Hasse principles for higher-dimensional fields

Abstract

For rather general excellent schemes $X$, K. Kato defined complexes of Gersten-Bloch-Ogus type involving the Galois cohomology groups of all residue fields of $X$. For arithmetically interesting schemes, he developed a fascinating web of conjectures on some of these complexes, which generalize the classical Hasse principle for Brauer groups over global fields, and proved these conjectures for low dimensions. We prove Kato’s conjecture over number fields in any dimension. This gives a cohomological Hasse principle for function fields $F$ over a number field $K$, involving the corresponding function fields $F_v$ over the completions $K_v$ of $K$. For global function fields $K$ we prove the part on injectivity for coefficients invertible in $K$. Assuming resolution of singularities, we prove a similar conjecture of Kato over finite fields, and a generalization to arbitrary finitely generated fields.

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      doi = {10.1007/978-1-4757-4269-5_10},
      zblnumber = {0745.11053},
      }
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    @misc{Ja5,
      author = {Jannsen, Uwe},
      title = {Rigidity theorems for {$\mathcal K$}- and {$\mathcal H$}-homology and other functors},
      year = {2015},
      arxiv = {1503.08742},
      }
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    @article{Ja6, mrkey = {2609323},
      author = {Jannsen, Uwe},
      title = {Weights in arithmetic geometry},
      journal = {Jpn. J. Math.},
      fjournal = {Japanese Journal of Mathematics},
      volume = {5},
      year = {2010},
      number = {1},
      pages = {73--102},
      issn = {0289-2316},
      mrclass = {14G40 (11G25 11G35 14F20)},
      mrnumber = {2609323},
      doi = {10.1007/s11537-010-0947-4},
      zblnumber = {1204.14011},
      }
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      title = {Hasse principles for higher-dimensional fields},
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      arxiv = {0910.2803},
      }
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    @article{JS1, mrkey = {2046606},
      author = {Jannsen, Uwe and Saito, Shuji},
      title = {Kato homology of arithmetic schemes and higher class field theory over local fields},
      note = {Kazuya Kato's fiftieth birthday},
      journal = {Doc. Math.},
      fjournal = {Documenta Mathematica},
      year = {2003},
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      pages = {479--538},
      issn = {1431-0635},
      mrclass = {11G45 (11G25 14F42 19D45)},
      mrnumber = {2046606},
      mrreviewer = {Tam{á}s Szamuely},
      zblnumber = {1092.14504},
      }
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    @article{JS2,
      author = {Jannsen, Uwe and Saito, Shuji},
      mrkey = {2648936},
      title = {Algebraic {$K$}-theory and motivic cohomology},
      note = {abstracts from the workshop held June 28--July 4, 2009, organized by Thomas Geisser, Annette Huber-Klawitter, Uwe Jannsen and Marc Levine},
      journal = {Oberwolfach Rep.},
      fjournal = {Oberwolfach Reports},
      volume = {6},
      year = {2009},
      number = {2},
      pages = {1731--1773},
      issn = {1660-8933},
      mrclass = {14F42 (14-06 14C15)},
      mrnumber = {2648936},
      doi = {10.4171/OWR/2009/31},
      zblnumber = {1177.14016},
      }
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    @article{JSS, mrkey = {3176615},
      author = {Jannsen, Uwe and Saito, Shuji and Sato, Kanetomo},
      title = {\'{E}tale duality for constructible sheaves on arithmetic schemes},
      journal = {J. reine angew. Math.},
      fjournal = {Journal für die Reine und Angewandte Mathematik. [Crelle's Journal]},
      volume = {688},
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      issn = {0075-4102},
      mrclass = {14G40 (14F20)},
      mrnumber = {3176615},
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      booktitle = {Algebraic Cycles and Motives. {V}ol. 2},
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      VOLUME = {24},
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      }

Authors

Uwe Jannsen

Universität Regensburg, Regensburg Germany