Abstract
We introduce a homology theory with supports and with coefficients in a sheaf. It has a very explicit description of the chains in terms of a triangulation of an ambient space, making the theory useful for integration purposes. We prove a Poincaré Duality Theorem that states that our homology modules are isomorphic to the classical sheaf cohomology modules with supports. This theorem is a main ingredient in the proof of a criterion on the vanishing of real principal value integrals in terms of cohomology. We briefly explain how real principal value integrals appear as residues of poles of distributions $|f|^{s}$ and as coefficients of asymptotic expansions of oscillating integrals.
Citation
Philippe Jacobs. "A sheaf homology theory with supports." Illinois J. Math. 44 (3) 644 - 666, Fall 2000. https://doi.org/10.1215/ijm/1256060422
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