The homotopy coniveau filtration, by Marc Levine

Following the construction of Friedlander-Suslin, we define the "homotopy coniveau tower" for a general cohomology theory on smooth k-schemes. Assuming the cohomology theory satisfies some natural axioms, we show that the homotopy coniveau tower is homotopy invariant, satisfes a localization property, and can be made functorial. We also give an interpretation of the layers in the tower in terms of a generalized version of Bloch's higher Chow groups. In particular, this gives a new proof that the layers of this tower for K-theory agree with motivic cohomology, independent of the results of Bloch-Lichtenbaum or Friedlander-Suslin. We show how these constructions lead to a tower of functors on the Morel-Voevodsky motivic stable homotopy category, and identify this stable homotopy coniveau tower with Voevodsky's slice filtration. 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. For the K-theory P^1-spectrum, one recovers the Bloch-Lichtenbaum/Friedlander-Suslin spectral sequence.

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