Cohomological Hasse principle for schemes over valuation rings of higher dimensional 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.