K-theory of division algebras over local fields

This is a report on joint work with Michael Larsen and Ayelet Lindenstrauss.

Let $K$ be complete discrete valuation field with finite residue field of characteristic $p > 0$, and let $D$ be a central division algebra of finite dimension $d$ over $K$. We show that for every positive integer $i$, there exists an isomorphism of $p$-adic $K$-groups $$K_i(D,\mathbb Z_p) \xrightarrow{ \operatorname{Nrd}_{D/K} } K_i(K,\mathbb Z_p)$$ such that $d \operatorname{Nrd}_{D/K} = N_{D/K}$ is the norm. The case where $p$ does not divide $d$ was proved thirty years ago by Suslin and Yufryakov, and the new ingredient that makes it possible now to prove the general case is the cyclotomic trace map.