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On Voevodsky’s algebraic Ktheory spectrum BGL
, 2007
"... Under a certain normalization assumption we prove that the P 1spectrum BGL of Voevodsky which represents algebraic Ktheory is unique over Spec(Z). Following an idea of Voevodsky, we equip the P 1spectrum BGL with the structure of a commutative ring P 1spectrum in the motivic stable homotopy cate ..."
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Cited by 13 (6 self)
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Under a certain normalization assumption we prove that the P 1spectrum BGL of Voevodsky which represents algebraic Ktheory is unique over Spec(Z). Following an idea of Voevodsky, we equip the P 1spectrum BGL with the structure of a commutative ring P 1spectrum in the motivic stable homotopy category. Furthermore, we prove that under a certain normalization assumption this ring structure is unique over Spec(Z). For an arbitrary Noetherian base scheme S we pull this structure back to get a distinguished monoidal structure on BGL. 1
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 11 (7 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.
Enriched Functors and Stable Homotopy Theory
 DOCUMENTA MATH.
, 2002
"... In this paper we employ enriched category theory to construct a convenient model for several stable homotopy categories. This is achieved in a threestep process by introducing the pointwise, homotopy functor and stable model category structures for enriched functors. The general setup is shown to d ..."
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Cited by 7 (3 self)
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In this paper we employ enriched category theory to construct a convenient model for several stable homotopy categories. This is achieved in a threestep process by introducing the pointwise, homotopy functor and stable model category structures for enriched functors. The general setup is shown to describe equivariant stable homotopy theory, and we recover Lydakis’ model category of simplicial functors as a special case. Other examples – including motivic homotopy theory – will be treated in subsequent papers.
Lstable Functors
, 2007
"... We generalize and greatly simplify the approach of Lydakis and DundasRöndigsØstvær to construct an Lstable model structure for small functors from a closed symmetric monoidal model category V to a Vmodel category M, where L is a small cofibrant object of V. For the special case V = M = S ∗ poi ..."
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Cited by 1 (0 self)
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We generalize and greatly simplify the approach of Lydakis and DundasRöndigsØstvær to construct an Lstable model structure for small functors from a closed symmetric monoidal model category V to a Vmodel category M, where L is a small cofibrant object of V. For the special case V = M = S ∗ pointed simplicial sets and L = S1 this is the classical case of linear functors and has been described as the first stage of the Goodwillie tower of a homotopy functor. We show, that our various model structures are compatible with a closed symmetric monoidal product on small functors. We compare them with other Lstabilizations described by Hovey, Jardine and others. This gives a particularly easy construction of the classical and the motivic stable homotopy category with the correct smash product. We
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 1 (1 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
Motivic strict ring . . .
, 2009
"... 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 u ..."
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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 by
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.