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https://en.wikipedia.org/wiki/Cohomology_with_compact_support
called the long exact sequence of cohomology with compact support. It has numerous applications, such as the Jordan curve theorem, which is obtained for X = R² and Z a simple closed curve in X. De Rham cohomology with compact support satisfies a covariant Mayer–Vietoris sequence: if U and V are open sets covering X, then
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/317640/invariants-in-relative-cohomology-and-compact-support-cohomology-of-the-quotient
Invariants in relative cohomology and compact support cohomology of the quotient. ... or point out the general results in algebraic topology with which one could prove it? ... Relative de rham cohomology with compact support. 3. Cohomology with compact support. Question feed
https://en.wikipedia.org/wiki/De_Rham_cohomology
In mathematics, de Rham cohomology (after Georges de Rham) is a tool belonging both to algebraic topology and to differential topology, capable of expressing basic topological information about smooth manifolds in a form particularly adapted to computation and the concrete representation of cohomology classes.It is a cohomology theory based on the existence of differential forms with ...
https://math.stackexchange.com/questions/39214/cohomology-with-compact-support
Furthermore, on any (connected orientable) manifold, closed or not, Poincare duality is true when expressed as a duality between cohomology and cohomology with compact support in the complementary dimension. Thus cohomology with compact support is a natural tool when working with non-closed manifolds.
https://en.wikipedia.org/wiki/Algebraic_de_Rham_cohomology
The hypercohomology of the de Rham complex of sheaves is called the algebraic de Rham cohomology of X over Y and is denoted by (/) or just () if Y is clear from the context. (In many situations, Y is the spectrum of a field of characteristic zero.) Algebraic de Rham cohomology was introduced by Grothendieck (1966).
http://www.math.unipd.it/~maurizio/mathps/PDfADRC.pdf
Poincar e Duality for Algebraic De Rham Cohomology. Francesco Baldassarri, Maurizio Cailotto, Luisa Fiorot Abstract. We discuss in some detail the algebraic notion of De Rham cohomology with compact supports for singular schemes over a eld of characteristic zero. We prove
http://math.columbia.edu/~dejong/seminar/note_on_algebraic_de_Rham_cohomology.pdf
Algebraic de Rham cohomology is a Weil cohomology theory with coe cients in K= kon smooth projective varieties over k. We do not assume kalgebraically closed since the most interesting case of …
http://www.few.vu.nl/~vdvorst/DeRham.pdf
on with a proof of de Rham’s theorem, which states that for smooth manifolds singular cohomology is identical to de Rham cohomology. A similar proof is used in chapter 10, where I proved Poincar´e duality, which gives a relation between de Rham cohomology and …
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
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