= Injective resolution
{title2=$0\to M\to I^0\to I^1\to\cdots$}
An exact augmented cochain complex with injective module terms. Module categories have enough injectives: coinduction $J=\operatorname{Hom}_{\mathbb Z}(A,\mathbb Q/\mathbb Z)$ is injective by its character adjunction, and the maps $m\mapsto(a\mapsto\phi(am))$, over all additive characters $\phi$ of $M$, embed $M$ into a product of copies of $J$. Products of injectives are injective by coordinatewise extension. Repeat on cokernels to obtain a resolution.
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