Structure of Chevalley groups over rings via universal localization.

In the current article we study the structure of a Chevalley group G ( R ) over a commutative ring R . We generalize and improve the following results on: • the standard, relative, and multi-relative commutator formulas; • the nilpotent structure of [relative] K 1 ; • the bounded word length of commutators. To this end we enlarge the elementary group, construct a generic element for the extended elementary group, and use localization in the universal ring. The key step is a construction of a generic element for the principal congruence subgroup, corresponding to a principal ideal. [ABSTRACT FROM AUTHOR]