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

TL;DR

This paper investigates algebraic invariants of Stanley-Reisner rings of Cohen-Macaulay simplicial complexes, establishing inequalities and constructing complexes with prescribed invariants to deepen understanding of their algebraic and combinatorial properties.

Contribution

It proves a new inequality relating dimension, regularity, and type of Cohen-Macaulay simplicial complexes and constructs complexes with specific invariants for given parameters.

Findings

Established the inequality d ≤ reg(Δ) * type(Δ) for certain Cohen-Macaulay complexes.

Constructed complexes with prescribed regularity and type for given dimensions.

Demonstrated the existence of complexes with specific algebraic invariants.

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 $d \leq \mathrm{reg}(\Delta) \cdot \mathrm{type}(\Delta)$ 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 $\mathrm{type}(\Delta(G))=t$.