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We prove the global triangulation conjecture for families of refined p-adicrepresentations under a mild condition. That is, for a refined family, theassociated family of (phi, Gamma)-modules admits a global triangulation on aZariski open and dense subspace of the base that contains all regularnon-critical points. We also determine a large class of points which belongs tothe locus of global triangulation. Furthermore, we prove that all thespecializations of a refined family are trianguline. In the case of theColeman-Mazur eigencurve, our results provide the key ingredient for showingits properness in a subsequent work.

Triangulation of refined families