(left) resolution of R-module

Definition

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

Explicitly, this means that

1. is surjective with kernel
2. for all .

Alternate definition

A resolution of is a complex supported in non-negative degrees such that

Non-uniqueness

Every module has trivial resolutions

and

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 is a resolution of such that each R-module has the property P.

Example

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

Then

is a finitely generated free resolution of .