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I found this article about Galois Cohomology.

In it, it says that, when $G$ is a profinite group and $\mathbf{C}_G$ is the category of all discrete $G$-modules, $\mathbf{C}_G$ doesn't have enough projectives. He explains this by saying that $\mathbb{Z}G$ is not in $\mathbf{C}_G$ (which I understand because there isn't an open subgroup $U$ of $G$ that fixes $e_1$, the 1 in $\mathbb{Z}G$), so no free module over $\mathbb{Z}G$ is in $\mathbf{C}_G$. Therefore, he concludes that $\mathbf{C}_G$ doesn't contain enough projectives. Why is that ?

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