Combinatorial Duality and Intersection Product: A Direct Approach

The proof of the combinatorial Hard Lefschetz Theorem for the ``virtual'' intersection cohomology of a not necessarily rational polytopal fan that has been presented by K. Karu completely establishes Stanley's conjectures for the generalized $h$-vector of an arbitrary polytope. The main ingredients, namely, Poincare Duality and the Hard Lefschetz Theorem, both rely on the intersection product. In the constructions of Barthel, Brasselet, Fieseler and Kaup and Bressler and Lunts, there remained an apparent ambiguity. The recent solution of this problem by Bressler and Lunts uses the formalism of derived categories. The present article gives a straightforward approach to combinatorial duality and a natural intersection product, completely within the framework of elementary sheaf theory and commutative algebra, thus avoiding derived categories.