Let a discrete group $G$ be a directed union of subgroups $G_i$, and $M$ a $G$-module. Under what conditions is it true that $H^i(G,M)=\varprojlim_j H^i(G_j,M)$? One can write down a spectral sequence $$ E_2^{p,q} = R^p\varprojlim_j H^q(G_j,M) \Rightarrow H^{p+q}(G,M), $$ so my question essentially boils down to when are the higher derived limit terms zero?
The specific examples I am interested in are $G=\mathbb{Q}$ or $\mathbb{Q}/\mathbb{Z}$ as the directed union of its finitely-generated subgroups. In this case, does $R\varprojlim$ vanish for every $G$-module? Every $R[G]$-module finitely-generated over $R$, $R$ Noetherian?