The Stacks project

Lemma 56.56.3. Let $G$ be a topological group. Let $R$ be a ring. For every $i \geq 0$ the diagram

\[ \xymatrix{ \text{Mod}_{R, G} \ar[rr]_{H^ i(G, -)} \ar[d] & & \text{Mod}_ R \ar[d] \\ \text{Mod}_ G \ar[rr]^{H^ i(G, -)} & & \textit{Ab} } \]

whose vertical arrows are the forgetful functors is commutative.

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 13.15.5). 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 13.29.4 that

\[ \text{Ext}^ i(\mathbf{Z}, F(M)) = \text{Ext}^ i(R, M) \]

for $M$ in $\textit{Mod}_{R, G}$. 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. $\square$


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