  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.
