Arithmetical ranks of Stanley-Reisner ideals of simplicial complexes with a cone

TL;DR

This paper explores how adding a cone to a simplicial complex affects the arithmetical rank of its Stanley-Reisner ideal, establishing conditions under which this rank equals the projective dimension of the associated Stanley-Reisner ring.

Contribution

It provides a relation between the arithmetical ranks of Stanley-Reisner ideals before and after adding a cone, extending understanding of their algebraic properties.

Findings

Arithmetical rank equals projective dimension under certain conditions

Relation between original and coned simplicial complexes

Conditions for equality of algebraic invariants

Abstract

When a cone is added to a simplicial complex $\Delta$ over one of its faces, we investigate the relation between the arithmetical ranks of the Stanley-Reisner ideals of the original simplicial complex and the new simplicial complex $\Delta'$. In particular, we show that the arithmetical rank of the Stanley-Reisner ideal of $\Delta'$ equals the projective dimension of the Stanley-Reisner ring of $\Delta'$ if the corresponding equality holds for $\Delta$.