Deriving some properties of Stanley-Reisner rings from their squarefree zero-divisor graphs

Year 2022, Volume: 31 Issue: 31, 121 - 133, 17.01.2022

Ashkan Nıkseresht

https://doi.org/10.24330/ieja.1058421

Abstract

Let $\Delta$ be a simplicial complex, $I_{\Delta }$ its Stanley-Reisner ideal and $R=K\left[\Delta \right]$ its Stanley-Reisner ring over a field $K$. In 2018, the author introduced the squarefree zero-divisor graph of $R$, denoted by ${\Gamma }_{sf}\left(R\right)$, and proved that if $\Delta$ and ${\Delta }^{\prime }$ are two simplicial complexes, then the graphs ${\Gamma }_{sf}\left(K\right[\Delta \left]\right)$ and ${}_{\mathrm{\Gamma s}f}\left(K\right[{\Delta }^{\prime }\left]\right)$ are isomorphic if and only if the rings $K\left[\Delta \right]$ and $K\left[{\Delta }^{\prime }\right]$ are isomorphic. Here we derive some algebraic properties of $R$ using combinatorial properties of ${\Gamma }_{sf}\left(R\right)$. In particular, we state combinatorial conditions on ${\Gamma }_{sf}\left(R\right)$ which are necessary or sufficient for $R$ to be Cohen-Macaulay. Moreover, we investigate when ${\Gamma }_{sf}\left(R\right)$ is in some well-known classes of graphs and show that in these cases, $I_{\Delta }$ has a linear resolution or is componentwise linear. Also we study the diameter and girth of ${\Gamma }_{sf}\left(R\right)$ and their algebraic interpretations.

Keywords

Squarefree monomial ideal, simplicial complex, squarefree zero-divisor graph, Cohen-Macaulay ring, linear resolution

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