Abstract

Let $\mathcal{ℱ}$ be a field of characteristic $2$. In this paper we determine the Kato–Milne cohomology of the rational function field $\mathcal{ℱ}(x)$ 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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