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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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