Stable free lattices in residually reducible Galois deformations and control theorem of Selmer groups

In this paper, we study the graph of homothety classes of stable free lattices in a two-dimensional representation over a local UFD. This generalizes a classical result of Serre in the case where the base ring is a discrete valuation ring. As applications, we consider the case where the representation comes from a residually reducible Hida family and we study the control theorem of Selmer groups. These results enable us to know the precise statement of the main conjecture in the residually reducible case as we will remark in § 4 and § 5..