# Spectral sequences in $K$-theory

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.

• Here is one reference: Levine's math.uiuc.edu/K-theory/336/mot.pdf (see Theorem 12.10). – Mikhail Bondarko Aug 2 '17 at 6:56
• 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math.illinois.edu/K-theory/0628 Also be aware that Marc Levine found a gap in the [BL] argument, which is still unpublished. – Dan Grayson Aug 3 '17 at 11:57
• 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? – user97068 Aug 3 '17 at 17:32