Building resolutions

We consider how a resolution of an abelian group M over 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 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 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.