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

When a cone is added to a simplicial complex Δ 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 Δ′. In particular, we show that the arithmetical rank of the Stanley–Reisner ideal of Δ′ equals the projective dimension of the Stanley–Reisner ring of Δ′ if the corresponding equality holds for Δ