Cohen-Macaulay and (S2) Properties of the Second Power of Squarefree Monomial Ideals

Do Trong Hoang,Naoki Terai,Giancarlo Rinaldo

doi:10.3390/math7080684

Do Trong Hoang, Naoki Terai + Show 1 more

Open Access

https://doi.org/10.3390/math7080684

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Abstract

We show that Cohen-Macaulay and (S 2 ) properties are equivalent for the second power of an edge ideal. We give an example of a Gorenstein squarefree monomial ideal I such that S / I 2 satisfies the Serre condition (S 2 ), but is not Cohen-Macaulay.

Highlights

S/I when I is a squarefree monomial ideal generated in degree two and S/I 2 has the Cohen-Macaulay (equivalently (S2 )) property
We recall some notation on simplicial complexes and their Stanley-Reisner ideals
To introduce a characterization of the (S2 ) property for the second symbolic power of a Stanley-Reisner ideal, we first define the diameter of a simplicial complex

Summary

Introduction

S/I when I is a squarefree monomial ideal generated in degree two and S/I 2 has the Cohen-Macaulay (equivalently (S2 )) property. We classify squarefree monomial ideals I generated in degree two with dim S/I ≤ 4 such that S/I 2 have the Cohen-Macaulay (equivalently (S2 )). See [6,8] for the fact that for a very well-covered graph G, the second power I ( G ) is not Cohen-Macaulay if the edge ideal I ( G ) of G is not a complete intersection.

Stanley-Reisner Ideals

Edge Ideals

The Second Power of Edge Ideals

Upper Bound of the Number of Vertices

Classification

Example