The Cohen-Macaulay Property of $f$-ideals

For positive integers $d<n$, let $[n]_d=\{A\in 2^{[n]}\mid |A|=d\}$ where $[n]=:\{1,2,\ldots, n\}$. For a pure $f$-simplicial complex $\Delta$ such that ${\rm dim}(\Delta)={\rm dim}(\Delta^c)$ and $\mathcal{F}(\Delta)\cap \mathcal{F}(\Delta^c)=\emptyset$, we prove that the facet ideal $I(\Delta)$ is Cohen-Macaulay if and only if it has linear resolution. For a $d$-dimensional pure $f$-simplicial complex $\Delta$ such that $\Delta'=:\langle F\mid F\in [n]_d\smallsetminus \mathcal F(\Delta)\rangle$ is an $f$-simplicial complex, we prove that $I(\Delta^c)$ is Cohen-Macaulay if and only if $I(\Delta')$ has linear resolution.