# On $F$-crystalline representations

### Bryden Cais

Department of Mathematics Department of Mathematics University of Arizona Purdue University Tucson, Arizona 85721 West Lafayette IN 47907 USA USA### Tong Liu

Department of Mathematics Department of Mathematics University of Arizona Purdue University Tucson, Arizona 85721 West Lafayette IN 47907 USA USA

## Abstract

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 $K_\infty/K$ 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.

## Cite this article

Bryden Cais, Tong Liu, On $F$-crystalline representations. Doc. Math. 21 (2016), pp. 223–270

DOI 10.4171/DM/532