# Betti numbers of the Stanley-Reisner ring of a simplicial complex which is the cycle on $n$-vertices

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 $$\dim_k \hat H^{j-i-1} (\Delta_W, k)$$.

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