ON F-VECTORS AND BETTI NUMBERS OF MULTICOMPLEXES

A multicomplex M is a collection of monomials closed under divisibilit y. For such M we construct a cell complex Delta(M) whose i-dimensional cells are in bijection with the f(i) monomials of M of degree i+1. Th e bijection is such that the inclusion relation of cells corresponds t o divisibility of monomials. We then study relations between the numbe rs f(i) and the Betti numbers of Delta(M). For squarefree monomials th e construction specializes to the standard geometric realization of a simplicial complex.