Generalization of the Artin-Hasse logarithm for the Milnor $K$-groups of $\delta$-rings

Abstract

Let $R$ be a $p$-adically complete ring equipped with a $\delta$-structure. We construct a functorial group homomorphism from the Milnor $K$-group $K^{M}_{n}(R)$ to the quotient of the $p$-adic completion of the module of differential forms $\widehat{\Omega}^{n-1}_{R}/d\widehat{\Omega}^{n-2}_{R}$. This homomorphism is a $p$-adic analogue of the Bloch map defined for the relative Milnor $K$-groups of nilpotent extensions of rings of nilpotency degree $N$ for which the number $N!$ is invertible. Bibliography: 12 titles.