Techniques of localization in the theory of algebraic cycles, by Marc Levine

We extend the technique used by Bloch in proving the localization property for cycle complexes to give similar results for simplicial spaces. As an application, we extend the localization property for Bloch's cycle complexes to mixed characteristic. We also apply the machinery to the G-theory niveau tower for the cosimplicial scheme X x Delta, proving a localization property for the terms in the tower. Combined with the fundamental reinterpretation of the Bloch-Lichtenbaum spectral sequence due to Friedlander and Suslin, this allows one to extend the Bloch-Lichtenbaum spectral sequence to a spectral sequence from motivic Borel-Moore homology to G-theory, for a scheme which is essentially of finite type over a regular base scheme of dimension at most one, including mixed characteristic. For a regular scheme, this gives a spectral sequence from motivic cohomology (defined in terms of Bloch's higher Chow groups) to K-theory.

Marc Levine <>