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Noncommutative geometry, quantum fields and motives
 Colloquium Publications, Vol.55, American Mathematical Society
, 2008
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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 44 (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.
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 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 38 (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.
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 31 (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.
Techniques of localization in the theory of algebraic cycles
 J. Algebraic Geom
"... Abstract. We extend the localization techniques of Bloch to simplicial spaces. As applications, we give an extension of Bloch’s localization theorem for the higher Chow groups to schemes of finite type over a regular scheme of dimension one (including mixed characteristic) and, relying on a fundamen ..."
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Cited by 26 (8 self)
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Abstract. We extend the localization techniques of Bloch to simplicial spaces. As applications, we give an extension of Bloch’s localization theorem for the higher Chow groups to schemes of finite type over a regular scheme of dimension one (including mixed characteristic) and, relying on a fundamental result of FriedlanderSuslin, we globalize the BlochLichtenbaum spectral sequence to give a spectral sequence converging to the Gtheory of a scheme X, of finite type over a regular scheme of dimension one, with E 1term the motivic BorelMoore homology (the same as the higher Chow groups of X, after a reindexing). 0.1. Bloch’s higher Chow groups. We recall Bloch’s definition of the higher Chow groups [1]. Fix a base field k. Let ∆N k denote the standard “algebraic Nsimplex”: = Spec k[t0,..., tN] / ∑