On differential modules associated to de Rham representations in the imperfect residue field case

Let K be a complete discrete valuation field of mixed characteristic (0, p) with possibly imperfect residue fields, and let G(K) the absolute Galois group of K. In the first part of this paper, we prove that Scholl's generalization of fields of norms over K is compatible with Abbes-Saito's ramification theory. In the second part, we construct a functor N-dR that associates a de Rham representation V to a (phi, del)-module in the sense of Kedlaya. Finally, we prove a compatibility between Kedlaya's differential Swan conductor of N-dR(V) and the Swan conductor of V, which generalizes Marmora's formula.