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 ΓW. Let N=ℓν be invertible in W. Choose an ordered Z/N-basis B of ΓW/NΓW. Products of suitable classes define an equivalence of complete filtered spectra ⨁substackJ⊆ B J finite Σ|J|Filmot•−|J| K(k;Z/N) xrightarrow≃ Filmot• K(L;Z/N) The summand indexed by ∅ is the generic restriction map, after the rigidity equivalence K(W;Z/N)≃ K(k;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 Kn(R;Z/N)→ Kn(Frac(R);Z/N) is injective for every n, provided N∈ R×. These injectivity consequences extend from prime-power to arbitrary finite invertible coefficients by primary decomposition.
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BibTeX
@misc{feld2026finitecoefficient,
title = {{Finite-coefficient K-theory of henselian valued fields and Gersten injectivity}},
author = {Niels Feld},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2608.18789},
eprint = {2608.18789},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.18789},
}
RIS
TY - UNPB
TI - Finite-coefficient K-theory of henselian valued fields and Gersten injectivity
AU - Niels Feld
PY - 2026
DA - 2026/11/05
PB - arXiv
DO - 10.48550/arxiv.2608.18789
UR - https://arxiv.org/abs/2608.18789
AB - Let W be a henselian valuation ring with fraction field L, residue field k, and value group Γ_W. Let N=ℓ^ν be invertible in W. Choose an ordered Z/N-basis B of Γ_W/NΓ_W. Products of suitable classes define an equivalence of complete filtered spectra ⨁_substackJ⊆ B J finite Σ^|J|Fil_mot^•−|J| K(k;Z/N) xrightarrow≃ Fil_mot^• K(L;Z/N) The summand indexed by ∅ is the generic restriction map, after the rigidity equivalence K(W;Z/N)≃ K(k;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;Z/N)→ K_n(Frac(R);Z/N) is injective for every n, provided N∈ R^×. These injectivity consequences extend from prime-power to arbitrary finite invertible coefficients by primary decomposition.
ER -