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There is an algebraic analogue of the Atiyah-Hirzebruch spectral sequence from singular cohomology to topological $K$-theory of a topological space.

For a field $k$, let $X$ be smooth variety $X$ over $k$.

The following spectral sequence will be referred to in the sequel as the motivic spectral sequence: $$E_2^{i,j} := H^{i-j}(X, \mathbf{Z}(-j)) \Rightarrow K_{-i-j}(X).$$

See:

  1. the Bloch-Lichtenbaum motivic spectral sequence in [BL], and the generalizations by Levine [L] and Friedlander-Suslin [FS] to smooth varieties over $k$.
  2. the Voevodsky motivic spectral sequence [V].
  3. the Grayson motivic spectral sequence [G].

For $k$ and $X$ as in the foregoing, we may form the étale hypercohomology of the Bloch complex $z^{j}(X,\bullet)$ ([B]) on $X_{\rm\acute{e}t}$, denoted $H^{\bullet}_{L}(X, \mathbf{Z}(j))$ and usually called Lichtenbaum cohomology.

Questions:

  1. Is an "étale analogue" of the motivic spectral sequence from the foregoing, i.e.: $$E_2^{i,j} := H_L^{i-j}(X, \mathbf{Z}(j))\Rightarrow K_{-i-j}^{\rm\acute{e}t}(X)$$ available?
  2. If the answer to $(1)$ is "yes", what is the currently known generality?
  3. If the answer to $(1)$ is "yes", references?

References:

[BL] S. Bloch, S. Lichtenbaum, A spectral sequence for motivic cohomology, K-theory, 1995.

[L] M. Levine, Techniques of localization in the theory of algebraic cycles, 2001.

[FS] E. M. Friedlander, A. Suslin, The spectral sequence relating algebraic K-theory to motivic cohomology, 2002.

[V] V. Voevodsky, A possible new approach to the motivic spectral sequence for algebraic K-theory, 2002.

[G] A. Suslin, On the Grayson spectral sequence, 2003.

[B] S. Bloch, Algebraic cycles and Higher $K$-theory, 1986.

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    $\begingroup$ Here is one reference: Levine's math.uiuc.edu/K-theory/336/mot.pdf (see Theorem 12.10). $\endgroup$ – Mikhail Bondarko Aug 2 '17 at 6:56
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    $\begingroup$ Here are some references, which don't answer your question: a survey: math.illinois.edu/K-theory/0981/book/1-039-070.pdf Levine's spectral sequence: math.illinois.edu/K-theory/0628 Also be aware that Marc Levine found a gap in the [BL] argument, which is still unpublished. $\endgroup$ – Dan Grayson Aug 3 '17 at 11:57
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    $\begingroup$ Then, it looks like the only motivic ss currently in the literature, is for the Zariski hypercohomology of Bloch's complex to $K$-theory. For the étale hypercohomology, only a mod $n$ version of the ss the OP asked about, is proved, so far. Am I summarizing correctly? $\endgroup$ – user97068 Aug 3 '17 at 17:32
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  1. Yes. Firstly, I apologize for the self-referencing. But the references for this is a joint paper with Marc Levine (basically finishing up what he had sketched in aforementioned paper), Markus Spitzweck and Paul Arne Ostvaer. It is currently under revision but has been available on the arxiv for sometime: https://arxiv.org/abs/1711.06258

  2. The spectral sequence itself is available over regular schemes since it is an incarnation of the slice spectral sequence in motivic homotopy theory. Over more general bases, we are computing the etale version of Weibel's homotopy K-theory

The paper also addresses convergence issues in extensive, if not terse, details.

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