The homotopy coniveau tower

By M. Levine. 2005

135 citations recorded by the source corpus.

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.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/6cf2650c0348fe22d721d009658ba1c2ebad97c3

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

A commutative P^1-spectrum representing motivic cohomology over Dedekind domains (2012, methodology): “We finally remark that it should be possible to generalize our strictification process in section 5 to define a homotopy coniveau tower over Dedekind domains as in [28].”
Mixed Weil cohomologies (2007, background): “that the map HБ → HQ is an isomorphism in DA1(S,Q): this follows from the coniveau spectral sequence of the K-theory spectrum, which degenerates rationally; see [21,37].”
Localization theorems in topological Hochschild homology and topological cyclic homology (2008, background): “Recent work of Levine [27] axiomatizes the role of localization and Mayer-Vietoris theorems in the construction of the motivic spectral sequence [2, 16], and such theorems for TR should provide key input to the construction of this conjectural “additive” motivic spectral sequence.”
Motivic slices and coloured operads (2010, methodology): “Over a perfect field, s01 = MZ by the work of Levine [17] and Voevodsky [33].”
Algebraic K‐theory, A1‐homotopy and Riemann–Roch theorems (2009, result): “It follows from known results (see [28]) that ch:HБ → HQ is an isomorphism; equivalently, cht:BGLQ → ⊕i∈ZHQ(i)[2i] is an isomorphism.”
The first stable homotopy groups of motivic spheres (2016, background): “The map s0(1Λ)→ s0(KGLΛ) is an isomorphism by [32]; see also [64, Th.”
Slices of hermitian $K$–theory and Milnor's conjecture on quadratic forms (2013, methodology): “Geometry & Topology, Volume 20 (2016) Theorem 4.1 (Levine, Voevodsky) The unit map 1 !”
RELATIVE CYCLES WITH MODULI AND REGULATOR MAPS (2014, background): “pe over k. When Xis smooth and quasi-projective, S.Bloch’s apparently na¨ıve deﬁnition of Higher Chow groups, given in terms of algebraic cycles, provides the right answer, as established in [17] and [32]. In larger generality, motivic 2010 Mathematics Subject Classiﬁcation. Primary 14C25; Secondary”
The generalized slices of Hermitian K‐theory (2016, background): “Work of Voevodsky [ 30 ] and Levine [ 9 ] shows that the explicit deﬁnitions of motivic cohomology mentioned above agree with the abstract deﬁnition H μ Z = s 0 S .”