Grigory Garkusha: Semi-local Milnor K-Theory

In this talk I will tell about semilocal Milnor K-theory of fields. A strongly convergent spectral sequence relating semilocal Milnor K-theory to semilocal motivic cohomology is constructed. In weight 2, the motivic cohomology groups HpZar(k,ℤ(2)), p≤1, are computed as semilocal Milnor K-theory groups KˆM2,3−p(k). The following applications are given: (i) several criteria for the Beilinson-Soulé Vanishing Conjecture; (ii) computation of K4 of a field; (iii) the Beilinson conjecture for rational K-theory of fields of prime characteristic is shown to be equivalent to vanishing of rational semilocal Milnor K-theory.