Geodesic
Non-Abelian $K$-theory for Chevalley groups over rings
Zapiski Nauchnykh Seminarov POMI, Problems in the theory of representations of algebras and groups. Part 26, Tome 423 (2014), pp. 244-263

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We announce some results on the structure of Chevalley groups $G(R)$ over a commutative ring $R$ recently obtained by the author. The following results are generalized and improved: (1) Relative local-global principle. (2) Generators of relative elementary subgroups. (3) Relative multi-commutator formulas. (4) Nilpotent structure of relative $\mathrm K_1$. (5) Boundedness of commutator length. \noindent The proof of first two items is based on computations with generators of the elementary subgroups translated into the language of parabolic subgroups. For the proof of the further ones we enlarge the relative elementary subgroup, construct a generic element, and use localization in a universal ring. 
@article{ZNSL_2014_423_a11,
     author = {A. V. Stepanov},
     title = {Non-Abelian $K$-theory for {Chevalley} groups over rings},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {244--263},
     publisher = {mathdoc},
     volume = {423},
     year = {2014},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2014_423_a11/}
}
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A. V. Stepanov. Non-Abelian $K$-theory for Chevalley groups over rings. Zapiski Nauchnykh Seminarov POMI, Problems in the theory of representations of algebras and groups. Part 26, Tome 423 (2014), pp. 244-263. http://geodesic.mathdoc.fr/item/ZNSL_2014_423_a11/