I have a basic question about the definition of group cohomology. Suppose $\Gamma$ is a discrete group, $R$ a commutative ring and $V$ is an $R\Gamma$ module. Then, the first definition of group cohomology I saw was via homogenuous resolution, where we put $C^n(\Gamma,V) =\{\ \Gamma^{n+1}\rightarrow V\}\ $, defined the differential in the obvious way to get a complex and checked that it commutes with the $\Gamma$-action. But then, we restricted to the $\Gamma$ - invariant subspaces to compute the cohomology. I'm a bit confused as to why we made that restriction. Why not just compute the cohomology of the complex $\cdots\rightarrow C^{n-1}(\Gamma,V)\rightarrow C^n(\Gamma,V)\rightarrow \cdots$ ?
Is it "just" to get the same result as when you use the inhomogenuous (bar-) resolution? We did see a bit later that you get an R-module isomorphism between the $\Gamma$-invariants of the homogenuous complex above and the n-cochains of the bar-complex. Or is there another reason why it's natural to restrict to $\Gamma$-invariants in the homogenuous complex?