A special case of a projective resolution. Every module $M$ over an associative ring $R$ is the quotient module $F_0/N_0$ of a free $R$-module $F_0$ by a submodule $N_0$. The submodule $N_0$ has a similar representation $F_1/N_1$, etc. As a result one obtains an exact sequence of free modules

$$F_0\leftarrow F_1\leftarrow\dotsb\leftarrow F_n\leftarrow\dotsb,$$

called the free resolution of $M$. The canonical homomorphism $F_0\to M$ is called a supplementing homomorphism (or augmentation).

#### Comments[edit]

See also Free module.