Suppose that $A$ is a Noetherian (not necessarily commutative) $\mathbb{C}[h]$-algebra equipped with a rational action of an affine reductive group $G$, i.e., $A$ is a $\mathbb{C}[G]$-comodule and $g(a b) = g(a)g(b)$ for all $g \in G$ and $a,b \in A$. Moreover, assume that $G$-acts on $h$ by some fixed character $\chi$. An $(A, G)$-module is an $A$-module $M$ equipped with a rational $G$-action satisfying $g(am) = g(a)g(m)$ for all $a \in A$, $m \in M$, and $g \in G$.

The $h$-adic completion $\hat{M}$ of an $(A, G)$-module $M$ carries a $G$-action but it is rarely the case that this action is rational.

Example: Suppose $G = \mathbb{C}^{*}$ and $A = M = \mathbb{C}[h]$ with the standard action. The representation $\hat{M} = \mathbb{C}[[h]]$ cannot be rational because it is not the direct sum of its weight spaces.

Let $(\hat{A}, G)-\text{mod}^{\text{pro}}$ be the set of finitely generated $A$-modules $M$ equipped with compatible $G$-action (not necessarily rational) such that the induced $G$-action on $M/h^{n}M$ is rational for all $n$.

Q1: How does the functor of $G$-invariants behave on $(\hat{A}, G)-\text{mod}^{\text{pro}}$? It is clearly left exact. Is it actually exact?

One can embed $G-\text{Mod}^{\text{rat}} = \mathbb{C}[G]-\text{coMod}$ into $G-\text{Mod} = \mathbb{C}[G]^*-\text{Mod}$. The inclusion has a right adjoint that takes a $G$-module $M$ to its largest rational submodule $M^{\text{rat}}$.

Q2: Is the functor $\_^{\text{rat}}:(\hat{A}, G)-\text{mod}^{\text{pro}} \to ((\hat{A})^{\text{rat}}, G)-\text{mod}$ part of an equivalence of categories? It seems likely since $(\hat{A}, G)-\text{mod}^{\text{pro}}$ should be generated by modules of the form $\hat{A} \otimes V$ where $V$ is finite dimensional rational representation and $(\hat{A} \otimes V)^{\text{rat}} = (\hat{A})^{\text{rat}} \otimes V$. This would imply Q1 since invariant vectors are always rational.

Q3: Does anyone know anything about $(\hat{A})^{\text{Rat}}$? In particular is this ring Noetherian? I have only seen equivariant completion studied in this paper.

Q4: More generally, let $G-\text{Mod}^{\text{pro}}$ be the subcategory of $G-\text{Mod}$ generated by cofiltered limits of rational $G$-modules. How do $\_^{\text{rat}}$ and $\_^G$ behave on $G-\text{Mod}^{\text{pro}}$? Is this category even abelian?