Computation of Betti numbers of monomial ideals associated with stacked polytopes (Q1360902)

Let \(\Delta P(v,d)\) be a stacked \(d\)-polytope with \(v\) vertices, \(\Delta \) its boundary complex, and \(A:=k[x_1,\dots,x_v]\) the graded polynomial algebra in \(v\) indeterminates over a field \(k\), with \(\deg x_i:=1,\) for \(i=1,\dots,v\). Associated with \(P(v,d)\) is the Stanley-Reisner algebra \(k[\Delta ]\) defined as the quotient algebra of \(A\) by the homogeneous monomial ideal \( I_\Delta \) generated by the square-free ``non-faces'' of \(\Delta \). The Betti numbers of \(k[\Delta ]\) are defined as the Betti numbers of \(k[\Delta ]=A/I_\Delta \) viewed as a graded module over the polynomial algebra \(A\). In the present paper, the authors explicitly determine the Betti numbers of \( k[\Delta ]\) and show that they are independent of the field \(k\) and of the combinatorial type of \(P(v,d)\). The proof uses Hochster's formula relating the Betti numbers with the ranks of the reduced simplicial homology groups of certain subcomplexes of \(\Delta \). It is also proved that the sequence of the Betti numbers is unimodal. [See also Zbl 0860.55018]