Localization-completion strikes again : relative K1 is nilpotent by abelian

Let G and E stand for one of the following pairs of groups: • Either G is the general quadratic group U (2n, R, Λ), n ≥ 3, and E its elementary subgroup EU (2n, R, Λ), for an almost commutative form ring (R, Λ), • or G is the Chevalley group G (Φ, R) of type Φ, and E its elementary subgroup E (Φ, R), where Φ is a reduced irreducible root system of rank ≥ 2 and R is commutative. Using Bak's localization-completion method in [A. Bak, Nonabelian K-theory: The nilpotent class of K1 and general stability, K-Theory 4 (4) (1991) 363-397], it was shown in [R. Hazrat, Dimension theory and nonstable K1 of quadratic modules, K-Theory 514 (2002) 1-35 and R. Hazrat, N. Vavilov, K1 of Chevalley groups are nilpotent, J. of Pure and Appl. Algebra 179 (2003) 99-116] that G / E is nilpotent by abelian, when R has finite Bass-Serre dimension. In this note, we combine localization-completion with a version of Stein's relativization [M.R. Stein, Relativizing functors on rings and algebraic K-theory, J. Algebra 19 (1) (1971) 140-152], which is applicable to our situation [A. Bak, N. Vavilov, Structure of hyperbolic unitary groups I, Elementary subgroups, Algebra Colloq. 7 (2) (2000) 159-196], and carry over the results in the latter of the two references cited above to the relative case. In other words, we prove that not only absolute K1 functors, but also the relative K1 functors, are nilpotent by abelian.