Injective resolution ($0\to M\to I^0\to I^1\to \cdots$)

An exact augmented cochain complex with injective module terms. Module categories have enough injectives: coinduction $J={\mathrm{Hom}}_{\mathbb{Z}}(A,\mathbb{Q}/\mathbb{Z})$ is injective by its character adjunction, and the maps $m\mapsto (a\mapsto \varphi (am))$, over all additive characters $\varphi$ 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.