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

TL;DR

This paper extends the theory of de Rham representations and differential modules to imperfect residue fields, establishing compatibility with ramification theory and generalizing formulas for Swan conductors.

Contribution

It proves compatibility of Scholl's fields of norms with Abbes-Saito ramification, constructs a functor linking de Rham representations to Kedlaya's modules, and generalizes Marmora's Swan conductor formula.

Findings

Compatibility of Scholl's fields of norms with ramification theory

Construction of a functor from de Rham representations to Kedlaya's modules

Generalization of Swan conductor formula for imperfect residue fields

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 $(\varphi,\nabla)$-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.