  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 the case where the base ring is a discrete valuation ring
due to Serre. As applications, we consider the case when 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 residually reducible case as we will remark in section 4 and
section 5.
