Is there a notion (for schemes or just locally ringed spaces) of cohomology with compact support? I guess there is for algebraic schemes over $\mathbf{C}$, but what about schemes in general? Does anybody have a good reference?



Cohomology with compact support for coherent sheaves on a scheme
Yes there is. Take $S$ a scheme and take $f \colon X \to S$ a compactifiable morphism of schemes. By definition this means that there exists a proper $S$-scheme Y which contains $X$ as an open subscheme. Then, given a compactification and a sheaf on $X$, you may define the cohomology with proper support of this sheaf as the cohomology of the pushforward of your sheaf to $Y$. 


I think this works for many different cohomologies, but you need to check that the compact support cohomology does not depend on the chosen compactification. At least for etale sheaves I know this is so. 


