In Milne's notes on Class Field Theory (http://www.jmilne.org/math/CourseNotes/CFT.pdf), he initially defines group cohomology in terms of injective resolutions, then he talks about computing cohomology using cochains. I don't see him mention anywhere that the group has to be finite in order for cochains to work, but this seems to be the case?

Later, he discusses profinite groups, in which he says that cohomology of profinite groups can be computed using continuous cochains. What isn't clear is the following: is the cohomology using continuous cochains a modified cohomology theory, different from the one using injective resolutions? In this case, then, do we get the cohomology theory using injective resolutions if we use all cochains, not just continuous ones? Or do the continuous cochains give the same cohomology as injective resolutions, and cochains which are not necessarily continuous only give cohomology in the case of finite groups? I.e., is there only one such cohomology theory? At the very least, using general cochains versus continuous cochains in the case of infinite profinite groups is different, for in one case $H^1$ is $\mathrm{Hom}(G,M)$ when M is trivial, and in the other case $H^1$ is $\mathrm{Hom}_{\mathrm{cts}}(G,M)$.

Assuming that set-theoretic cochains only work for finite groups, why is it the case? It seems that the proof that cochains compute cohomology (i.e. by looking at a projective resolution of $\mathbb{Z}$) fails because the modules used in the case of finite groups, i.e. tensor powers of $\mathbb{Z}[G]$, aren't necessarily projective when $G$ is infinite (in the case when $G$ is finite, they are free). Is this correct?

discrete$G$-modules, so take injective resolutions there. In general, for $G$-module $M$ its "discretization" is maximal discrete $G$-submodule (i.e., elts with open stabilizer). This has univ. mapping property, so discretization of injective $G$-mod is inj. in category of discrete $G$-mods. So latter category has enough injectives. For finite discrete $G$ it's the old thing. For general $G$ recovers "cont. cochain" construction, via work; dropping continuity does matter. $\endgroup$ – BCnrd May 9 '10 at 8:12discreteG-modules, and state that it can be computed usingcontinuouscochains. A finite group can be regarded as a profinite group with the discrete topology, in which case the two definitions coincide (obviously). $\endgroup$ – JS Milne May 9 '10 at 19:53