear framed presheaves is established. As a consequence a variant of Gabber rigidity theorem is obtained for a cohomology theory representable in the motivic stable homotopy category by a φ-torsion spectrum with φ ∈ GW(k) of rank coprime to the (exponential) characteristic of the base field k. It is shown that the values of such cohomology theories at an essentially smooth Henselian ring and its residue field coincide. The result is applicable to cohomology theories representable by n-torsion spectra as well as to the ones representable by η-periodic spectra and spectra related to Witt groups.Research is supported by the Russian Science Foundation grant 14-21-00035. Preliminaries on framed correspondencesAll schemes are supposed to be separated and noetherian.Definition 2.1. Let X be a scheme and Z be a closed subscheme of X. Anétale neighborhood of Z in X is a pair of morphisms (p : U → X, r : Z → U ) where p isétale and p • r = i for the closed

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Framed motives of algebraic varieties (after V. Voevodsky)

Cited by 25 publications

References 0 publications

Group schemes and motivic spectra

Group schemes and motivic spectra

Framed and MW-transfers for homotopy modules

Rigidity for linear framed presheaves and generalized motivic cohomology theories

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