On the product of cocycles in a polyhedral complex

We construct an algorithm for multiplying cochains in a polyhedral complex. It depends on the choice of a linear functional on the ambient space. The cocycles form a subring in the ring of cochains, the coboundaries form an ideal in the ring of cocycles, and the quotient ring is the cohomology ring. The multiplication algorithm depends on the geometry of the cells of the complex. For simplicial complexes (the simplest geometry of cells), it reduces to the well-known Čech algorithm. Our algorithm is of geometric origin. For example, it applies in the calculation of mixed volumes of polyhedra and the construction of stable intersections of tropical varieties. In geometry it is customary to multiply cocycles with values in the exterior algebra of the ambient space. Therefore we assume that the ring of values is supercommutative.