Semilocal Milnor K-theory

2020/07/17 by Garkusha, Grigory
#Algebraic Geometry (math.AG) #FOS: Mathematics #K-Theory and Homology (math.KT) #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2007.09044

Abstract

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^p_Zar(k,\mathbb Z(2)), p≤ 1, are computed as semilocal Milnor K-theory groups \widehatK^M_2,3-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.

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