  A product of cochains in a polyhedral complex is constructed. The
multiplication algorithm depends on the choice of a parameter. The parameter is
a linear functional on the ambient space. Cocycles form a subring of the ring
of cochains, cobounds form an ideal of the ring of cocycles, and the quotient
ring is a ring of cohomologies. If the complex is simplicial, then the
algorithm reduces to the well known algorithm of ${\rm\breve{C}}$ech. Similar
algorithms are used in geometry of polyhedra for multiplying of cocycles taking
values in the exterior algebra of the ambient space. Therefore, we assume the
ring of values to be supercommutative. This text contains the basic
definitions, the statements of theorems and some of their applications in
convex geometry. The full text will be offered to "Izvestiya: Mathematics".
