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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://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/Lectures+on+%C3%89tale+Cohomology
18. Cohomology groups with compact support. cohomology with compact support; 19. Finiteness theorem; Sheaves of ℤ l \mathbb{Z}_l-modules. ℓ-adic cohomology; 20. The smooth base change theorem. proper base change theorem; 21. The comparison theorem. complex analytic topology. Riemann existence theorem. comparison theorem (étale cohomology ...
https://www.encyclopediaofmath.org/index.php/Etale_cohomology
The cohomology of sheaves in the étale topology (cf. Etale topology). It is defined in the standard manner by means of derived functors. Let be a scheme and let be the étale topology on . Then the category of sheaves of Abelian groups on is an Abelian category with a sufficient collection of ...
https://mathoverflow.net/questions/97789/lemmas-on-etale-cohomology-with-compact-support-from-the-book-arithmetic-dualit
I was reading Milne's book "Arithmetic Duality Theorems". On page 166 there are a lot of useful lemmas on the etale cohomology with compact support on S-integers. However, I get confused when I tried to look at Prop 2.3 (a) and (d) at once.
https://www.math.ru.nl/~bmoonen/Seminars/EtCohConrad.pdf
Etale cohomology´ This chapter summarizes the theory of the ´etale topology on schemes, culmi-nating in the results on ‘-adic cohomology that are needed in the construction of Galois representations and in the proof of the Ramanujan–Petersson conjecture. In §1.1 we discuss the basic properties of the ´etale topology on a scheme, includ-
https://www.jmilne.org/math/CourseNotes/LEC.pdf
Sheaf theory Etale cohomology is modelled on the cohomology theory of sheaves in the usual topological sense. Much of the material in these notes parallels that in, for example, Iversen, B., Cohomology of Sheaves, Springer, 1986. Algebraic geometry I shall assume familiarity with the theory of algebraic varieties, for
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/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
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