Semilocal Milnor K-theory

In this paper, semilocal Milnor $K$-theory of fields is introduced and studied. A strongly convergent spectral sequence relating semilocal Milnor $K$-theory to semilocal motivic cohomology is constructed. In weight 2, the motivic cohomology groups $H_{Zar}^p(k,\mathbb{Z}(2))$, $p\le 1$, are computed as semilocal Milnor $K$-theory groups ${\widehat{K}}_{2,3-p}^M(k)$. The following applications are given: (i) several criteria for the Beilinson-Soul'e Vanishing Conjecture; (ii) computation of $K_4$ 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.