2016/05/26 by Patrick Forré, Forré, Patrick
Mathematics · Pharmacology, Toxicology and Pharmaceutics · #11G25 #11G45 #14E15 #14F42 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #Alkaloids: synthesis and pharmacology #FOS: Mathematics #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.1605.08344
openalex publication_date 2016/05/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
K. Kato's conjecture about the cohomological Hasse principle for regular\nconnected schemes mathfrak X which are flat and proper over the complete\ndiscrete valuation rings mathcal ON of higher local fields FN is proven.\nThis generalizes the work of M. Kerz, S. Saito and U. Jannsen for finite fields\nto the case of all higher local fields. For that purpose a p-alteration\ntheorem for the local uniformization of schemes over valuation rings of\narbitrary finite rank and a corresponding Bertini theorem is developed\nextending the results of O. Gabber, J. deJong, L. Illusie, M. Temkin, S. Saito,\nU. Jannsen to the non-noetherian world. As an application it is shown that\ncertain motivic cohomology groups of varieties over higher local fields are\nfinite. This is one of the rare cases where such a result could be shown for\nschemes without finite or separably closed residue fields. Furthermore, it will\nbe derived that the kernels of the reciprocity map \ρX : \SKN(X) \→\n\π1^\ab(X) and norm map NX|F: \SKN(X) \→ KNM(FN)\nmodulo maximal p'-divisible subgroups are finite for regular X which are\nproper over a higher local field FN with final residue characteristic p.\nThis generalizes results of S. Bloch, K. Kato, U. Jannsen, S. Saito from\nvarieties over finite and local fields to varieties over higher local fields,\nboth of arbitrary dimensions.\n