Relative Milnor $K$-groups and differential forms of split nilpotent extensions

Abstract

Let $R$ be a commutative ring and $I\subset R$ be a nilpotent ideal such that the quotient $R/I$ splits out of $R$ as a ring. Let $N$ be a natural number such that $I^N=0$. We establish a canonical isomorphism between the relative Milnor $K$-group $K_{n+1}^M(R,I)$ and the quotient of the relative module of differential forms ${\Omega }_{R,I}^n/d{\Omega }_{R,I}^{n-1}$ assuming that $N!$ is invertible in $R$ and that the ring $R$ is weakly $5$-fold stable. The latter means that any $4$-tuple of elements in $R$ can be shifted by an invertible element to become a $4$-tuple of invertible elements.

37 pages