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Izv. RAN. Ser. Mat., 2018, Volume 82, Issue 5, Pages 23–60 (Mi izv8762)  

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

S. O. Gorchinskiyab, D. N. Tyurinb

a Steklov Mathematical Institute of Russian Academy of Sciences, Moscow
b National Research University Higher School of Economics, Moscow

Abstract: 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 $N\ge 1$ 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.

Funding Agency Grant Number
Ministry of Education and Science of the Russian Federation 14.641.31.0001
The authors were partially supported by Laboratory of Mirror Symmetry NRU HSE, RF Government grant no. 14.641.31.0001.


DOI: https://doi.org/10.4213/im8762

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English version:
Izvestiya: Mathematics, 2018, 82:5, 880–913

Bibliographic databases:

Document Type: Article
UDC: 512.6+512.7
MSC: 19D45, 13F25
Received: 29.01.2018

Citation: S. O. Gorchinskiy, D. N. Tyurin, “Relative Milnor $K$-groups and differential forms of split nilpotent extensions”, Izv. RAN. Ser. Mat., 82:5 (2018), 23–60; Izv. Math., 82:5 (2018), 880–913

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