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17
Motivic Landweber Exactness
 DOCUMENTA MATH.
, 2009
"... We prove a motivic Landweber exact functor theorem. The main result shows the assignment given by a Landwebertype formula involving the MGLhomology of a motivic spectrum defines a homology theory on the motivic stable homotopy category which is representable by a Tate spectrum. Using a universal ..."
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Cited by 23 (8 self)
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We prove a motivic Landweber exact functor theorem. The main result shows the assignment given by a Landwebertype formula involving the MGLhomology of a motivic spectrum defines a homology theory on the motivic stable homotopy category which is representable by a Tate spectrum. Using a universal coefficient spectral sequence we deduce formulas for operations of certain motivic Landweber exact spectra including homotopy algebraic Ktheory. Finally we employ a Chern character between motivic spectra in order to compute rational algebraic cobordism groups over fields in terms of rational motivic cohomology groups and the Lazard ring.
A universality theorem for voevodsky’s algebraic cobordism spectrum
, 709
"... An algebraic version of a theorem due to Quillen is proved. More precisely, for a ground field k we consider the motivic stable homotopy category SH(k) of P 1spectra, equipped with the symmetric monoidal structure described in [PPR1]. The algebraic cobordism P 1spectrum MGL is considered as a comm ..."
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Cited by 19 (2 self)
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An algebraic version of a theorem due to Quillen is proved. More precisely, for a ground field k we consider the motivic stable homotopy category SH(k) of P 1spectra, equipped with the symmetric monoidal structure described in [PPR1]. The algebraic cobordism P 1spectrum MGL is considered as a commutative monoid equipped with a canonical orientation th MGL ∈ MGL 2,1 (Th(O(−1))). For a commutative monoid E in the category SH(k) it is proved that assignment ϕ ↦ → ϕ(th MGL) identifies the set of monoid homomorphisms ϕ: MGL → E in the motivic stable homotopy category SH(k) with the set of all orientations of E. The result was stated originally in a slightly different form by G. Vezzosi in [Ve]. 1 Oriented commutative ring spectra We refer to [PPR1, Appendix] for the basic terminology, notation, constructions, definitions, results. For the convenience of the reader we recall the basic definitions. Let S be a Noetherian scheme of finite Krull dimension. One may think of S being the spectrum of
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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Cited by 9 (2 self)
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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
Around the Gysin Triangle II.
 DOCUMENTA MATH.
, 2008
"... The notions of orientation and duality are well understood in algebraic topology in the framework of the stable homotopy category. In this work, we follow these lines in algebraic geometry, in the framework of motivic stable homotopy, introduced by F. Morel and V. Voevodsky. We use an axiomatic tre ..."
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Cited by 3 (0 self)
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The notions of orientation and duality are well understood in algebraic topology in the framework of the stable homotopy category. In this work, we follow these lines in algebraic geometry, in the framework of motivic stable homotopy, introduced by F. Morel and V. Voevodsky. We use an axiomatic treatment which allows us to consider both mixed motives and oriented spectra over an arbitrary base scheme. In this context, we introduce the Gysin triangle and prove several formulas extending the traditional panoply of results on algebraic cycles modulo rational equivalence. We also obtain the Gysin morphism of a projective morphism and prove a duality theorem in the (relative) pure case. These constructions involve certain characteristic classes (Chern classes, fundamental classes, cobordism classes) together with their usual properties. They imply statements in motivic cohomology, algebraic Ktheory (assuming the base is regular) and ”abstract” algebraic cobordism as well as the dual statements in the corresponding homology theories. They apply
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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Cited by 2 (1 self)
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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.
1 Existence and uniqueness of E∞structures on motivic Ktheory spectra
, 2011
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Voevodsky’s mixed motives versus . . . mixed motives
, 2014
"... Following an insight of Kontsevich, we prove that the quotient of Voevodsky’s category of geometric mixed motives DMgm by the endofunctor −⊗Q(1)[2] embeds fullyfaithfully into Kontsevich’s category of noncommutative mixed motives KMM. We show also that this embedding is compatible with the one ..."
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Following an insight of Kontsevich, we prove that the quotient of Voevodsky’s category of geometric mixed motives DMgm by the endofunctor −⊗Q(1)[2] embeds fullyfaithfully into Kontsevich’s category of noncommutative mixed motives KMM. We show also that this embedding is compatible with the one between pure motives. As an application, we obtain a precise relation between the Picard groups Pic(−), the Grothendieck groups, the Schurfiniteness, and the Kimurafiniteness of the categories DMgm and KMM. In particular, the quotient of Pic(DMgm) by the subgroup of Tate twists Q(i)[2i] injects into Pic(KMM). Along the way, we relate KMM with