Building resolutions

We consider how a resolution of an abelian group M over ℤ could be lifted to a free resolution of the trivial module R over R[M], where R is the field of the rationals. The extended resolution is defined in terms of the exterior and divided powers algebras. Furthermore if the resolution of M is in fact a free resolution over ℤ[G] for some group G then the extended resolution will provide a free resolution of the augmentation ideal of R[M] over R[M ⋊ G]. Furthermore if R is a subring of the rationals containing ℤ and all j ≤ i are invertible in R then the extended complex can be defined up to dimension (i + 1) and is exact up to dimension i.