The homotopy coniveau tower

Abstract

We examine the ‘homotopy coniveau tower’ for a general cohomology theory on smooth k-schemes and give a new proof that the layers of this tower for K-theory agree with motivic cohomology. In addition, we show that the homotopy coniveau tower agrees with Voevodsky's slice tower for S1-spectra, giving a proof of a connectedness conjecture of Voevodsky. The homotopy coniveau tower construction extends to a tower of functors on the Morel–Voevodsky stable homotopy category, and we identify this ℙ1-stable homotopy coniveau tower with Voevodsky's slice tower for ℙ1-spectra. We also show that the zeroth layer for the motivic sphere spectrum is the motivic cohomology spectrum, which gives the layers for a general ℙ1-spectrum the structure of a module over motivic cohomology. This recovers and extends results of Voevodsky on the zeroth layer of the slice filtration, and yields a spectral sequence that is reminiscent of the classical Atiyah–Hirzebruch spectral sequence.