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

S. O. Gorchinskiy ; D. N. Tyurin

Izvestiya. Mathematics , Tome 82 (2018) no. 5, pp. 880-913

Résumé

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

Détail

Keywords: Milnor $K$-groups, differential forms.