Arithmetic and differential Swan conductors of rank one representations with finite local monodromy

We consider a complete discrete valuation field of characteristic $p$, with possibly nonperfect residue field. Let ${\rm V}$ be a rank one continuous representation of its absolute Galois group with finite local monodromy. We will prove that the {\it arithmetic Swan conductor\/} of ${\rm V}$ (defined after K.~Kato, which fits in the more general theory of Abbes-Saito) coincides with the {\it differential Swan conductor\/} of the associated differential module ${\rm D}^{\dag}({\rm V})$ defined by K.~Kedlaya. This construction is a generalization to the nonperfect residue case of the Fontaine's formalism as presented in the work of N.~Tsuzuki. Our method of proof will allow us to give a new interpretation of the refined Swan conductor.