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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 45 (5 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.
Lecture notes on motivic cohomology
 of Clay Mathematics Monographs. American Mathematical Society
, 2006
"... From the point of view taken in these lectures, motivic cohomology with coefficients in an abelian group A is a family of contravariant functors H p,q (−, A) : Sm/k → Ab from smooth schemes over a given field k to abelian groups, indexed by ..."
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Cited by 21 (2 self)
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From the point of view taken in these lectures, motivic cohomology with coefficients in an abelian group A is a family of contravariant functors H p,q (−, A) : Sm/k → Ab from smooth schemes over a given field k to abelian groups, indexed by
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 15 (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).
Failure Of Brown Representability In Derived Categories
"... Let T be a triangulated category with coproducts, T c T the full subcategory of compact objects in T. If T is the homotopy category of spectra, Adams proved the following in [1]: All homological functors fT c g op ! Ab are the restrictions of representable functors on T, and all natural tr ..."
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Cited by 14 (0 self)
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Let T be a triangulated category with coproducts, T c T the full subcategory of compact objects in T. If T is the homotopy category of spectra, Adams proved the following in [1]: All homological functors fT c g op ! Ab are the restrictions of representable functors on T, and all natural transformations are the restrictions of morphisms in T. It has been something of a mystery, to what extent this generalises to other triangulated categories. In [36], it was proved that Adams' theorem remains true as long as T c is countable, but can fail in general. The failure exhibited was that there can be natural transformations not arising from maps in T. A puzzling open problem remained: Is every homological functor the restriction of a representable functor on T? In a recent paper, Beligiannis [5] made some progress. But in this article, we settle the problem. The answer is no. There are examples of derived categories T = D(R) of rings, and homological functors fT c g op ! Ab which are not restrictions of representables. Contents
Higher Chow Groups And Etale Cohomology
 In: Cycles, Transfers and Motivic Homology Theories, Annals of Math Studies, Princeton Univ
"... Introduction The main purpose of the present paper is to relate the higher Chow groups of varieties over an algebraically closed eld introduced by S.Bloch [B1] to etale cohomology. We follow the approach suggested by the auther in 1987 during the Lumini conference on algebraic Ktheory. The rst and ..."
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Introduction The main purpose of the present paper is to relate the higher Chow groups of varieties over an algebraically closed eld introduced by S.Bloch [B1] to etale cohomology. We follow the approach suggested by the auther in 1987 during the Lumini conference on algebraic Ktheory. The rst and most important step in this direction was done in [SV1], where singular cohomology of any qfhsheaf were computed in terms of Extgroups. The diculty in the application of the results of [SV1] to higher Chow groups lies in the fact that a priori higher Chow are not dened as singular homology of a sheaf. To overcome this diculty we prove that for an ane varietie X higher Chow groups CH i (X; n) of codimension i dimX may be computed using equidimensional cycles only(this is done in the rst two sections of the paper). In section 3 we generalize this result to all quasiprojective varieties over a eld of char