Cohomology relative to a G-set and finiteness conditions

We shall consider a cohomology theory relative to group actions on sets and develop a completion analogous to Mislin, Benson and Carlson. Later we restrict ourselves to cohomology relative to all finite subgroups. There we shall study relative analogues to finiteness conditions on kG-modules such as finite relative projective dimension and being of relative type FP∞.