Results 1  10
of
33
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 ..."
Abstract

Cited by 45 (5 self)
 Add to MetaCart
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 ..."
Abstract

Cited by 21 (2 self)
 Add to MetaCart
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
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 ..."
Abstract

Cited by 13 (0 self)
 Add to MetaCart
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
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 ..."
Abstract

Cited by 10 (0 self)
 Add to MetaCart
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
Motivic cohomology are isomorphic to higher Chow groups
, 1999
"... In this short paper we show that the motivic cohomology groups defined in [3],[4] are isomorphic to the motivic cohomology groups defined in [1, ] for smooth schemes over any field. In view of [1, Corollary 11.2] it implies that motivic cohomology of [3],[4] are isomorphic to higher Chow groups. Bec ..."
Abstract

Cited by 10 (4 self)
 Add to MetaCart
In this short paper we show that the motivic cohomology groups defined in [3],[4] are isomorphic to the motivic cohomology groups defined in [1, ] for smooth schemes over any field. In view of [1, Corollary 11.2] it implies that motivic cohomology of [3],[4] are isomorphic to higher Chow groups. Because of the homotopy invariance property of higher Chow groups this implies that the Cancellation Theorem [4, Theorem 2.1] holds over any field. Both facts were previously known only under the resolution of singularities assumption. The only new element in the proof is Proposition 5. Recall that the motivic complex Z(q) of weight q was defined in [4],[3] as C∗(Ztr(G∧q m))[−q]. In [1, §11] Friedlander and Suslin defined complexes which we will denote ZSF tr (q) as C∗(zequi(Aq, 0))[−2q]. The hypercohomology with coefficients in ZSF tr (q) are shown in [1] to coincide with the higher Chow groups for smooth varieties over all fields. In this paper we prove the following result. Theorem 1 For any field k the complexes of sheaves with transfers Ztr(q)