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 Poincare 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.
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机译：我们介绍了一种具有支撑和系数的同源性理论。它根据环境空间的三角剖分对链进行了非常明确的描述，从而使该理论可用于集成目的。我们证明了庞加莱对偶定理，该定理指出我们的同源模块与带有支持的经典捆同构模块是同构的。该定理是证明关于同调的实际主值积分消失的准则的主要成分。我们简要地解释了真实的主值积分是如何作为分布极点| f |〜s的残差以及振荡积分的渐近展开系数出现的。
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