摘要 :
The category of framed correspondences Fr-*(k), framed presheaves and framed sheaves were invented by Voevodsky in his unpublished notes [20]. Based on the notes [20] a new approach to the classical Morel-Voevodsky motivic stable ...
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The category of framed correspondences Fr-*(k), framed presheaves and framed sheaves were invented by Voevodsky in his unpublished notes [20]. Based on the notes [20] a new approach to the classical Morel-Voevodsky motivic stable homotopy theory was developed in [8]. This approach converts the classical motivic stable homotopy theory into an equivalent local theory of framed bispectra. The main result of the paper is the core of the theory of framed bispectra. It states that for any homotopy invariant quasi-stable radditive framed presheaf of Abelian groups F, the associated Nis-nevich sheaf F-nis, is strictly homotopy invariant and quasi-stable whenever the base field k is infinite perfect of characteristic different from 2.
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It is shown that the category of enriched functors [C,V] is Grothendieck whenever V is a closed symmetric monoidal Grothendieck category and C is a category enriched over V. Localizations in [C,V] associated to collections of obje...
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It is shown that the category of enriched functors [C,V] is Grothendieck whenever V is a closed symmetric monoidal Grothendieck category and C is a category enriched over V. Localizations in [C,V] associated to collections of objects of C are studied. Also, the category of chain complexes of generalized modules Ch(C-R) is shown to be identified with the Grothendieck category of enriched functors [mod R, Ch(Mod R)] over a commutative ring R, where the category of finitely presented R-modules mod R is enriched over the closed symmetric monoidal Grothendieck category Ch(Mod R) as complexes concentrated in zeroth degree. As an application, it is proved that Ch(C-R) is a closed symmetric monoidal Grothendieck model category with explicit formulas for tensor product and internal Hom-objects. Furthermore, the class of unital algebraic almost stable homotopy categories generalizing unital algebraic stable homotopy categories of Hovey-Palmieri-Strickland 14] is introduced. It is shown that the derived category of generalized modules D(C-R) over commutative rings is a unital algebraic almost stable homotopy category which is not an algebraic stable homotopy category. (C) 2015 Elsevier Inc. All rights reserved.
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摘要 :
We investigate certain categorical aspects of Voevodsky's triangulated categories of motives. For this, various recollements for Grothendieck categories of enriched functors and their derived categories are established. In order t...
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We investigate certain categorical aspects of Voevodsky's triangulated categories of motives. For this, various recollements for Grothendieck categories of enriched functors and their derived categories are established. In order to extend these recollements further with respect to Serre's localization, the concept of the (strict) Voevodsky property for Serre localizing subcategories is introduced. This concept is inspired by the celebrated Voevodsky theorem on homotopy invariant presheaves with transfers. As an application, it is shown that Voevodsky's triangulated categories of motives fit into recollements of derived categories of associated Grothendieck categories of Nisnevich sheaves with specific transfers. (C) 2021 Elsevier Inc. All rights reserved.
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摘要 :
The category of framed correspondences Fr-*(k) and framed sheaves were invented by Voevodsky in his unpublished notes [Notes on framed correspondences, 2001, https://www.math.ias.edu/vladimir/publications]. Based on the theory, fr...
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The category of framed correspondences Fr-*(k) and framed sheaves were invented by Voevodsky in his unpublished notes [Notes on framed correspondences, 2001, https://www.math.ias.edu/vladimir/publications]. Based on the theory, framed motives are introduced and studied in Garkusha and Panin [J. Amer. Math. Soc. 34 (2021), pp. 261-313]. These are Nisnivich sheaves of S-1-spectra and the major computational tool of Garkusha and Panin. The aim of this paper is to show the following result which is essential in proving the main theorem of Garkusha and Panin: given an infinite perfect base field k, any k-smooth scheme X and any n >= 1, the map of simplicial pointed Nisnevich sheaves (-, A(1)//G(m))(+)(boolean AND n) -> T-n induces a Nisnevich local level weak equivalence of S-1-spectra
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Using a recent computation of the rational minus part of S H(k) by Ananyevskiy, Levine and Panin, a theorem of Cisinski and Deglise and a version of the Rondigs and Ostvar theorem, rational stable motivic homotopy theory over an i...
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Using a recent computation of the rational minus part of S H(k) by Ananyevskiy, Levine and Panin, a theorem of Cisinski and Deglise and a version of the Rondigs and Ostvar theorem, rational stable motivic homotopy theory over an infinite perfect field of characteristic different from 2 is recovered in this paper from finite Milnor Witt correspondences in the sense of Calmes and Fasel.
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