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 $\Gamma$ dense in $\mathbb{R}$ and residue field $k$. We compute all the graded pieces of the filtration induced by $v$ on the Milnor $\mathrm{K}$-theory modulo $p$ of $K$, in terms of the exterior powers of the ${\mathbb{F}}_p$-vector space $\Gamma /p\Gamma$ 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 ${\dim }_p(K)\ge {\dim }_{{\mathbb{F}}_p}(\Gamma /p\Gamma )+{\dim }_p(k)$ between Kato's $p$-dimensions, and we show that it becomes an equality as soon as $(K,v)$ is maximally complete.

arXiv comment42 pages. Comments welcome :)