Bloch-Kato conjecture for Z/2-coefficients and algebraic Morava K-theories

In this paper we show that the existence of algebro-geometrical analogs of the higher Morava K-theories satisfying some basic properties would imply the Bloch-Kato conjecture with Z/2-coefficients for fields which admit resolution of singularities (see [2] for a precise formulation of this condition). Our approach is inspired by two different ideas. The first is the use of algebraic K-theory and norm varieties in the proof of Bloch-Kato conjecture with Z/2-coefficients in weight three given by A. Merkurjev and A. Suslin in [4] and independently by M. Rost in [6]. The second is the “chromatic” approach to algebraic topology which was developed by Jack Morava, Mike Hopkins, Douglas Ravenel and others. The Bloch-Kato conjecture in its original form asserts that for any field k and any prime l not equal to char(k) the canonical homomorphisms K M(k)/l → H n et(k, μ ⊗n l ) Preliminary version. June 1995.