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 $(0,p)$, whose residue field may not be perfect, and $G_K$ 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 ${\mathbb{N}}_{\mathrm{dR}}(V)$ associating a de Rham representation $V$ with a $(\phi ,\nabla )$-module in the sense of Kedlaya. Finally, we prove a compatibility between Kedlaya's differential Swan conductor of ${\mathbb{N}}_{\mathrm{dR}}(V)$ and Swan conductor of $V$, which generalizes Marmora's formula.

50pages; v4: minor corrections

References in corpus (4)

• Slope filtrations revisited
• Frobenius modules over multivariate Robba rings
• Galois theory of B_dR^+ in the imperfect residue field case
• Ramification theory and perfectoid spaces