1
$\begingroup$

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?

Improve this question
$\endgroup$
4
  • 1
    $\begingroup$ It's constructed in ``Etale motives" by Cisinski and Deglise, proposition 7.1.6. Here's a link to the arxiv version. $\endgroup$
    – Eoin
    Commented Mar 13, 2018 at 1:49
  • 1
    $\begingroup$ 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. $\endgroup$
    – Eoin
    Commented Mar 13, 2018 at 1:55
  • $\begingroup$ 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.:) $\endgroup$
    – Mikhail Bondarko
    Commented Mar 13, 2018 at 13:12
  • $\begingroup$ 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. $\endgroup$
    – Matthias Wendt
    Commented Mar 13, 2018 at 14:29

0

Reset to default

You must log in to answer this question.