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

Abstract

Let $K$ be a complete discrete valuation field of mixed characteristic $\left(0,p\right)$ 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 ${\mathbb{N}}_{\mathrm{\text{ dR}}}$ that associates a de Rham representation $V$ to a $\left(\phi ,\nabla \right)$-module in the sense of Kedlaya. Finally, we prove a compatibility between Kedlaya’s differential Swan conductor of ${\mathbb{N}}_{\mathrm{\text{ dR}}}\left(V\right)$ and the Swan conductor of $V$, which generalizes Marmora’s formula.