Appears in collection : A conference in honor of Arthur Ogus on the occasion of his 70th birthday

Let $R$ be an $n$-dimensional excellent henselian local domain of mixed characteristic $(0,p)$ with residue field $k$ of $p$-rank $r$. Let $X$ be the punctured spectrum of $R$ and $j$ the inclusion of $X-V(p)$ in $X$. Using work with Orgogozo I show that the $p$-cohomological dimension of $X$ is $\dim_{p}(k)+2n-1$ and study the top cohomologies. In particular I construct isomorphisms for normal $R$ $$ H^{2n+r}(X,j_{!}\mathbb{Z}/p^{s}(n+r))\xrightarrow{\sim}H^{1}(k,\ W_{s}\Omega_{\log}^{r}) $$ which generalize a result of Kato in dimension 1. I also discuss facts about prime to $p$ cohomology and multiplicities associated to certain local ring maps.