On the K-theory of division algebras over local fields

Let $K$ be a complete discrete valuation field with finite residue field of characteristic $p$, and let $D$ be a central division algebra over $K$ of finite index $d$. Thirty years ago, Suslin and Yufryakov showed that for all prime numbers $\ell$ different from $p$ and integers $j \geq 1$ , there exists a "reduced norm" isomorphism of $\ell$-adic $K$-groups $\operatorname{Nrd}_{D/K} \colon K_j(D,\mathbb{Z}_{\ell}) \to K_j(K,\mathbb{Z}_{\ell})$ such that $d \cdot \operatorname{Nrd}_{D/K}$ is equal to the norm homomorphism $N_{D/K}$. The purpose of this paper is to prove the analogous result for the $p$-adic $K$-groups. To do so, we employ the cyclotomic trace map to topological cyclic homology and show that there exists a "reduced trace" equivalence $\operatorname{Trd}_{A/S} \colon \operatorname{THH}(A\,|\,D,\mathbb{Z}_p) \to \operatorname{THH}(S\,|\,K,\mathbb{Z}_p)$ between two $p$-complete cyclotomic spectra associated with $D$ and $K$, respectively. Interestingly, we show that if $p$ divides $d$, then it is not possible to choose said equivalence such that, as maps of cyclotomic spectra, $d \cdot \operatorname{Trd}_{A/S}$ agrees with the trace $\operatorname{Tr}_{A/S}$, although this is possible as maps of spectra with $\mathbb{T}$-action.