# Free resolution

From Encyclopedia of Mathematics

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

See also Free module.

**How to Cite This Entry:**

Free resolution.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Free_resolution&oldid=44580

This article was adapted from an original article by V.E. Govorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article