Change of Selmer group for big Galois representations and application to normalization (Q651272)

Let \(O\) be the ring of integers of a \(p\)-adic field, and let \(R\) be a ring which is finite and free over the power series ring \(O[[X_1,\dots,X_n]]\). Let \(\rho : \mathrm{Gal}(\overline{\mathbb Q}/\mathbb Q)\rightarrow\Aut_R(T)\), where \(T\) is a finitely-generated \(R\)-module (i.e. a `big Galois representation' in the sense of [\textit{J. Nekovář}, Selmer complexes. Astérisque 310. Paris: Société Mathématique de France (2007; Zbl 1211.11120)]). One can, under suitable hypotheses, attach a Selmer group \(\mathrm {Sel}(\rho)\) to such \(\rho\) which is a finitely-generated \(R\)-module canonically defined in terms of the Galois cohomology of \(\rho\). The article investigates the following question: given representations \(\rho_1\) and \(\rho_2\) as above on \(R\)-modules \(T_1\) and \(T_2\) which are isogenous, how are the Selmer groups \(\text{Sel}(\rho_1)\) and \(\text{Sel}(\rho_2)\) related? Under certain conditions on the \(T_i\), the authors prove a formula relating the Selmer groups. Under additional hypotheses, they also study how the formation of Selmer groups interacts with normalization of the coefficient ring and discuss how an Iwasawa main conjecture for a big Galois representation over a non-normal ring follows from a corresponding conjecture over the normalization.