The goal of the present paper is the study of some algebraic invariants of Stanley-Reisner rings of Cohen-Macaulay simplicial complexes of dimension $d - 1$. We prove that the inequality $d \leq \mathrm{ reg}(\Delta) \cdot \mathrm{ type}(\Delta)$ holds for any $(d-1)$-dimensional Cohen-Macaulay simplicial complex $\Delta$ satisfying $\Delta=\mathrm{ core}(\Delta)$, where $\mathrm{ reg}(\Delta)$ (resp. $\mathrm{ type}(\Delta)$) denotes the Castelnuovo-Mumford regularity (resp. Cohen-Macaulay type) of the Stanley-Reisner ring $\Bbbk[\Delta]$. Moreover, for any given integers $d,r,t$ satisfying $r,t \geq 2$ and $r \leq d \leq rt$, we construct a Cohen-Macaulay simplicial complex $\Delta(G)$ as an independent complex of a graph $G$ such that $\dim(\Delta(G))=d-1$, $\mathrm{ reg}(\Delta(G))=r$ and $\mathrm{ type}(\Delta(G))=t$.
Categories: math.AC math.CO