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Cristalline representations in the case of an imperfect residue field. (Représentations cristallines dans le cas d’un corps résiduel imparfait.) (French) Zbl 1168.11051
Summary: Let \(K\) be a complete discrete valuation field of characteristic 0, with residue field \(k_K\) of characteristic \(p\). We assume that \(k_K\) admits a finite \(p\)-basis. Let \(\overline{K}\) be an algebraic closure of \(K\) and \(G_K= \text{Gal}(\overline{K}/K)\). We construct and study \(p\)-adic periods rings \(B_{\text{cris}}\subset B_{dR}\) generalizing those defined by J.-M. Fontaine when \(k_K\) is perfect. Those rings are endowed with the usual extra structures plus a connection. They allow to extend the notions of crystalline and de Rham \(p\)-adic representations of \(G_K\) to the case of non perfect \(k_K\). The main result of this work, generalizing a theorem of P. Colmez and J.-M. Fontaine, is the fact that the category of crystalline \(p\)-adic representations of \(G_K\) is equivalent to the category of weakly admissible \(F\)-isocrystals filtered over \(K\).

MSC:
11S20 Galois theory
11F80 Galois representations
11F85 \(p\)-adic theory, local fields
11S15 Ramification and extension theory
11S25 Galois cohomology
14F30 \(p\)-adic cohomology, crystalline cohomology
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