Centrality of K2-functor revisited

Abstract

Abstract We prove that St ( n , A ) is a crossed module over GL ( n , A ) under a local stable rank condition on an algebra A over a commutative ring. Our proof uses only elementary localization techniques in terms of pro-groups and stability results for K 1 and K 2 . We also prove similar result for the Steinberg group associated with any sufficiently isotropic general linear group constructed by a quasi-finite algebra.