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In this paper, we study the graph of homothety classes of stable freelattices in a two-dimensional representation over a local UFD. This generalizesa classical result of the case where the base ring is a discrete valuation ringdue to Serre. As applications, we consider the case when the representationcomes from a residually reducible Hida family and we study the control theoremof Selmer groups. These results enable us to know the precise statement of themain conjecture in residually reducible case as we will remark in section 4 andsection 5.

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