Assume that $\Delta$ is a simplicial complex and $\Delta ^v$ is its Alexander dual.

*Let in addition $\Delta$ be pure, then what happens to betti-numbers of $I_{\Delta}$ or $I_{\Delta^v}$*?

Is there a known fact about it?

Where does this question come from?

If $I_{\Delta} = P_{F_1}\cap \dots \cap P_{F_m}$ is the standard primary decomposition of $I_{\Delta}$, then $I_{\Delta^v}$ is generated by $\{x_{F_1}\cap \dots \cap x_{F_m}\}.$ Now let $\Delta$ be pure. Then $I_{\Delta^v}$ is generated in one degree. So if $\beta_{0j}\neq 0$ and $\beta_{0k}\neq 0$, then $j=k$.

Thank you.