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 ${\Gamma }_W$. Let $N={\ell }^{\nu }$ be invertible in $W$. Choose an ordered $\mathbf{Z}/N$-basis $B$ of ${\Gamma }_W/N{\Gamma }_W$. Products of suitable classes define an equivalence of complete filtered spectra $$\underset{\begin{array}{c}J\subseteq B\\ J\text{ finite}\end{array}}{⨁}{\Sigma }^{|J|}{\mathrm{Fil}}_{\mathrm{mot}}^{\bullet -|J|}K(k;\mathbf{Z}/N)\stackrel{\simeq }{\to }{\mathrm{Fil}}_{\mathrm{mot}}^{\bullet }K(L;\mathbf{Z}/N)$$ 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 $K_n(R;\mathbf{Z}/N)\to K_n(\mathrm{Frac}(R);\mathbf{Z}/N)$ 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.

42 pages