Inequalities of invariants on Stanley-Reisner rings of Cohen-Macaulay simplicial complexes

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 $mathrm{type}(Delta(G))=t$.