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Brauer groups of varieties over local fields of finite characteristic

Amalendu Krishna, Subhadip Majumder

TL;DR

The paper proves that for varieties over positive characteristic local fields, the non-log Kato ramification filtration on the Brauer group ${ m Br}(X)$ coincides with the evaluation filtration, extending Bright–Newton’s char-zero results to characteristic $p$. The authors develop and deploy a comprehensive toolkit—non-log Kato filtrations, a generalized refined Swan conductor, Matsuda filtration for higher cohomology, and the Kato complex—together with Bertini-type theorems to relate ramification, evaluation, and specialization across fibers and hypersurfaces. They derive positive-characteristic analogues of key results due to Ieronymou, Saito–Sato, and Kai, including mixed-characteristic and higher-dimensional Brauer–Manin dualities, and they establish the exhaustiveness and comparability of the evaluation and Matsuda filtrations. The work yields applications to Brauer–Manin obstructions, Albanese dualities for 0-cycles, and finiteness statements, thereby advancing understanding of arithmetic of varieties over positive-characteristic local fields and their Brauer groups. The methodology blends de Rham–Witt theory, Milnor K-theory, Kato complexes, refined ramification theory, and Bertini-type reductions to control ramification and evaluation in broad geometric settings.

Abstract

We show that the non-log version of Kato's ramification filtration on the Brauer group of a separated and finite type regular scheme over a positive characteristic local field coincides with the evaluation filtration. This extends a recent result of Bright-Newton to positive characteristics. Among several applications, we extend some results of Ieronymou, Saito-Sato and Kai to positive characteristics.

Brauer groups of varieties over local fields of finite characteristic

TL;DR

The paper proves that for varieties over positive characteristic local fields, the non-log Kato ramification filtration on the Brauer group coincides with the evaluation filtration, extending Bright–Newton’s char-zero results to characteristic . The authors develop and deploy a comprehensive toolkit—non-log Kato filtrations, a generalized refined Swan conductor, Matsuda filtration for higher cohomology, and the Kato complex—together with Bertini-type theorems to relate ramification, evaluation, and specialization across fibers and hypersurfaces. They derive positive-characteristic analogues of key results due to Ieronymou, Saito–Sato, and Kai, including mixed-characteristic and higher-dimensional Brauer–Manin dualities, and they establish the exhaustiveness and comparability of the evaluation and Matsuda filtrations. The work yields applications to Brauer–Manin obstructions, Albanese dualities for 0-cycles, and finiteness statements, thereby advancing understanding of arithmetic of varieties over positive-characteristic local fields and their Brauer groups. The methodology blends de Rham–Witt theory, Milnor K-theory, Kato complexes, refined ramification theory, and Bertini-type reductions to control ramification and evaluation in broad geometric settings.

Abstract

We show that the non-log version of Kato's ramification filtration on the Brauer group of a separated and finite type regular scheme over a positive characteristic local field coincides with the evaluation filtration. This extends a recent result of Bright-Newton to positive characteristics. Among several applications, we extend some results of Ieronymou, Saito-Sato and Kai to positive characteristics.
Paper Structure (53 sections, 114 theorems, 213 equations)

This paper contains 53 sections, 114 theorems, 213 equations.

Table of Contents

  1. Introduction
  2. The Bright-Newton theorem
  3. Theorems of Ieronymou
  4. The theorem of Saito-Sato
  5. Theorems of Lichtenbaum and Kai
  6. Future works
  7. Outline of proofs
  8. Common notations
  9. Kato's filtration for discrete valued fields
  10. de Rham-Witt forms, Milnor $K$-theory and and the Brauer group
  11. The cohomology groups
  12. The canonical lifting
  13. Kato filtration for hdvf
  14. Filtrations of the Brauer group
  15. Kato filtration for schemes
  16. ...and 38 more sections

Key Result

Theorem 1.1

Let ${\mathcal{X}}$ be as above and let $Y^o$ denote the regular locus of $Y$. Then there exists a canonical map $\partial_X \colon {\rm fil}_0 {\operatorname{Br}}(X) \to H^1_{\textnormal{\'et}}(Y^o, {{\mathbb Q}}/{{\mathbb Z}})$. Moreover, we have the following.

Theorems & Definitions (233)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Remark 1.5
  • Theorem 1.6
  • Remark 1.7
  • Theorem 1.8
  • Theorem 1.9
  • Corollary 1.10
  • ...and 223 more