Localization–completion strikes again: Relative K1K1 is nilpotent by abelian

Let GG and EE stand for one of the following pairs of groups:• Either GG is the general quadratic group U(2n,R,Λ)U(2n,R,Λ), n≥3n≥3, and EE its elementary subgroup EU(2n,R,Λ), for an almost commutative form ring (R,Λ)(R,Λ),• or GG is the Chevalley group G(Φ,R)G(Φ,R) of type ΦΦ, and EE its elementary subgroup E(Φ,R)E(Φ,R), where ΦΦ is a reduced irreducible root system of rank ≥2≥2 and RR is commutative.Using Bak’s localization–completion method in [A. Bak, Nonabelian KK-theory: The nilpotent class of K1K1 and general stability, KK-Theory 4 (4) (1991) 363–397], it was shown in [R. Hazrat, Dimension theory and nonstable K1K1 of quadratic modules, KK-Theory 514 (2002) 1–35 and R. Hazrat, N. Vavilov, K1K1 of Chevalley groups are nilpotent, J. of Pure and Appl. Algebra 179 (2003) 99–116] that G/EG/E is nilpotent by abelian, when RR 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 KK-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 K1K1 functors, but also the relative K1K1 functors, are nilpotent by abelian.