$F$-simplicial complexes which are minimal Cohen-Macaulay.

Let $\D$ be a $(d-1)$-dimensional pure $f$-simplicial complex over vertex set $[n]$. Assume further that $\D$ is minimal Cohen-Macaulay. In this paper, it is proved that $\D$ is acyclic \iff $n=2d$. It is also indicated that the recent work of \cite{Dao2020} implies that shellable condition on a pure simplicial complex $\D$ is identical with CM properties of a full series of subcomplexes of $\D$.