Let $\Delta$ be a simplicial complex with $r$-skeleton $\Delta_r$ (i.e. the set of all faces $F\in \Delta$ such that $\dim F\leq r$). Show $\operatorname{depth}k[\Delta]=\text{max}\{r:\Delta_r \text { is Cohen-Macaulay over $k$}\}+1$.
This is exercise 5.1.23 of Burns&Herzog's Cohen-Macaulay Rings. The hint suggests to do induction on the number of faces.
I suppose $R=k[\Delta]$ is not Cohen-Macaulay with $n$ faces and $\dim R=d$. By removing one facet of dimension $d$. We get $k[\Delta']=R'=R/\text{\{a monomial ideal corresponding to the facet\}}$ where $\Delta'$ is a simplicial complex consists of $n-1$ faces and we may use inductive hypothesis. But I don't know how to show the depth remains unchanged under the quotient. Or maybe the way I tried wasn't right.
