On ramification in transcendental extensions of local fields

Abstract

Let $L/K$ be an extension of complete discrete valuation fields, and assume that the residue field of $K$ is perfect and of positive characteristic. The residue field of $L$ is not assumed to be perfect. In this paper, we prove a formula for the Swan conductor of the image of a character $χ\in H^1(K, \mathbb{Q}/\mathbb{Z})$ in $H^1(L, \mathbb{Q}/\mathbb{Z})$ for $χ$ sufficiently ramified. Further, we define generalizations $ψ_{L/K}^{\mathrm{ab}}$ and $ψ_{L/K}^{\mathrm{AS}}$ of the classical Hasse-Herbrand $ψ$-function and prove a formula for $ψ_{L/K}^{\mathrm{ab}}(t)$ for sufficiently large $t\in \mathbb{R}$.