Cohomology with compact support for coherent sheaves on a scheme 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?
 A: 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. 
A: SGA 2 is the original reference. You can find it here 
http://www.math.polytechnique.fr/~laszlo/sga2/sga2-smf.pdf
A: If X is a smooth scheme over complex numbers then you can consider $X_{an}$ as an complex analytic manifold and compute  singular/ deRham/ simplicial cohomology with compact supports (this will be different from usual cohomology if X is not proper) 
On the Algebraic side there is etale cohomology with compact supports (which is defined by embedding X into a proper scheme...). 
Comparison theorems tell you that etale cohomology with torsion coefficients agree with singular cohomology (with torsion coeff).
Any reference on etale cohomology will discuss this.
Ref:SGA 4.5, Milne: Etale Cohomology.
