Poincaré series of monomial rings

Abstract

Let k be a field, let I be an ideal generated by monomials in the polynomial ring k [ x 1 , … , x t ] and let R = k [ x 1 , … , x t ] / I be the associated monomial ring. The k-vector spaces Tor i R ( k , k ) are N t -graded. We derive a formula for the multigraded Poincaré series of R, P k R ( x , z ) = ∑ i ⩾ 0 , α ∈ N t dim k Tor i , α R ( k , k ) x α z i , 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.