This almost final version of the authors' extension of the spectral sequence exhibited by Spencer Bloch and Steve Lichtenbaum differs from 0360 in that many "well known" issues are considered in some detail (for example, strict functoriality of K-theory spaces, multiplicative structures of exact couples, etc.)

Beginning with the Bloch-Lichtenbaum exact couple relating the motivic cohomology (i.e., Bloch's higher Chow groups) of a field F to the algebraic K-theory of F, the authors construct a spectral sequence for any smooth scheme X of finite type over F whose E-2 term is the motivic cohomology of X and whose abutment is the Quillen K-theory of X. A multiplicative structure is exhibited on this spectral sequence. For not necessarily smooth schemes, a similar spectral sequence is constructed which converges to the Quillen K-theory of coherent sheaves on X.

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Eric M. Friedlander <eric@math.nwu.edu>

Andrei Suslin <suslin@math.nwu.edu>