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 [7], it was shown in [18] and [19] 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 [34], which is applicable to our situation [11], and carry over the results in [18] and [19] 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.