that the order complex of $P \setminus \{\hat 0\}$ is a Cohen-Macaulay simplicial complex. On face rings of simplicial posets, the following properties are known.

-   The Krull dimension of $K[P]$ is $\dim P +1$.

-   $K[P]$ has an l.s.o.p.

-   $H_{K[P]}(t)=(h_0+h_1t+ \cdots + h_dt^d) /(1-t)^d$, where $d=\dim P +1$ and where $(h_0,h_1,\dots,h_d)=h(P)$.

-   If $\Gamma(P)$ is homeomorphic to a ball or a sphere then $P$ is Cohen-Macaulay.

See [@St pp. 325–326] for the first three properties. The last property follows from a topological criterion of Cohen-Macaulay simplicial complexes [@St2 II, Corollary 4.2 and Proposition 4.3].

A subset $I$ of a simplicial poset $P$ is said to be an *order ideal of* $P$ if $\sigma \in I$ and $\tau \leq \sigma$ imply $\tau \in I$. Thus an order ideal of a simplicial poset is again a simplicial poset. For a simplicial poset $P$ and elements $\sigma_1,\dots,\sigma_k \in P$, we write $\langle\sigma_1,\dots,\sigma_k \rangle$ for the order ideal of $P$ generated by $\sigma_1,\dots,\sigma_k$, in other words, $$\langle \sigma_1,\dots,\sigma_k\rangle=\{\tau \in P: \mbox{ there is $1 \leq j \leq k$ such that } \tau \leq \sigma_j\}.$$ For elements $f_1,\dots,f_m$ in a ring $A$, we write $(f_1,\dots,f_m)$ for the ideal of $A$ generated by $f_1,\dots,f_m$. 