English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

K. Kato's conjecture about the cohomological Hasse principle for regularconnected schemes $\mathfrak X$ which are flat and proper over the completediscrete 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 fieldsto the case of all higher local fields. For that purpose a $p$-alterationtheorem for the local uniformization of schemes over valuation rings ofarbitrary finite rank and a corresponding Bertini theorem is developedextending 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 thatcertain motivic cohomology groups of varieties over higher local fields arefinite. This is one of the rare cases where such a result could be shown forschemes without finite or separably closed residue fields. Furthermore, it willbe 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 areproper over a higher local field $F_N$ with final residue characteristic $p$.This generalizes results of S. Bloch, K. Kato, U. Jannsen, S. Saito fromvarieties over finite and local fields to varieties over higher local fields,both of arbitrary dimensions.

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