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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?

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If you really mean the word "coherent" in the title, then the answer is no. E.g., you would want a $j_!$ morphism for $j$ an open embedding, but one can see that there's no left adjoint to $j^*$ in this setting (since it would imply tensor products commute with infinite products). – Moosbrugger Oct 13 '11 at 1:54
up vote 7 down vote accepted

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.

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SGA 2 is the original reference. You can find it here

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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.

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For a discussion of "Sheaf cohomology wih compact supports over any topological space" you may want to look up "Sheaf theory - Breadon" and " Sheaves in topology-A.Dimca". It should be noted that: Etale cohomology with compact support requires the existence of proper embedding (one shows that result is independent of a chosen embedding), so it is not a straightforward generalization of "Sheaf cohomology wih compact supports" to the etale site. – isildur Oct 13 '11 at 0:22

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