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23
Triangulated categories of mixed motives
"... Abstract. We construct triangulated categories of mixed motives over a noetherian scheme of finite dimension, extending Voevodsky’s definition of motives over a field. We prove that motives with rational coefficients satisfy the formalism of the six operations of Grothendieck. This is achieved by st ..."
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Abstract. We construct triangulated categories of mixed motives over a noetherian scheme of finite dimension, extending Voevodsky’s definition of motives over a field. We prove that motives with rational coefficients satisfy the formalism of the six operations of Grothendieck. This is achieved by studying descent properties of motives, as well as by comparing different presentations of these categories, following insights and constructions of Beilinson, Morel and Voevodsky. Finally, we associate with any mixed Weil cohomology a system of categories of coefficients and well behaved realization functors.
On the Motivic Spectra Representing Algebraic Cobordism and Algebraic KTheory
 DOCUMENTA MATH.
, 2009
"... We show that the motivic spectrum representing algebraic Ktheory is a localization of the suspension spectrum of P∞, and similarly that the motivic spectrum representing periodic algebraic cobordism is a localization of the suspension spectrum of BGL. In particular, working over C and passing to sp ..."
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We show that the motivic spectrum representing algebraic Ktheory is a localization of the suspension spectrum of P∞, and similarly that the motivic spectrum representing periodic algebraic cobordism is a localization of the suspension spectrum of BGL. In particular, working over C and passing to spaces of Cvalued points, we obtain new proofs of the topological versions of these theorems, originally due to the second author. We conclude with a couple of applications: first, we give a short proof of the motivic ConnerFloyd theorem, and second, we show that algebraic Ktheory and periodic algebraic cobordism are E∞ motivic spectra.
Relations between slices and quotients of the algebraic cobordism spectrum
"... Abstract. We prove a relative statement about the slices of the algebraic cobordism spectrum. If the map from MGL to a certain quotient of MGL introduced by Hopkins and Morel is the map to the zeroslice then a relative version of Voevodsky’s conjecture on the slices of MGL holds true. We outline th ..."
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Cited by 13 (6 self)
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Abstract. We prove a relative statement about the slices of the algebraic cobordism spectrum. If the map from MGL to a certain quotient of MGL introduced by Hopkins and Morel is the map to the zeroslice then a relative version of Voevodsky’s conjecture on the slices of MGL holds true. We outline the picture for Ktheory and rational slices.
Motivic strict ring models for Ktheory
 Proc. Amer. Math. Soc
"... It is shown that the Ktheory of every noetherian base scheme of finite Krull dimension is represented by a strict ring object in the setting of motivic stable homotopy theory. The adjective ‘strict ’ is used to distinguish between the type of ring structure we construct and one which is valid only ..."
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It is shown that the Ktheory of every noetherian base scheme of finite Krull dimension is represented by a strict ring object in the setting of motivic stable homotopy theory. The adjective ‘strict ’ is used to distinguish between the type of ring structure we construct and one which is valid only up to homotopy. Both the categories of motivic functors and motivic symmetric spectra furnish convenient frameworks for constructing the ring models. Analogous topological results follow
Periodizable motivic ring spectra
, 2009
"... We show that the cellular objects in the module category over a motivic E∞ring spectrum E can be described as the module category over a graded topological spectrum if E is strongly periodizable in our language. A similar statement is proven for triangulated categories of motives. Since MGL is st ..."
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Cited by 3 (3 self)
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We show that the cellular objects in the module category over a motivic E∞ring spectrum E can be described as the module category over a graded topological spectrum if E is strongly periodizable in our language. A similar statement is proven for triangulated categories of motives. Since MGL is strongly periodizable we obtain topological incarnations of motivic Landweber spectra. Under some categorical assumptions the unit object of the model category for triangulated motives is as well strongly periodizable giving motivic cochains whose module category models integral triangulated
Motivic Brown–Peterson invariants of the rationals
"... Let BPhni, 0 n 1, denote the family of motivic truncated Brown–Peterson spectra over Q. We employ a “localtoglobal ” philosophy in order to compute the bigraded homotopy groups of BPhni. Along the way, we produce a computation of the homotopy groups of BPhni over Q2, prove a motivic Hasse principl ..."
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Let BPhni, 0 n 1, denote the family of motivic truncated Brown–Peterson spectra over Q. We employ a “localtoglobal ” philosophy in order to compute the bigraded homotopy groups of BPhni. Along the way, we produce a computation of the homotopy groups of BPhni over Q2, prove a motivic Hasse principle for the spectra BPhni, and reprove several classical and recent theorems about the K –theory of particular fields in a streamlined fashion. We also compute the bigraded homotopy groups of the 2–complete algebraic cobordism spectrum MGL over Q. 55T15; 19D50, 19E15 1
MOTIVIC INVARIANTS OF pADIC FIELDS
"... ABSTRACT. I provide a complete analysis of the motivic Adams spectral sequences converging to the bigraded coefficients of the 2completions of the motivic spectra BPGL and kgl over padic fields, p> 2. The former spectrum is the algebraic BrownPeterson spectrum at the prime 2 (and hence is part ..."
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ABSTRACT. I provide a complete analysis of the motivic Adams spectral sequences converging to the bigraded coefficients of the 2completions of the motivic spectra BPGL and kgl over padic fields, p> 2. The former spectrum is the algebraic BrownPeterson spectrum at the prime 2 (and hence is part of the study of algebraic cobordism), and the latter is a certain BPGLmodule that plays the role of a “connective ” motivic algebraic Ktheory spectrum. This is the first in a series of two papers investigating motivic invariants of padic fields, and it lays the groundwork for an understanding of the motivic AdamsNovikov spectral sequence over such base fields. CONTENTS
Stable motivic pi1 of lowdimensional fields
"... ABSTRACT. Let k be a field with cohomological dimension less than 3; we call such fields lowdimensional. Examples include algebraically closed fields, finite fields and function fields thereof, local fields, and number fields with no real embeddings. We determine the 1column of the motivic Adams ..."
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ABSTRACT. Let k be a field with cohomological dimension less than 3; we call such fields lowdimensional. Examples include algebraically closed fields, finite fields and function fields thereof, local fields, and number fields with no real embeddings. We determine the 1column of the motivic AdamsNovikov spectral sequence over k. Combined with rational information we use this to compute pi1S, the first stable motivic homotopy group of the sphere spectrum over k. Our main result affirms Morel’s pi1conjecture in the case of lowdimensional fields. We also determine pi1+nαS for weights n ∈ Z r {−2,−3,−4}. 1.
Motivic twisted Ktheory
, 2010
"... This paper sets out basic properties of motivic twisted Ktheory with respect to degree three motivic cohomology classes of weight one. Motivic twisted Ktheory is defined in terms of such motivic cohomology classes by taking pullbacks along the universal principal BGmbundle for the classifying spa ..."
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This paper sets out basic properties of motivic twisted Ktheory with respect to degree three motivic cohomology classes of weight one. Motivic twisted Ktheory is defined in terms of such motivic cohomology classes by taking pullbacks along the universal principal BGmbundle for the classifying space of the multiplicative group scheme. We show a Künneth isomorphism for homological motivic twisted Kgroups computing the latter as a tensor product of Kgroups over the Ktheory of BGm. The proof employs an Adams Hopf algebroid and a trigraded Torspectral sequence for motivic twisted Ktheory. By adopting the notion of an E∞ring spectrum to the motivic homotopy theoretic setting, we construct spectral sequences relating motivic (co)homology groups to twisted Kgroups. It generalizes various spectral sequences computing the algebraic Kgroups of schemes over fields. Moreover, we construct a Chern character between motivic twisted Ktheory and twisted periodized rational motivic cohomology, and show that it is a rational isomorphism. The paper includes a discussion of some open problems.