  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
$\Delta$.
