A combinatorial proof of a formula for Betti numbers of a stacked polytope

الأصل · EN

For a simplicial complex Δ, the graded Betti number βᵢ,ⱼ(k[Δ]) of the Stanley-Reisner ring k[Δ] over a field k has a combinatorial interpretation due to Hochster. Terai and Hibi showed that if Δ is the boundary complex of a d-dimensional stacked polytope with n vertices for d≥3, then βₖ₋₁,ₖ(k[Δ])=(k-1)n-dk. We prove this combinatorially.