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THE SPECTRAL SEQUENCE RELATING ALGEBRAIC KTHEORY TO MOTIVIC COHOMOLOGY
"... The purpose of this paper is to establish in Theorem 13.13 a spectral sequence from the motivic cohomology of a smooth variety X over a field F to the algebraic Ktheory of X: E p,q 2 = Hp−q (X, Z(−q)) = CH −q (X, −p − q) ⇒ K−p−q(X). (13.13.1) Such a spectral sequence was conjectured by A. Beilins ..."
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Cited by 64 (6 self)
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The purpose of this paper is to establish in Theorem 13.13 a spectral sequence from the motivic cohomology of a smooth variety X over a field F to the algebraic Ktheory of X: E p,q 2 = Hp−q (X, Z(−q)) = CH −q (X, −p − q) ⇒ K−p−q(X). (13.13.1) Such a spectral sequence was conjectured by A. Beilinson [Be] as a natural analogue of the AtiyahHirzebruch spectral sequence from the singular cohomology to the topological Ktheory of a topological space. The expectation of such a spectral sequence has provided much of the impetus for the development of motivic cohomology (e.g., [B1], [V2]) and should facilitate many computations in algebraic Ktheory. In the special case in which X equals SpecF, this spectral sequence was established by S. Bloch and S. Lichtenbaum [BL]. Our construction depends crucially upon the main result of [BL], the existence of an exact couple relating the motivic cohomology of the field F to the multirelative Ktheory of coherent sheaves on standard simplices over F (recalled as Theorem 5.5 below). A major step in generalizing the work of Bloch and Lichtenbaum is our reinterpretation of their spectral sequence in terms of the “topological filtration ” on the Ktheory of the standard cosimplicial scheme ∆ • over F. We find that the spectral sequence arises from a tower of Ωprespectra K( ∆ • ) = K 0 ( ∆ • ) ← − K 1 ( ∆ • ) ← − K 2 ( ∆ • ) ← − · · · Thus, even in the special case in which X equals SpecF, we obtain a much clearer understanding of the BlochLichtenbaum spectral sequence which is essential for purposes of generalization. Following this reinterpretation, we proceed using techniques introduced by V. Voevodsky in his study of motivic cohomology. In order to do this, we provide an equivalent formulation of Ktheory spectra associated to coherent sheaves on X with conditions on their supports K q ( ∆ • × X) which is functorial in X. We then Partially supported by the N.S.F. and the N.S.A.
Universal homotopy theories
 Adv. Math
"... Abstract. Begin with a small category C. The goal of this short note is to point out that there is such a thing as a ‘universal model category built from C’. We describe applications of this to the study of homotopy colimits, the DwyerKan theory of framings, to sheaf theory, and to the homotopy the ..."
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Cited by 52 (3 self)
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Abstract. Begin with a small category C. The goal of this short note is to point out that there is such a thing as a ‘universal model category built from C’. We describe applications of this to the study of homotopy colimits, the DwyerKan theory of framings, to sheaf theory, and to the homotopy theory of schemes. Contents
Patching the norm residue isomorphism theorem
, 2007
"... Abstract. We provide a patch to complete the proof of the VoevodskyRost Theorem, that the norm residue map is an isomorphism. (This settles the motivic BlochKato conjecture). ..."
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Cited by 38 (2 self)
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Abstract. We provide a patch to complete the proof of the VoevodskyRost Theorem, that the norm residue map is an isomorphism. (This settles the motivic BlochKato conjecture).
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.
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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Cited by 17 (0 self)
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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.
Motivic Functors
 DOCUMENTA MATH.
, 2003
"... The notion of motivic functors refers to a motivic homotopy theoretic analog of continuous functors. In this paper we lay the foundations for a homotopical study of these functors. Of particular interest is a model structure suitable for studying motivic functors which preserve motivic weak equivale ..."
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Cited by 15 (9 self)
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The notion of motivic functors refers to a motivic homotopy theoretic analog of continuous functors. In this paper we lay the foundations for a homotopical study of these functors. Of particular interest is a model structure suitable for studying motivic functors which preserve motivic weak equivalences and a model structure suitable for motivic stable homotopy theory. The latter model is Quillen equivalent to the category of motivic symmetric spectra. There is a symmetric monoidal smash product of motivic functors, and all model structures constructed are compatible with the smash product in the sense that we can do homotopical algebra on the various categories of modules and algebras. In particular, motivic cohomology
Applications of AtiyahHirzebruch spectral sequence for motivic cobordism
 Department of Mathematics, Faculty of Education, Ibaraki University
"... Abstract. We study applications of AtiyahHirzebruch spectral sequences for motivic cobordisms found by Hopkins and Morel. 1. ..."
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Cited by 15 (8 self)
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Abstract. We study applications of AtiyahHirzebruch spectral sequences for motivic cobordisms found by Hopkins and Morel. 1.
TYPE THEORY AND HOMOTOPY
"... The purpose of this informal survey article is to introduce the reader to a new and surprising connection between Geometry, Algebra, and Logic, which has recently come to light in the form of an interpretation of the constructive type theory of Per MartinLöf into homotopy ..."
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Cited by 14 (0 self)
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The purpose of this informal survey article is to introduce the reader to a new and surprising connection between Geometry, Algebra, and Logic, which has recently come to light in the form of an interpretation of the constructive type theory of Per MartinLöf into homotopy
Multiplicative Properties of the Slice Filtration
"... Let S be a Noetherian separated scheme of finite Krull dimension, and let SH(S) denote the motivic stable homotopy category of Morel and Voevodsky. In order to get a motivic version of the Postnikov tower, Voevodsky [Voe02] constructs a filtered family of triangulated subcategories of SH(S): ..."
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Cited by 13 (5 self)
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Let S be a Noetherian separated scheme of finite Krull dimension, and let SH(S) denote the motivic stable homotopy category of Morel and Voevodsky. In order to get a motivic version of the Postnikov tower, Voevodsky [Voe02] constructs a filtered family of triangulated subcategories of SH(S):
A 1Homotopy of Chevalley Groups
, 2008
"... In this paper, we describe the sheaves of A1homotopy groups of a simplyconnected Chevalley group G. The A1homotopy group sheaves can be identified with the sheafification of the unstable KaroubiVillamayor Kgroups. ..."
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Cited by 10 (4 self)
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In this paper, we describe the sheaves of A1homotopy groups of a simplyconnected Chevalley group G. The A1homotopy group sheaves can be identified with the sheafification of the unstable KaroubiVillamayor Kgroups.