#### Vol. 4, No. 7, 2010

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On families of $\varphi,\Gamma$-modules

### Kiran Kedlaya and Ruochuan Liu

Vol. 4 (2010), No. 7, 943–967
##### Abstract

Berger and Colmez (2008) formulated a theory of families of overconvergent étale $\left(\phi ,\Gamma \right)$-modules associated to families of $p$-adic Galois representations over $p$-adic Banach algebras. In contrast with the classical theory of $\left(\phi ,\Gamma \right)$-modules, the functor they obtain is not an equivalence of categories. In this paper, we prove that when the base is an affinoid space, every family of (overconvergent) étale $\left(\phi ,\Gamma \right)$-modules can locally be converted into a family of $p$-adic representations in a unique manner, providing the “local” equivalence. There is a global mod $p$ obstruction related to the moduli of residual representations.

##### Keywords
$p$-adic Galois representations, $(\varphi,\Gamma)$-modules
Primary: 11F80
Secondary: 11S20
##### Milestones
Received: 10 December 2009
Accepted: 10 January 2010
Published: 29 January 2011
##### Authors
 Kiran Kedlaya Department of Mathematics Massachusetts Institute of Technology Cambridge, MA 02139 United States http://www-math.mit.edu/~kedlaya/ Ruochuan Liu Department of Mathematics University of Michigan 1844 East Hall 530 Church Street Ann Arbor, MI 48109-1043 United States http://www-personal.umich.edu/~ruochuan