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 $\mathrm{\ell }$ different from $p$ and integers $j\ge 1$ , there exists a “reduced norm” isomorphism of $\mathrm{\ell }$-adic $K$-groups $\operatorname{Nrd}{D/K} \colon K_j(D,$\mathbb{Z}${\ell}) \to K_j(K,$\mathbb{Z}${\ell})$suchthat$d \cdot \operatorname{Nrd}{D/K}$ is equal to the norm homomorphism $N_{D\mathrm{/}K}$. 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}$_p)$betweentwo$p$-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, $d\cdot {\mathrm{Trd}}_{A\mathrm{/}S}$ agrees with the trace ${\mathrm{Tr}}_{A\mathrm{/}S}$, although this is possible as maps of spectra with $\mathbb{T}$-action.