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/Q_p, and an arbitrary finite extension K/F, we construct a general class of infinite and totally wildly ramified extensions K_∞/K so that the functor V 7 V =(pipe)_GK∞ is fully-faithfull on the category of Fcrystalline 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.