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https://math.stackexchange.com/questions/39214/cohomology-with-compact-support
For a closed manifold, cohomology and cohomology with compact supports coincide, but for open manifolds they do not. The top dimensional cohomology with compact support is always one-dimensional for a connected orientable manifold, regardless of whether or not the manifold is closed.
https://link.springer.com/chapter/10.1007%2F978-3-642-82783-9_3
Let x denote a locally compact space, i.e. a Hausdorff topological space in which every point has a compact neighbourhood. Let us prove two simple facts about locally compact spaces. Cohomology with Compact Support SpringerLinkAuthor: Birger Iversen
https://mathoverflow.net/questions/59017/group-cohomology-with-compact-support
This response is a little late, but I have thought about the same question recently. I don't think there is a way to define cohomology with compact support in a purely group theoretic way. The problem is that compact cohomology will distinguish between multiple cusps, but cocycles can only capture one cusp.
https://mathoverflow.net/questions/17466/cohomology-with-compact-support-for-coherent-sheaves-on-a-scheme
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. $\endgroup$ – …
https://ncatlab.org/nlab/show/compactly+supported+cohomology
In compactly supported cohomology cocycles and coboundaries on some space are required to have compact support: to be non-trivial only over a compact subspace/compact subobject of the base. References General. James Milne, section 18 of Lectures on Étale Cohomology; Compactly supported de Rham cohomology
https://www.physicsforums.com/threads/cohomology-with-compact-support.570117/
Jan 23, 2012 · So in singular cohomology with compact support, you consider the subcomplex of C^k(X) of those singular k-cochains f having the property (P) there exists a compact subset K of X such that f(c)= 0 for all k-chains c sitting outside of K.
https://en.wikipedia.org/wiki/%C3%89tale_cohomology
Poincaré duality and cohomology with compact support. The étale cohomology groups with compact support of a variety X are defined to be (,) = (,!) where j is an open immersion of X into a proper variety Y and j! is the extension by 0 of the étale sheaf F to Y.
http://www.math.wisc.edu/~maxim/Topnotes9.pdf
1 Cohomolgy with Compact Support Let Xbe a topological space and Kbe a compact subset of X, ... is called the cohomology of Xwith compact support. De nition of Direct Limit of Groups (G ) 2I Let G be abelian groups indexed by some directed set I, namely, Iis partially ordered and
https://link.springer.com/chapter/10.1007/978-3-663-09991-8_6
In this chapter we will define cohomology with compact support for rigid analytic varieties and adic spaces. Let me give a summary of this definition.
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