On the cohomological dimension of Henselian valued fields
Abstract
Let (K,v) be a Henselian valued field of rank one and of mixed characteristic (0,p), with value group Γ dense in ℝ and residue field k. We compute all the graded pieces of the filtration induced by v on the Milnor K-theory modulo p of K, in terms of the exterior powers of the 𝔽p-vector space Γ/pΓ and of the modules of differential forms of k. This extends to the non-discrete setting a classical theorem of Kato concerning complete discretely valued fields. We also prove that, when (K,v) is maximally complete, no class escapes this filtration by accumulating at a limit. As an application, we prove the inequality dimp(K)≥dim𝔽p(Γ/pΓ)+dimp(k) between Kato's p-dimensions, and we show that it becomes an equality as soon as (K,v) is maximally complete.
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BibTeX
@misc{izquierdo2026cohomological,
title = {{On the cohomological dimension of Henselian valued fields}},
author = {Diego Izquierdo and Giancarlo Lucchini Arteche and Gonzalo Manzano Flores},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.37961},
eprint = {2610.37961},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.37961},
}
RIS
TY - UNPB
TI - On the cohomological dimension of Henselian valued fields
AU - Diego Izquierdo
AU - Giancarlo Lucchini Arteche
AU - Gonzalo Manzano Flores
PY - 2026
DA - 2026/10/27
PB - arXiv
DO - 10.48550/arxiv.2610.37961
UR - https://arxiv.org/abs/2610.37961
AB - Let (K,v) be a Henselian valued field of rank one and of mixed characteristic (0,p), with value group Γ dense in ℝ and residue field k. We compute all the graded pieces of the filtration induced by v on the Milnor K-theory modulo p of K, in terms of the exterior powers of the 𝔽_p-vector space Γ/pΓ and of the modules of differential forms of k. This extends to the non-discrete setting a classical theorem of Kato concerning complete discretely valued fields. We also prove that, when (K,v) is maximally complete, no class escapes this filtration by accumulating at a limit. As an application, we prove the inequality dim_p(K)≥dim_𝔽_p(Γ/pΓ)+dim_p(k) between Kato's p-dimensions, and we show that it becomes an equality as soon as (K,v) is maximally complete.
ER -