Is it useful to consider cohomology of group representations? In group representation theory, one attempts to explain and classify (some of) the modules over the group ring $k[G]$, for some field $k$. In group cohomology, one develops the machinery of the cohomology of groups out of the category of modules over the ring $\mathbb{Z}[G]$. In both cases, one has tensor products, restriction maps, induced modules, etc, and all these constructions look very similar in both cases, with $k$ replaced by $\mathbb{Z}$ in group cohomology.
My question is, might one develop a theory of cohomology of $G$-representations over a field $k$ analogous to the theory of cohomology of $G$-modules? Essentially, one would take a representation $M$, take an injective resolution of $M$ in the category of $k[G]$-modules, then find the fixed elements to get the cohomology. I believe this would be $\mathrm{Ext}_{k[G]}(k,M)$, where $k$ is given the trivial $G$-action.
Given that we know a lot about the structure of finite-dimensional representations of finite groups (character tables), what would the character table tell us about the cohomology? What might the cohomology tell us about the character table? What happens if we consider representations of infinite groups, such as Lie groups, algebraic groups, or Galois groups?
 A: You have already used the following fact in your statement:
For every representation $\rho: G\to \mathrm{Aut}_R(A)$ ($A$ an $R$-module) we can give $A$ the structure of an $R[G]$-module by $g\cdot a = \rho_g(a)$. Conversely, any torsion free $R[G]$-module $A$ admits a representation $\rho: G \to \mathrm{ Aut}_R(A)$ given by $\rho_g(a) := g\cdot a$.
So studying $R[G]$-modules IS studying representations of $G$. My answer would be if you want to study representations study $R[G]$-modules. $R[G]$-modules aren't like a cure-all, because they don't give you an action to start with. Does anyone know how to classify actions?


*

*Studying a fixed $R[G]$-module is the same as studying a fixed representation. So it won't tell you anything about the character tables which involves different representations.

*You can still use $R[G]$-modules to study representations of infinite groups and galois groups. You can use it to derive properties about galois groups acting on roots of unity or torsion points of elliptic curves. http://en.wikipedia.org/wiki/Kummer_theory

*The wikipedia article http://en.wikipedia.org/wiki/Group_cohomology tells you that the Ext construction from resolutions is the same as the "standard" construction of group cohomology by cochain complexes.


I'm not sure if my answer was helpful. Hopefully it saved you some time. I don't think I understood what you meant about "taking the fixed elements to get the cohomology".
