# Coniveau in étale motivic cohomology

Let $X$ be a smooth variety over a field.

Is there a spectral sequence:

$$E_1^{p,q} := \bigoplus_{x\in X^{(p)}}H^{q-p}(\kappa(x)_{\rm ét},\mathbf{Z}(n))\Rightarrow H^{q-p}(X_{\rm ét},\mathbf{Z}(n))$$

with the abutement being étale motivic cohomology?

Here is my guess: since after rationalizing, $H^{r}(*,\mathbf{Q}(n)) = H^r(*_{\rm ét},\mathbf{Q}(n))$, one can invoke the known analog of the above spectral sequence for usual motivic cohomology, rationally.

One is then reduced to show this for mod $\ell^t$ étale motivic cohomology, which is, depending on $\ell$, either $(\mathbf{Z}/\ell^t)(n)$ étale cohomology, or level $p^t$ logarithmic de Rham Witt cohomology.

For both, the result is true.

So my expectation is that this is true and has been shown in the literature. Any reference, please?

• It's constructed in Etale motives" by Cisinski and Deglise, proposition 7.1.6. Here's a link to the arxiv version. – Eoin Mar 13 '18 at 1:49
• Although, I haven't read the paper in really any detail so I would be cautious to say it's the same \'etale motivic cohomology as that constructed by Voevodsky. – Eoin Mar 13 '18 at 1:55
• The reference is fine and the question appears to be easy. However, I would like to note that one cannot obtain anything like de Rham Witt cohomology using Voevodsky theory since he didn't incorporate these things into his framework.:) – Mikhail Bondarko Mar 13 '18 at 13:12
• I think this should drop out of (a suitable hypercohomology generalization of) the general methods in Colliot-Thelene, Hoobler, Kahn: "The Bloch-Ogus-Gabber theorem". The methods there yield the spectral sequence with the right abutment, and the identification of the $E_1$-term could be done using the statements mentioned in the question. – Matthias Wendt Mar 13 '18 at 14:29