Lemma 19.3.1. Let $G$ be a topological group. Let $R$ be a ring. The category $\text{Mod}_{R, G}$ of $R\text{-}G$-modules, see Étale Cohomology, Definition 59.57.1, has functorial injective embeddings. In particular this holds for the category of discrete $G$-modules.
Proof. By the remark above the lemma the category $\text{Mod}_{R[G]}$ has functorial injective embeddings. Consider the forgetful functor $v : \text{Mod}_{R, G} \to \text{Mod}_{R[G]}$. This functor is fully faithful, transforms injective maps into injective maps and has a right adjoint, namely
\[ u : M \mapsto u(M) = \{ x \in M \mid \text{stabilizer of }x\text{ is open}\} \]
Since $v(M) = 0 \Rightarrow M = 0$ we conclude by Homology, Lemma 12.29.5. $\square$
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