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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.
https://mathoverflow.net/questions/141467/sheaf-cohomology-with-compact-supports-and-verdier-duality
I'd like to know the relationship between the cohomology of this complex and the cohomology of the formal adjoint complex with compact supports (for the de Rham complex, this is again the de Rham complex, but with compact supports, and the relationship is given by Poincaré duality).
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://www.seas.upenn.edu/~jean/sheaves-cohomology.pdf
A Gentle Introduction to Homology, Cohomology, and Sheaf Cohomology Jean Gallier and Jocelyn Quaintance Department of Computer and Information Science
https://math.stackexchange.com/questions/1174503/cohomology-with-compact-support-for-sheaves-in-separated-schemes-of-finite-type
Besides, in the context of étale cohomology (which was developed in order to have a cohomology theory like the singular one), the only definition that would make sense is something that can be compared to the topological cohomology with compact support. Hence your second and third definitions cannot be used.
https://math.stackexchange.com/questions/tagged/sheaf-cohomology
I am under the impression there are two (related?) definitions of sections with compact support. ... vector-bundles sheaf-cohomology etale-cohomology. asked Jul 3 '19 at 13:22. Matthew Levy. ... Newest sheaf-cohomology questions feed Subscribe to RSS Newest sheaf-cohomology questions feed To subscribe to this RSS feed, copy and paste this URL ...
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://arxiv.org/pdf/1612.09526.pdf
sheaf will be isomorphic to the ordinary singular cohomology of the polyhedral complex. The analogous statement holds for the compactly supported versions. See [5, Example 6.2.4] for a simple example and more details. De nition 5. The cellular sheaf cohomology (with compact support) of Gis the coho-
https://www.math.ucla.edu/~jens/data/sheafcohomology.pdf
theorem to see sheaf cohomology with compact support, derived functors, distin-guished triangles and an instance of Verdier duality in action. All these concepts will be studied in detail in the following talks. 1
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