Poincaré series of monomial rings

AbstractLet k be a field, let I be an ideal generated by monomials in the polynomial ring k[x1,…,xt] and let R=k[x1,…,xt]/I be the associated monomial ring. The k-vector spaces ToriR(k,k) are Nt-graded. We derive a formula for the multigraded Poincaré series of R,PkR(x,z)=∑i⩾0,α∈NtdimkTori,αR(k,k)xαzi, in terms of the homology of certain simplicial complexes associated to subsets of the minimal set of generators for I. The homology groups occuring in the formula can be interpreted as the homology groups of lower intervals in the lattice of saturated subsets of the generators for I