Techniques of localization in the theory of algebraic cycles

Abstract. We extend the localization techniques developed by Bloch to sim-plicial spaces. As applications, we give an extension of Bloch’s localization theorem for the higher Chow groups to schemes of finite type over a regu-lar scheme of dimension at most one (including mixed characteristic) and, relying on a fundamental result of Friedlander-Suslin, we globalize the Bloch-Lichtenbaum spectral sequence to give a spectral sequence converging to the G-theory of a scheme X, of finite type over a regular scheme of dimension one, with E1-term the motivic Borel-Moore homology. 1