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

Abstract

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 analogue of the Milnor exact sequence [4] 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 [17, Theorem A.3] that gives the norm theorem for inseparable polynomials.