The Stacks project

Proof. Let us denote the forgetful functor $F : \text{Mod}_{R, G} \to \text{Mod}_ G$. Then $F$ has a left adjoint $H : \text{Mod}_ G \to \text{Mod}_{R, G}$ given by $H(M) = M \otimes _\mathbf {Z} R$. Observe that every object of $\text{Mod}_ G$ is a quotient of a direct sum of modules of the form $\mathbf{Z}[G/U]$ where $U \subset G$ is an open subgroup. Here $\mathbf{Z}[G/U]$ denotes the $G$-modules of finite $\mathbf{Z}$-linear combinations of right $U$ congruence classes in $G$ endowed with left $G$-action. Thus every bounded above complex in $\text{Mod}_ G$ is quasi-isomorphic to a bounded above complex in $\text{Mod}_ G$ whose underlying terms are flat $\mathbf{Z}$-modules (Derived Categories, Lemma 13.15.4). Thus it is clear that $LH$ exists on $D^-(\text{Mod}_ G)$ and is computed by evaluating $H$ on any complex whose terms are flat $\mathbf{Z}$-modules; this follows from Derived Categories, Lemma 13.15.7 and Proposition 13.16.8. We conclude from Derived Categories, Lemma 13.30.2 that

for $M$ in $\text{Mod}_{R, G}$. Observe that $H^0(G, -) = \mathop{\mathrm{Hom}}\nolimits (\mathbf{Z}, -)$ on $\text{Mod}_ G$ where $\mathbf{Z}$ denotes the $G$-module with trivial action. Hence $H^ i(G, -) = \text{Ext}^ i(\mathbf{Z}, -)$ on $\text{Mod}_ G$. Similarly we have $H^ i(G, -) = \text{Ext}^ i(R, -)$ on $\text{Mod}_{R, G}$. Combining everything we see that the lemma is true. $\square$