A possible new approach to the motivic spectral sequence for algebraic K-theory

Despite the considerable progress in motivic cohomology and motivic homotopy theory achieved in recent years we still do not have a simple construction of the spectral sequence relating motivic cohomology and algebraic K-theory. The construction invented by Dan Grayson (see [3]) is simple and elegant but we are still unable to identify the E2-term of the resulting spectral sequence with the motivic cohomology groups. The approach pioneered by Spencer Bloch and Steven Lichtenbaum in [1] and further developed by Eric Friedlander and Andrei Suslin in [2] gives a spectral sequence of the required form but is technically and conceptually very involved. In [9] we suggested a different approach to this problem. Its first ingredient is a construction of a canonical Postnikov tower for any motivic spectrum E. The quotients of this tower si(E) are called the slices of E and, by construction, there is a spectral sequence whose E2-term is given by the cohomology theories represented by the slices and which attempts to