(left) resolution of R-module

Definition

A (left) resolution of an R-module $M$ is an acyclic complex of $R$-modules of the form

$$\cdots \stackrel{f_2}{\to }{P}_1\stackrel{f_1}{\to }{P}_0\stackrel{\pi }{\to }M\to 0$$

Explicitly, this means that

1. $\pi$ is surjective with kernel $\text{im }{f}_1$
2. $\text{im }{f}_{i+1}=\ker f_i$ for all $i\ge 1$.

Alternate definition

A resolution of $M$ is a complex $(P_{\bullet },f_{\bullet })$ supported in non-negative degrees such that

$$H_{\bullet }(P_{\bullet },f_{\bullet })=\{\begin{array}{ll}M& \text{if }\bullet =0\\ 0& \text{else}\end{array}$$

Non-uniqueness

Every module $M$ has trivial resolutions

$$\cdots ⟶0⟶0⟶M⟶0$$

and

$$\cdots ⟶0⟶M\stackrel{\text{id}}{\to }M⟶0⟶M⟶0$$

so usually extra conditions are attached.

P-resolutions

Let P be a property of modules. For example, P could be free, finitely generated, flat, projective, etc. A P-resolution of $M$ is a resolution $(P_{\bullet },f_{\bullet })$ of $M$ such that each R-module $P_i$ has the property P.

Example

A finitely generated module $M$ over a PID $R$ has a finitely generated resolution of length at most one. From the classification we can write

$$M\simeq R^{\oplus b}\oplus \underset{i=1}{\overset{N}{⨁}}R/(q_i)$$

Then

$$R^{\oplus N}\stackrel{\left[\begin{array}{ccc}-& 0& -\\ q_1& & \\ & \ddots & \\ & & q_N\end{array}\right]}{\to }{R}^{\oplus b}\oplus R^{\oplus N}$$

is a finitely generated free resolution of $M$.