I am pretty familiar with deRham cohomology. for me, deRham cohomology is synonymous with cohomology in general. but I often run into that cocycle condition, you know, where c(g,h)+c(gh,k)=c(g,hk)+c(h,k) you need to look at conditions of this type when deciding whether a projective representation of a group can straightened into a linear rep, or a rep of a central extension. I know this is a question of cohomology, and the language (cocycle) is very suggestive of stuff I know from deRham cohomology, but beyond that, I don't know much. so does someone want to give me a 50 cent sketch of the details? what is the "d" operator in this cohomology? does the group have to be a manifold? there is such a thing as cohomology of discrete groups, isn't there? and Lie algebras? is there a cup product? what is the name of the cohomology where this cocycle condition lives?