Computation of Betti numbers of monomial ideals associated with stacked polytopes

Let P( v, d) be a stacked d-polytope with v vertices, .6.(P( v, d)) the boundary complex of P( v, d), and k[.6.(P( v, d))] = A/ IA(P(v,d)) the Stanley-Reisner ring of .6.(P( v, d)) over a field k. We comﾂｭpute the Betti numbers which appear in a minimal free resolution of k[.6.(P(v,d))] over A, and show that every Betti number depends only on v and d and is independent of the base field k