This might be very basic, but the short exact sequence $$ 0 \to \mathbb{Z} \to \mathbb{Q} \to \mathbb{Q}/\mathbb{Z} \to 0 $$ is both an injective resolution of $\mathbb{Z}$, and a flat resolution of $\mathbb{Q}/\mathbb{Z}$, making it a very useful exact sequence in many homological computations.