#### Vol. 7, No. 8, 2013

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The $p$-adic monodromy theorem in the imperfect residue field case

### Shun Ohkubo

Vol. 7 (2013), No. 8, 1977–2037
##### Abstract

Let $K$ be a complete discrete valuation field of mixed characteristic $\left(0,p\right)$ and ${G}_{K}$ the absolute Galois group of $K$. In this paper, we will prove the $p$-adic monodromy theorem for $p$-adic representations of ${G}_{K}$ without any assumption on the residue field of $K$, for example the finiteness of a $p$-basis of the residue field of $K$. The main point of the proof is a construction of $\left(\phi ,{G}_{K}\right)$-module ${\stackrel{˜}{ℕ}}_{rig}^{\nabla +}\left(V\right)$ for a de Rham representation $V$, which is a generalization of Pierre Colmez’s ${\stackrel{˜}{ℕ}}_{rig}^{+}\left(V\right)$. In particular, our proof is essentially different from Kazuma Morita’s proof in the case when the residue field admits a finite $p$-basis.

We also give a few applications of the $p$-adic monodromy theorem, which are not mentioned in the literature. First, we prove a horizontal analogue of the $p$-adic monodromy theorem. Secondly, we prove an equivalence of categories between the category of horizontal de Rham representations of ${G}_{K}$ and the category of de Rham representations of an absolute Galois group of the canonical subfield of $K$. Finally, we compute ${H}^{1}$ of some $p$-adic representations of ${G}_{K}$, which is a generalization of Osamu Hyodo’s results.

##### Keywords
$p$-adic Hodge theory, $p$-adic representations
##### Mathematical Subject Classification 2010
Primary: 11F80
Secondary: 11F85, 11S15, 11S20, 11S25
##### Milestones
Received: 1 July 2012
Revised: 2 April 2013
Accepted: 2 May 2013
Published: 24 November 2013
##### Authors
 Shun Ohkubo Department of Mathematical Sciences University of Tokyo Tokyo 153-8914 Japan