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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://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://www.jmilne.org/math/CourseNotes/lec.html
pdf file for the current version (2.21) These are the notes for a course taught at the University of Michigan in 1989 and 1998. In comparison with my book, the emphasis is on heuristic arguments rather than formal proofs and on varieties rather than schemes.
https://math.stackexchange.com/questions/521265/understanding-cohomology-with-compact-support
Understanding cohomology with compact support. Ask Question Asked 6 years, 1 month ago. Active 6 years, 1 month ago. Viewed 3k times 6. 1 $\begingroup$ I am trying to understand the definition of (singular) cohomology with compact supports. My understanding of singular cohomology goes like this. ...
https://www.encyclopediaofmath.org/index.php/Etale_cohomology
Similar theorems are true for any morphism of finite type, provided that one uses cohomology with compact support. If is an algebraic variety over an algebraically closed field, then for any constructible sheaf on the cohomology groups with compact support are finite and vanish for . If, in addition, is an affine variety, then for .
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://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. $\endgroup$ – isildur Oct 13 '11 at 0:22
https://stacks.math.columbia.edu/download/etale-cohomology.pdf
ÉTALE COHOMOLOGY 5 03N5 A family of morphismsDefinition 4.1. {ϕ i: U i →X} i∈I is called an étale coveringS if each ϕ i is an étale morphism and their images cover X, i.e., X = i∈I ϕ i(U i). This“defines”theétaletopology. Inotherwords,wecannowsaywhatthesheaves are.
https://en.wikipedia.org/wiki/Cohomology_base_change_theorem
Proper base change theorems for quasi-coherent sheaves apply in the ... This fact and the proper base change suggest to define the direct image functor with compact support for a map ... referred to as cohomology with compact support. It is an important variant of usual étale cohomology. Similar ideas are also used to construct ...
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