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

Abstract

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 $d\le \mathrm{reg}(\Delta )\cdot \mathrm{type}(\Delta )$ 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 $\mathbb{k}[\Delta ]$. Moreover, for any given integers $d,r,t$ satisfying $r,t\ge 2$ and $r\le d\le rt$, 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 $\mathrm{type}(\Delta (G))=t$.

9 pages

References in corpus (1)

• Homological invariants of Cameron--Walker graphs