The Cohen-Macaulay Property of $f$-ideals

Abstract

For positive integers $d<n$, let $[n{]}_d=\{A\in 2^{[n]}\mid |A|=d\}$ where $[n]=:\{1,2,\dots ,n\}$. For a pure $f$-simplicial complex $\Delta$ such that $\dim (\Delta )=\dim ({\Delta }^c)$ and $\mathcal{F}(\Delta )\cap \mathcal{F}({\Delta }^c)=\varnothing$, 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 ${\Delta }^{\prime }=:⟨F\mid F\in [n{]}_d\setminus \mathcal{F}(\Delta )⟩$ is an $f$-simplicial complex, we prove that $I({\Delta }^c)$ is Cohen-Macaulay if and only if $I({\Delta }^{\prime })$ has linear resolution.

Example 3.2 is revised. An new example (3.3) is added

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