Semilocal Milnor K-Theory - Grigory Garkusha (International Mathematics Research Notices, 2023)
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 HpZar(k,Z(2))⁠, p⩽1⁠, are computed as semilocal Milnor K-theory groups ˆK M2,3−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.