Cohomological Hasse principle for schemes over valuation rings of higher dimensional local fields

TL;DR

This paper proves Kato's conjecture on the cohomological Hasse principle for schemes over valuation rings of higher local fields, extending previous results to more general fields and establishing finiteness of certain motivic cohomology groups.

Contribution

It generalizes the cohomological Hasse principle to all higher local fields and develops new alteration and Bertini theorems for schemes over valuation rings of arbitrary finite rank.

Findings

Proof of Kato's conjecture for higher local fields.

Finiteness of certain motivic cohomology groups.

Finiteness of kernels of reciprocity and norm maps for schemes over higher local fields.

Abstract

K. Kato's conjecture about the cohomological Hasse principle for regular connected schemes $\mathfrak X$ which are flat and proper over the complete discrete valuation rings $\mathcal O_N$ of higher local fields $F_N$ is proven. This generalizes the work of M. Kerz, S. Saito and U. Jannsen for finite fields to the case of all higher local fields. For that purpose a $p$-alteration theorem for the local uniformization of schemes over valuation rings of arbitrary finite rank and a corresponding Bertini theorem is developed extending the results of O. Gabber, J. deJong, L. Illusie, M. Temkin, S. Saito, U. Jannsen to the non-noetherian world. As an application it is shown that certain motivic cohomology groups of varieties over higher local fields are finite. This is one of the rare cases where such a result could be shown for schemes without finite or separably closed residue fields. Furthermore, it will be derived that the kernels of the reciprocity map $\rho^X : \text{SK}_N(X) \to \pi_1^\text{ab}(X)$ and norm map $N_{X|F}: \text{SK}_N(X) \to K_N^M(F_N)$ modulo maximal $p'$-divisible subgroups are finite for regular $X$ which are proper over a higher local field $F_N$ with final residue characteristic $p$. This generalizes results of S. Bloch, K. Kato, U. Jannsen, S. Saito from varieties over finite and local fields to varieties over higher local fields, both of arbitrary dimensions.