K-theory and motivic cohomology of schemes, by Marc Levine

We examine the properties of the niveau tower studied in our earlier paper "Techiniques of localization in the theory of algebraic cycles". The various properties of Bloch's cycle complexes extend to the terms in the tower, giving the homotopy property, functoriality, products, and a related etale theory for the associated spectral sequence. The conivea tower for a regular scheme admits lambda-operations, which in turn endows the spectral sequence with Adams operations, acting by k^q on the E_2-terms. The K-theory spectral sequence degenerates after inverting "small" primes, and the G-theory sequence degenerates rationally. The filtration on K-theory induced by the coniveau tower contains the gamma-filtration, and agrees with the gamma filtration up to small torsion. Comparing the mod n motivic and etale spectral sequences gives a direct relation of the Beilinson-Lichtenbaum conjectures with the Quillen-Lichtenbaum conjectures. Voevodsky's solution of the Milnor conjecture enables a computation of the 2-local motivic cohomology of finite fields, curves over a finite field, and number rings, with the expected relation with 2-adic etale cohomology. The multiplicative properties of the spectral sequence allow the arguments of Kahn to go through, giving a computation of the 2-primary K-theory of rings of integers.

Marc Levine <marc@neu.edu>