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 $d \leq \mathrm{reg}(\Delta) \cdot \mathrm{type}(\Delta)$ 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 $\mathrm{type}(\Delta(G))=t$.
- Publication:
-
arXiv e-prints
- Pub Date:
- December 2020
- DOI:
- 10.48550/arXiv.2012.13725
- arXiv:
- arXiv:2012.13725
- Bibcode:
- 2020arXiv201213725H
- Keywords:
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- Mathematics - Commutative Algebra;
- Mathematics - Combinatorics;
- Primary 13F55;
- Secondary 13D02;
- 13D40;
- 05C70;
- 05E40
- E-Print:
- 9 pages