Let $R$ be a unital ring and let $M$ be a left $R$-module. There exist free left $R$-modules $F_n$ for all integers $n \geq 0$, an $R$-linear surjection $\varepsilon: F_0 \to M$, and $R$-linear maps $d_n: F_n \to F_{n-1}$ for all integers $n \geq 1$ such that the augmented sequence