We prove the global triangulation conjecture for families of refined p-adic
representations under a mild condition. That is, for a refined family, the
associated family of (phi, Gamma)-modules admits a global triangulation on a
Zariski open and dense subspace of the base that contains all regular
non-critical points. We also determine a large class of points which belongs to
the locus of global triangulation. Furthermore, we prove that all the
specializations of a refined family are trianguline. In the case of the
Coleman-Mazur eigencurve, our results provide the key ingredient for showing
its properness in a subsequent work.Comment: Final versio
