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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.
The Ktheory of fields in characteristic p
, 1996
"... Abstract. The purpose of this paper is to study the ppart of motivic cohomology and algebraic Ktheory in characteristic p (we use higher Chow groups as our definition of motivic cohomology). The main theorem states that for a field k of characteristic p, Hi (k, Z(n)) is uniquely pdivisible for i ..."
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Cited by 39 (3 self)
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Abstract. The purpose of this paper is to study the ppart of motivic cohomology and algebraic Ktheory in characteristic p (we use higher Chow groups as our definition of motivic cohomology). The main theorem states that for a field k of characteristic p, Hi (k, Z(n)) is uniquely pdivisible for i ̸ = n. This implies that the natural map KM n (k) − → Kn(k) from Milnor Ktheory to Quillen Ktheory is an isomorphism up to uniquely pdivisible groups, and that Kn(k) is ptorsion free. As a consequence, one can calculate the Ktheory mod p of smooth varieties over perfect fields of characteristic p in terms of cohomology of logarithmic de Rham Witt sheaves, for example Kn(X, Z/pr) = 0 for n> dimX. Another consequence is Gersten’s conjecture with mod pcoefficients for smooth varieties over discrete valuation rings with residue characteristic p. As the last consequence, Bloch’s cycle complexes localized at p satisfy all BeilinsonLichtenbaumMilne axioms for motivic complexes, except the vanishing conjecture. 1.
Twoprimary algebraic Ktheory of rings of integers in number fields
, 1997
"... We relate the algebraic Ktheory of a totally real number field F to its étale cohomology. We also relate it to the zetafunction of F when F is Abelian. This establishes the twoprimary part of the “Lichtenbaum conjectures.” To do this we compute the twoprimary Kgroups of F and of its ring of i ..."
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Cited by 38 (8 self)
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We relate the algebraic Ktheory of a totally real number field F to its étale cohomology. We also relate it to the zetafunction of F when F is Abelian. This establishes the twoprimary part of the “Lichtenbaum conjectures.” To do this we compute the twoprimary Kgroups of F and of its ring of integers, using recent results of Voevodsky and the Bloch–Lichtenbaum spectral sequence, modified for finite coefficients in an appendix. A second appendix, by M. Kolster, explains the connection to the zetafunction and Iwasawa theory.
The BlochKato conjecture and a theorem of SuslinVoevodsky
 J. reine angew. Math
"... Abstract. We give a new proof of the theorem of SuslinVoevodsky which shows that the BlochKato conjecture implies a portion of the BeilinsonLichtenbaum conjectures. Our proof does not rely on resolution of singularities, and thereby extends the SuslinVoevodsky to positive characteristic. 1. ..."
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Cited by 32 (4 self)
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Abstract. We give a new proof of the theorem of SuslinVoevodsky which shows that the BlochKato conjecture implies a portion of the BeilinsonLichtenbaum conjectures. Our proof does not rely on resolution of singularities, and thereby extends the SuslinVoevodsky to positive characteristic. 1.
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).
Motivic Cohomology Of Smooth Geometrically Cellular Varieties
, 1997
"... . We construct spectral sequences converging to the motivic cohomology of a smooth variety X over a eld F of characteristic 0. In case X is geometrically cellular, i.e. has a cellular decomposition over the algebraic closure of F , the spectral sequences take an especially simple form. Introduction ..."
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Cited by 11 (7 self)
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. We construct spectral sequences converging to the motivic cohomology of a smooth variety X over a eld F of characteristic 0. In case X is geometrically cellular, i.e. has a cellular decomposition over the algebraic closure of F , the spectral sequences take an especially simple form. Introduction Let F be a eld and X a projective homogeneous variety over F , i.e. a smooth projective variety whose geometric bre is isomorphic to the quotient G=P of a reductive group G by a parabolic subgroup P . Let K be the function eld of X. A basic question is the study of the maps H n+1 (F; Z=m(n)) n ! H n+1 nr (K=F; Z=m(n)): Here m is an integer prime to the characteristic of F , the cohomology is Galois cohomology and the index nr denotes unramied cohomology. The classical strategy to study n can be described as follows: 1. Over a separable closure F s of F , the variety X admits a cellular decomposition. In particular, the cycle map from its Chow ring modulo m to its etale...
On the Grayson spectral sequence
 Chisel, Algebra i Algebr. Geom.):218–253
, 2003
"... The main purpose of these notes is to show that Grayson’s motivic cohomology coincides with the usual definition of motivic cohomology see [V2, SV] for example and hence Grayson’s spectral sequence [Gr] for a smooth semilocal scheme X essentially of finite type over a field F takes the form ..."
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Cited by 10 (0 self)
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The main purpose of these notes is to show that Grayson’s motivic cohomology coincides with the usual definition of motivic cohomology see [V2, SV] for example and hence Grayson’s spectral sequence [Gr] for a smooth semilocal scheme X essentially of finite type over a field F takes the form