Milnor $K$-groups of nilpotent extensions

The talk is based on a series of common works with Dimitrii Tyurin and with Denis Osipov. We prove a version of the famous Goodwillie's theorem with algebraic $K$-groups being replaced by Milnor $K$-groups. Namely, given a commutative ring $R$ with a nilpotent ideal $I$, $I^N=0$, such that the quotient $R/I$ splits, we study relative Milnor $K$-groups $K^M_{n+1}(R,I)$, $n\geqslant 0$. Provided that the ring $R$ has enough invertible elements in a sense, these groups are related to the quotient of the module of relative differential forms $\Omega^n_{R,I}/d\,\Omega^{n-1}_{R,I}$. This holds in two different cases: when $N!$ is invertible in $R$ and when $R$ is a complete $p$-adic ring with a lift of Frobenius. However, the approaches and constructions are different in these cases.