A Combinatorial Proof of a Formula for Betti Numbers of a Stacked Polytope
A Combinatorial Proof of a Formula for Betti Numbers of a Stacked Polytope
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DOI：
10.37236/281
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发表时间：
2009-02
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影响因子：
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通讯作者：
Suyoung Choi;J. Kim
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文献类型：
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作者：
Suyoung Choi;J. Kim
对于单纯复形$\Delta$，域$k$上的Stanley-Reisner环$k[\Delta]$的分次Betti数$\beta_{i，j}（k[\Delta]）$由于Hochster而具有组合解释。Terai和Hibi证明了如果$\Delta$是一个有$n$个顶点的$d$维堆叠多面体的边界复形，对于$d\geq3$，则$\beta_{k-1，k}（k[\Delta]）=（k-1）\binom{n-d}{k}$。我们证明这一组合。
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 $d\geq3$, then $\beta_{k-1,k}(k[\Delta])=(k-1)\binom{n-d}{k}$. We prove this combinatorially.