Centrality of odd unitary $K_2$-functor

Abstract

Let $(R,\Delta )$ be an odd form algebra. We show that the unitary Steinberg group $\mathrm{StU}(R,\Delta )$ is a crossed module over the odd unitary group $\mathrm{U}(R,\Delta )$ in two major cases: if the odd form algebra has a free orthogonal hyperbolic family satisfying a local stable rank condition and if the odd form algebra is sufficiently isotropic and quasi-finite. The proof uses only elementary localization techniques in terms of pro-groups.