extension module

Definition

Let be an R-module. The ith extension module of by is

where is any projective resolution of .

Explicitly, note that is the complex

This is a complex because

Well-definedness

is well-defined (independent of the choice of projective resolution of ).

Proof

Let be two projective resolutions of . Then and are homotopy equivalent, let the chain maps be

We then get chain maps

Say is a chain homotopy from to . Define

This is a chain homotopy from to .

We can do this in the other direction to show that is homotopic to . Therefore,