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

Abstract

For a simplicial complex $\Delta$, the graded Betti number ${\beta }_{i,j}(k[\Delta ])$ of the Stanley-Reisner ring $k[\Delta ]$ over a field $k$ has a combinatorial interpretation due to Hochster. Terai and Hibi showed that if $\Delta$ is the boundary complex of a $d$-dimensional stacked polytope with $n$ vertices for $d\ge 3$, then ${\beta }_{k-1,k}(k[\Delta ])=(k-1)\left(\genfrac{}{}{0ex}{}{n-d}{k}\right)$. We prove this combinatorially.

arXiv comment7 pages