Higher local duality in Galois cohomology

A field $K$ is quasi-classical $d$-local if there exist fields $K=k_d,\dots ,k_0$ with $k_{i+1}$ Henselian admissible discretely valued with residue field $k_i$, and $k_0$ quasi-finite. We prove a duality theorem for the Galois cohomology of such $K$ with many coefficients, including finite coefficients of any order. Previously, such duality was only known in few cases : as a perfect pairing of finite groups for finite coefficients prime to $\mathrm{c}\mathrm{h}\mathrm{a}\mathrm{r}{k}_0$ in general, or for any finite coefficients when $k_1$ is $p$-adic ; or as a perfect pairing of locally compact Hausdorff groups for the $\mathrm{f}\mathrm{p}\mathrm{p}\mathrm{f}$ cohomology of finite group schemes when $K$ is local. With no obvious reasonable topology available, we abandon perfectness altogether and instead obtain nondegenerate pairings of abstract abelian groups. This is done with new diagram-chasing results for pairings of torsion groups, allowing a dévissage approach which reduces our results to the study of $K_r^M(K)\mathrm{/}p\times H_p^{d+1-r}(K)\to \mathbb{Z}\mathrm{/}p$ using results of Kato.