On the second powers of Stanley-Reisner ideals

TL;DR

This paper investigates properties of the second power of Stanley-Reisner ideals, establishing conditions for Cohen-Macaulayness and Gorensteinness, and providing new examples of Cohen-Macaulay second powers.

Contribution

It proves that if the second power of a Stanley-Reisner ideal is Cohen-Macaulay, then the quotient is Gorenstein, and offers criteria for symbolic powers and new examples.

Findings

$S/I_{ riangle}^2$ Cohen-Macaulay implies $S/I_{ riangle}$ is Gorenstein

Criteria for second symbolic power to satisfy $(S_2)$ and match the ordinary power

New examples of Stanley-Reisner ideals with Cohen-Macaulay second powers

Abstract

In this paper, we study several properties of the second power $I_{\Delta}^2$ of a Stanley-Reisner ideal $I_{\Delta}$ of any dimension. As the main result, we prove that $S/I_{\Delta}$ is Gorenstein whenever $S/I_{\Delta}^2$ is Cohen-Macaulay over any field $K$. Moreover, we give a criterion for the second symbolic power of $I_{\Delta}$ to satisfy $(S_2)$ and to coincide with the ordinary power, respectively. Finally, we provide new examples of Stanley-Reisner ideals whose second powers are Cohen-Macaulay.