#### DOCUMENTA MATHEMATICA, Vol. 21 (2016), 223-270

Bryden Cais, Tong Liu

On F-Crystalline Representations

We extend the theory of Kisin modules and crystalline representations to allow more general coefficient fields and lifts of Frobenius. In particular, for a finite and totally ramified extension $F/\Qp$, and an arbitrary finite extension $K/F$, we construct a general class of infinite and totally wildly ramified extensions $K_\infty/K$ so that the functor $V\mapsto V|G_{K_\infty}$ is fully-faithfull on the category of $F$-crystalline representations $V$. We also establish a new classification of $F$-Barsotti-Tate groups via Kisin modules of height 1 which allows more general lifts of Frobenius.

2010 Mathematics Subject Classification: Primary 14F30,14L05

Keywords and Phrases: F-crystalline representations, Kisin modules

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