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Limits of crystalline representations. (Limites de représentations cristallines.) (French. English summary) Zbl 1071.11067
Let $$F$$ be the fraction field of the ring of Witt vectors over a perfect field of characteristic $$p$$. Let $$G_F$$ denote the absolute Galois group of $$F$$. The author proves the following conjecture of Fontaine: a $$p$$-adic representation of $$G_F$$, which is a limit of subquotients of crystalline representations with Hodge-Tate weights in an interval $$[a,b]$$, is itself crystalline with Hodge-Tate weights in $$[a,b]$$ (Theorem 1).
In a case $$b-a\leq p-1$$ this result follows immediately from the construction of J.-M. Fontaine and G. Laffaille [Ann. Sci. Éc. Norm. Supér., IV. Sér. 15, 547–608 (1982; Zbl 0579.14037)]. If $$b-a> p-1$$, then the theory of Fontaine-Laffaille doesn’t apply, and in order to prove Theorem 1 the author studies $$(\varphi,\Gamma)$$-modules attached to crystalline representations. He improves some results of Fontaine, Wach and Colmez concerning characterization of crystalline representations in terms of $$(\varphi,\Gamma)$$-modules (Theorems 2 and 3). He also gives new proofs of some results of Fontaine and Laffaille (V.2).

##### MSC:
 11S25 Galois cohomology 11F80 Galois representations 11R23 Iwasawa theory 13K05 Witt vectors and related rings (MSC2000) 14F30 $$p$$-adic cohomology, crystalline cohomology
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