## THE SPECTRAL SEQUENCE RELATING ALGEBRAIC K-THEORY TO MOTIVIC COHOMOLOGY

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### BibTeX

@MISC{Friedlander_thespectral,

author = {Eric M. Friedlander and Andrei Suslin},

title = {THE SPECTRAL SEQUENCE RELATING ALGEBRAIC K-THEORY TO MOTIVIC COHOMOLOGY},

year = {}

}

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### Abstract

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.