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:

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

- 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?
- If the answer to $(1)$ is "yes", what is the currently known generality?
- 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.