Let $\Delta$ be a simplicial complex which is the cycle on $n$-vertices $V=\{x_1,...,x_n\}$ (say) i.e. the facets of $\Delta$ are $\{x_i, x_{i+1}\}$ for $1\le i\le n$ with $x_{n+1}=x_1$. Let $S=k[x_1,...,x_n]$ and $I_{\Delta}$ be the Stanley-Reisner ideal of $\Delta$ and consider $M:=S/I_{\Delta}$ as an $S$-module. How to find the Betti numbers $\beta_{i,j}^S(M)$ ?

I know that by Hochster's formula, $\beta_{i,j}^S(M)=\sum_{W\subseteq V, |W|=j} \dim_k \hat H^{j-i-1} (\Delta_W, k)$, where $\Delta_W$ is the simplicial complex whose faces are those faces of $\Delta$ whose vertices only come from $W$, and $\hat H^n(-,k)$ is the $n$-th reduced simplicial cohomology with coefficients in $k$.

However, I have no idea on how to calculate $\dim_k \hat H^{j-i-1} (\Delta_W, k)$.

Please help.

  • $\begingroup$ Most of the complexes $\Delta_W$ have trivial homology except dimension $0$. $\endgroup$ – Michal Adamaszek Apr 19 at 20:12

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