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Let $f : G' \subset G$ be an injection between profinite groups such that $G'$ is normal in $G$ (typical situation which I deal with : $G$ the absolute Galois group of a local field, $G'$ an open subgroup).

Let $k$ be a finite field, $\mathcal{O}$ the ring of integers of a finite extension of $\mathbb{Q}_p$ with residue field $k$ and $\mathcal{C}$ the category whose objects are of complete noetherian local $\mathcal{O}$-algebras with residue fields $k$ and whose morphisms are local $\mathcal{O}$-algebra homomorphisms inducing the identity on residue fields.

Let $\bar{\rho} : G \longrightarrow GL_n(k)$ be a continuous representation and let $\bar{\rho}': G' \longrightarrow GL_n(k)$ the continuous representation obtained via $f$.

Let $D^{\Box}$ be the functor $\mathcal{C} \longrightarrow Sets$ which assign to $A \in \mathcal{C}$ the set of lifts of $\bar{\rho}$ to $A$. Define $D'^{\Box}$ in the same way (with respect to $\bar{\rho}'$)

We know that $D^{\Box}$ and $D'^{\Box}$ are representable by $R^{\Box}$ and $R'^{\Box}$ and that there is a ring homomorphism $R'^{\Box} \to R^{\Box}$. Is $R^{\Box}$ a finite $R'^{\Box}$-algebra ? (there is no assumption on the reducibility of $\bar{\rho}$ and $\bar{\rho}'$).

I think this is true when you assume $\bar{\rho}$ and $\bar{\rho}'$ to be absolutely irreducible and instead of $R^{\Box}$ you consider $R$, the ring representing the deformation functor (i.e. conjugacy classes of lifts of $\bar{\rho}$ with conjugating matrices reducing to the identity matrix).

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There are stupid counterexamples. If you take $G$ to be a finite group with faithful noncentral representation $\rho:G \rightarrow GL_n(\mathbb{Z}_p)$ and $G' = 1.$ Then $R^{'\square} = \mathbb{Z}_p$ whereas $Spec(R^{\square})$ has infinitely many $\mathbb{\overline{Q}}_p$ points. It follows $R^{\square}[1/p]$ is of positive dimension and hence not finite over $R'^{\square}[1/p].$ The same then holds for $R$ and $R'.$ – JSpecter Mar 2 '13 at 23:16

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