I am trying to understand the proof of the Geisser-Levine theorem (Thm 8.4 here ) which claims that for a smooth variety $X$ over a perfect field of characteristic $p$ we have an isomorphism $$H^s(X, W_r \Omega^t_{log} )\cong H^{s+t}(X,\mathbb{Z}/p^r(t) ).$$ The proof of this ought to come from a local to global spectral sequence for motivic cohomology, but I'm not sure how. For the $r=1$ case, I think the argument is that since we have Zariski descent for both sides, it suffices to study the case $X=\text{Spec } R$, where $R$ is the localisation of a regular local ring at some prime ideal. Then you use the exact sequences $$0\to \Omega^n_{log} (R)\to \bigoplus_{x\in R^{(0)}} \Omega^n_{log}(k(x))\to \bigoplus_{x\in R^{(1)}} \Omega^{n-1}_{log}(k(x)) $$ and $$0\to H^i(R,\mathbb{Z}/p(n))\to \bigoplus_{x\in R^{(0)}}H^i(k(x),\mathbb{Z}/p(n)) \to \bigoplus_{x\in R^{(1)}}H^{i-1}(k(x),\mathbb{Z}/p(n))$$ to reduce the proposition to the case where $R$ is a field, where Bloch-Gabber-Kato tells us that both of these are isomorphic to Milnor $K$-theory (mod $p$) .
Is that the argument? And does that generalize to the $r>1$ case?
I'd also be interested if there are other proofs which do not use Gersten resolution and the intermediary of Milnor $K$-theory.
