Let $K$ be a $p$-adic field and $K'$ be a finite extension of $K$. Let $\Omega_{(K',\mathcal{O}_{K'})}$ be the sheaf of relative log differentials of the pair $(K',\mathcal{O}_{K'})$ over $\mathcal{O}_{K'}$ with the trivial log structure. Then Beilinson states ('p-adic periods and derived de rham cohomology', p. 726 §3.2) that the cokernel of the canonical map $$\Omega_{\mathcal{O}_{K'}}\rightarrow \Omega_{(K',\mathcal{O}_{K'})}$$ has cokernel isomorphic to the residue field $\mathcal{O}_{K'}/\mathbb{m}_{K'}$. How can this be seen?

When passing to the inductive limit, how can one show that, $\Omega_{\mathcal{O}_{\bar{K}}}\rightarrow \Omega_{(\bar{K},\mathcal{O}_{\bar{K}})}$ is an isomorphim using that $\Omega_{\mathcal{O}_{\bar{K}}}$ is p-divisible? Does one need to show that $\Omega_{(\bar{K},\mathcal{O}_{\bar{K}})}$ is also p-divisible?


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