Finite-coefficient K-theory of henselian valued fields and Gersten injectivity

Abstract

Let $W$ be a henselian valuation ring with fraction field $L$, residue field $k$, and value group $Γ_W$. Let $N=\ell^ν$ be invertible in $W$. Choose an ordered $\mathbf Z/N$-basis $B$ of $Γ_W/NΓ_W$. Products of suitable classes define an equivalence of complete filtered spectra $$ \bigoplus_{\substack{J\subseteq B\\J\text{ finite}}} Σ^{|J|}\operatorname{Fil}_{\mathrm{mot}}^{\bullet-|J|} K(k;\mathbf Z/N) \xrightarrow{\simeq} \operatorname{Fil}_{\mathrm{mot}}^{\bullet} K(L;\mathbf Z/N) $$ The summand indexed by $\varnothing$ is the generic restriction map, after the rigidity equivalence $K(W;\mathbf Z/N)\simeq K(k;\mathbf Z/N)$, and is therefore split injective. Independently of this splitting, excision yields a lifting theorem for regular henselian pairs. In particular, this yields finite-coefficient Gersten injectivity for noetherian henselian regular local rings. Applications to completions along regular primes give relative and sometimes nonhenselian examples. More generally, if $P$ is a Prüfer ring and $R$ is a henselian local ind-smooth $P$-algebra, then $R$ is a domain and $K_n(R;\mathbf Z/N)\to K_n(\operatorname{Frac}(R);\mathbf Z/N)$ is injective for every $n$, provided $N\in R^\times$. These injectivity consequences extend from prime-power to arbitrary finite invertible coefficients by primary decomposition.