free resolution

Definition

A free resolution of an R-module $M$ is a resolution such that every module is free.

Existence

Let $M$ be an $R$-module.

1. $M$ has a free resolution
2. If $R$ is Noetherian and $M$ is finitely generated, then $M$ has a finitely generated free resolution.

Proof

Let $X_0$ be a set of generators of $M$ (this may not be finite, for example $X_0=M$ always works). Then there is a surjective $R$-module homomorphism $R^{\oplus X_0}\stackrel{\pi }{\to }M$. Let $K_1$ be its kernel. Repeating the above argument, we get a surjective $R$-module homomorphism $R^{\oplus X_1}\stackrel{{\pi }_1}{\to }{K}_1$ which fits into the diagram

This means $\text{im }{f}_i=\ker \pi$ by the way we constructed. Now you can repeat this process for $\ker f_1$, and onward. In this way the horizontal subdiagram of

is a free resolution of $M$.

Proof for part 2:

The Neotherian condition on $R$ implies that any submodule of a finitely generated R-module is again finitely generated. This means that if we apply the process above then we know that we can find a finite generating set i.e. $|X_0|<\infty$.

Then, $K_1$ is a submodule of a finitely generated free module $R^{\oplus X_0}$ so that means there is a finite generating set $|X_1|<\infty$. Keep doing that…