Relative unitary commutator calculus, and applications

This note revisits localisation and patching method in the setting of generalised unitary groups. Introducing certain subgroups of relative elementary unitary groups, we develop relative versions of the conjugation calculus and the commutator calculus in unitary groups, which are both more general, and substantially easier than the ones available in the literature. For the general linear group such relative commutator calculus has been recently developed by the first and the third authors. As an application we prove the mixed commutator formula,[EU(2n,I,Γ),GU(2n,J,Δ)]=[EU(2n,I,Γ),EU(2n,J,Δ)],[EU(2n,I,Γ),GU(2n,J,Δ)]=[EU(2n,I,Γ),EU(2n,J,Δ)], for two form ideals (I,Γ)(I,Γ) and (J,Δ)(J,Δ) of a form ring (A,Λ)(A,Λ). This answers two problems posed in a paper by Alexei Stepanov and the second author.