## The $$p$$-adic monodromy theorem in the imperfect residue field case.(English)Zbl 1312.11046

Summary: Let $$K$$ be a complete discrete valuation field of mixed characteristic (0,p) 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 $$(\phi,G_K$$)-module $$\tilde{\mathbb N}_{\mathrm{rig}}^{\nabla +}(V)$$ for a de Rham representation $$V$$, which is a generalization of P. Colmez’s $$\tilde{\mathbb N}_{\mathrm{rig}}^+(V)$$ [in: Représentation $$p$$-adiques de groupes $$p$$-adiques I. Représentations galoisiennes et $$(\varphi, \Gamma)$$-modules. Paris: Société Mathématique de France. 117–186 (2008; Zbl 1168.11021)]. 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 O. Hyodo’s results [Adv. Stud. Pure Math. 12, 287–314 (1987; Zbl 0649.12011)].

### MSC:

 11F80 Galois representations 11F85 $$p$$-adic theory, local fields 11S15 Ramification and extension theory 11S20 Galois theory 11S25 Galois cohomology

### Keywords:

$$p$$-adic Hodge theory; $$p$$-adic representations

### Citations:

Zbl 1168.11021; Zbl 0649.12011
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