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Crystalline representations and \(F\)-crystals: the case of an imperfect residue field. (Représentations cristallines et \(F\)-cristaux: le cas d’un corps résiduel imparfait.) (French) Zbl 1217.11111

This article concerns \(p\)-adic Hodge theory, and, more particularly, the theory of crystalline representations of \(G_K = \text{Gal} (\overline{K}/K)\), where \(K\) is a \(p\)-adic field – a discretely valued field of characteristic zero whose residue field \(k\) is of characteristic \(p\).
When \(k\) is perfect, we know that crystalline representations of \(G_K\) can be constructed from Breuil-Kisin modules, and that those which are crystalline with Hodge-Tate weights in \(\{0;1\}\) come from \(p\)-divisible groups. The aim of the article under review is to generalize these two results to the case when \(k\) is not necessarily perfect, but has a finite \(p\)-basis.
Crystalline representations in this setting have been studied by the first author. The method of proof of the two results is similar to the classical case, once one has the necessary technical tools. The second result should be compared to Theorem 14 in [G. Faltings, “Coverings of \(p\)-adic period domains”, J. Reine Angew. Math. 643, 111–139 (2010; Zbl 1208.14039)].

MSC:

11S20 Galois theory
11S15 Ramification and extension theory
14F30 \(p\)-adic cohomology, crystalline cohomology

Citations:

Zbl 1208.14039
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References:

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