Kato-Milne cohomology group over rational function fields in characteristic 2, II
Ahmed Laghribi, Trisha Maiti
TL;DR
The work extends Milnor-Scharlau type exact sequences to the characteristic $2$ Kato-Milne cohomology setting for rational function fields, by constructing explicit residue maps and transfer maps that interrelate $H_2^{m+1}$ over the base field, its function field, and residue fields. It combines a detailed decomposition of $W_1(H_2^{m+1}(F_p))$, separable and inseparable transfer maps, and a reciprocity law to prove an exact sequence linking $H_2^{m+1}(\mathcal{F})$, $H_2^{m+1}(F)$, $\bigoplus_p W_1(H_2^{m+1}(F_p))$, and $H_2^m(\mathcal{F})$. Key contributions include the explicit Pfister-based decomposition, well-defined transfer constructions, and the Milnor-Scharlau type exact sequence in the Kato-Milne framework, thereby connecting Witt-group theory with differential forms via Teichmüller lifting in characteristic $2$. This advances the cohomological toolkit for char $2$ fields and enriches the understanding of quadratic forms through $H_2$-cohomology.
Abstract
Our aim in this paper is to prove in the setting of Kato-Milne cohomology in characteristic 2 an exact sequence which is analogue to the Milnor-Scharlau sequence [8, Theorem 6.2]. This is an extension of the Milnor exact sequence proved in [6].