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

When a cone is added to a simplicial complex $\mathrm{\Delta }$ 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 ${\mathrm{\Delta }}^{\mathrm{\prime }}$. In particular, we show that the arithmetical rank of the Stanley-Reisner ideal of ${\mathrm{\Delta }}^{\mathrm{\prime }}$ equals the projective dimension of the Stanley-Reisner ring of ${\mathrm{\Delta }}^{\mathrm{\prime }}$ if the corresponding equality holds for $\mathrm{\Delta }$.