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 K-theory 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 Atiyah-Hirzebruch spectral sequence from the singular cohomology to the topological K-theory 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 K-theory. In the special case in which X equals SpecF, this spectral sequence was established by S. Bloch and S. Lichtenbaum [B-L]. Our construction depends crucially upon the main result of [B-L], the existence of an exact couple relating the motivic cohomology of the field F to the multirelative K-theory 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 K-theory of the standard cosimplicial scheme ∆ • over F. We find that the spectral sequence arises from a tower of Ω-prespectra 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 Bloch-Lichtenbaum 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 K-theory 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.

We relate the algebraic K-theory of a totally real number field F to its étale cohomology. We also relate it to the zeta-function of F when F is Abelian. This establishes the two-primary part of the “Lichtenbaum conjectures.” To do this we compute the two-primary K-groups 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 zeta-function and Iwasawa theory.

Abstract. The purpose of this paper is to study the p-part of motivic cohomology and algebraic K-theory 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 p-divisible for i ̸ = n. This implies that the natural map KM n (k) − → Kn(k) from Milnor K-theory to Quillen K-theory is an isomorphism up to uniquely p-divisible groups, and that Kn(k) is p-torsion free. As a consequence, one can calculate the K-theory 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 p-coefficients for smooth varieties over discrete valuation rings with residue characteristic p. As the last consequence, Bloch’s cycle complexes localized at p satisfy all Beilinson-Lichtenbaum-Milne axioms for motivic complexes, except the vanishing conjecture. 1.

Abstract. We give a new proof of the theorem of Suslin-Voevodsky which shows that the Bloch-Kato conjecture implies a portion of the Beilinson-Lichtenbaum conjectures. Our proof does not rely on resolution of singularities, and thereby extends the Suslin-Voevodsky to positive characteristic. 1.

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

. 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...

Abstract. We provide a patch to complete the proof of the Voevodsky-Rost Theorem, that the norm residue map is an isomorphism. (This settles the motivic Bloch-Kato conjecture).

2. Continuous étale cohomology 6

The main purpose of these notes is to show that Grayson’s motivic cohomology coincides with the usual definition of motivic cohomology- see [V2, S-V] 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

Abstract not found