Duality for higher local fields after Kato and Suzuki

A field $K$ is $d$-local if there exist fields $K=k_d,\mathrm{.}\mathrm{.}\mathrm{.},k_0$ with $k_{i+1}$ complete discrete valuation with residue field $k_i$, and $k_0$ finite of characteristic $p$. By work of Deninger and Wingberg, the Galois cohomology of such fields with finite coefficients satisfies a duality generalizing Tate duality when either $d=0$, $\mathrm{c}\mathrm{h}\mathrm{a}\mathrm{r}{k}_1=0$ or the coefficients have no $p$-torsion. Reviewing and synthesizing results of Suzuki and Kato, we obtain $p$-torsion duality statements under the weaker assumption that either $d\le 1$ or $\mathrm{c}\mathrm{h}\mathrm{a}\mathrm{r}{k}_2=0$, as well as for varieties over $K$, where duality is stated in terms of locally compact Hausdorff topologies on the étale cohomology groups. More generally we obtain results for any perfect $k_0$, endowing the totally unramified cohomology groups of $K$ with the structure of ind-pro-quasi-algebraic $k_0$-groups.