Kato–Milne cohomology group over rational function fields in characteristic 2, I

Let $ℱ$ be a field of characteristic $2$ . In this paper we determine the Kato–Milne cohomology of the rational function field $ℱ (x)$ in one variable $x$ . This will be done by proving an analog of the Milnor exact sequence (“Algebraic $K$ -theory and quadratic forms”, Invent. Math. 9 (1970), 318–344) in the setting of Kato–Milne cohomology. As an application, we answer the open case of the norm theorem for Kato–Milne cohomology that concerns separable irreducible polynomials in many variables. This completes a result of Mukhija (Theorem A.3 of “Transfer for Kato–Milne cohomology over purely inseparable extensions”, J. Pure Appl. Algebra 226:8 (2022), art. id. 106930) that gives this norm theorem only for inseparable polynomials.

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Keywords

Kato–Milne cohomology, quadratic form, norm theorem

Mathematical Subject Classification

Primary: 11E04, 11E81, 13N05

Milestones

Received: 26 February 2025

Revised: 20 June 2025

Accepted: 8 August 2025

Published: 6 October 2025

Authors

Ahmed Laghribi
Université d’Artois, UR2462 Laboratoire de Mathématiques de Lens (LML) F-62300 Lens France

Trisha Maiti
Université d’Artois, UR2462 Laboratoire de Mathématiques de Lens (LML) F-62300 Lens France

Vol. 10, No. 3, 2025

Ahmed Laghribi and Trisha Maiti

Abstract

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Keywords

Mathematical Subject Classification

Milestones

Authors