Аннотация:
I will tell about the connection between relative Milnor $K$-groups $K^{M}_{n}(R,I)$ of a split nilpotent extension $(R,I)$ and it's modules of Kahler differential forms. Namely, let $R$ be a ring and $I\subset R$ be its nilpotent ideal such that $R/I$ splits and $I^{N}=0$ for some natural $N$ such that $N!$ is invertible in $R$. I will show that in this case there exists functorial homomorphism called Bloch map (since it was originally developed by S.Bloch) from relative Milnor $K$–group $K^{M}_{n+1}(R,I)$ to the quotient group $\Omega^n_{R,I}/d\,\Omega^{n-1}_{R,I}$ wich can be considered as a canonical integral for the map $d\log$. Moreover, under the assumption that $R$ is weakly $5$-fold stable this map is an isomorphism.
If time permits I will also tell about a particulary interesting geleralization of the Bloch map $B$ to the case of $p$-adically complete ring $R$ with a $\delta$-structure.