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https://en.wikipedia.org/wiki/Support_(mathematics)
In mathematics, the support of a real-valued function f is the subset of the domain containing those elements which are not mapped to zero. If the domain of f is a topological space, the support of f is instead defined as the smallest closed set containing all points not mapped to zero. This concept is used very widely in mathematical analysis
https://en.wikipedia.org/wiki/Direct_image_with_compact_support
Direct image with compact support. Jump to navigation Jump to search. In mathematics, in the theory of sheaves the direct image with compact (or proper) support is an image functor for sheaves. Definition. Image functors for sheaves; direct ... sends a sheaf F on X to f! (F) ...
https://en.wikipedia.org/wiki/Sheaf_cohomology
In mathematics, sheaf cohomology is the application of homological algebra to analyze the global sections of a sheaf on a topological space.Broadly speaking, sheaf cohomology describes the obstructions to solving a geometric problem globally when it can be solved locally. The central figure of this study is Alexander Grothendieck and his 1957 Tohoku paper.
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://math.stackexchange.com/questions/1174503/cohomology-with-compact-support-for-sheaves-in-separated-schemes-of-finite-type
usually there are three notions of cohomology with compact (proper) support. The first one usually done in the étale site. However the second one is used in Verdier duality. The third one is done in
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. $\endgroup$ – …
http://www.math.ucla.edu/~jens/data/sheafcohomology.pdf
X. This is a soft resolution of the constant sheaf R X. We immediately get that H dR (X;R) = H (X;R X) and analogously for cohomology with compact support. The singular cochain complex (of sheaves) on X also gives a resolution of the constant sheaf. Use this to prove de Rham’s theorem comparing de Rham and singular cohomology (and also with ...
https://math.stackexchange.com/questions/3281856/sections-with-compact-support
(For a sheaf, this is possible if the sheaf itself has proper support). Of course, in the continuous or differential world, there is a lot of sections with compact support. You can construct them the same way you would construct a function with compact support. $\endgroup$ – Roland Jul 6 at 13:45
https://www.encyclopediaofmath.org/index.php/Sheaf_theory
Some basic notions of sheaf theory and spectral sequences appeared in the work of J. Leray (1945 and later) in connection with the study of homological properties of continuous mappings of locally compact spaces, and he also gave the definition of cohomology (with compact support) with coefficients in a sheaf.
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
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