2 problems

Let $D=(D,\varphi_D,N,\operatorname{Fil}^\bullet D_K)$ be a weakly admissible filtered $(\varphi,N)$-module such that $\operatorname{gr}^wD_K=0$ for all $w\notin[0,p-1]$. Let…

Let $k$ be a perfect field of characteristic $p$, let $W(k)$ be its ring of Witt vectors, let $K_0=W(k)[1/p]$, and let $K$ be a finite totally ramified extension of $K_0$ with ring…