Paper detail

The homotopy coniveau tower

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, the homotopy coniveau tower agrees with Voevodsky's slice tower for $S^1$-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 $P^1$-stable homotopy coniveau tower with Voevodsky's slice filtration for $P^1$-spectra. We also show that the 0th layer for the motivic sphere spectrum is the motivic cohomology spectrum, which gives the layers for a general $P^1$-spectrum the structure of a module over motivic cohomology. This recovers and extends recent results of Voevodsky on the 0th layer of the slice filtration, and yields a spectral sequence that is reminiscent of the classical Atiyah-Hirzebruch spectral sequence.