Geodesic
Relative Milnor $K$-groups and differential forms of split nilpotent extensions
Izvestiya. Mathematics , Tome 82 (2018) no. 5, pp. 880-913

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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.
Keywords: Milnor $K$-groups, differential forms. 
@article{IM2_2018_82_5_a1,
     author = {S. O. Gorchinskiy and D. N. Tyurin},
     title = {Relative {Milnor} $K$-groups and differential forms of split nilpotent extensions},
     journal = {Izvestiya. Mathematics },
     pages = {880--913},
     publisher = {mathdoc},
     volume = {82},
     number = {5},
     year = {2018},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_2018_82_5_a1/}
}
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S. O. Gorchinskiy; D. N. Tyurin. Relative Milnor $K$-groups and differential forms of split nilpotent extensions. Izvestiya. Mathematics , Tome 82 (2018) no. 5, pp. 880-913. http://geodesic.mathdoc.fr/item/IM2_2018_82_5_a1/