Let recall that the arithmetical rank (ara) of an ideal \(I\) in a commutative noetherian ring \(R\) is the minimal number \(s\) of elements \(a_1,\dots,a_s\in R\) such that \(\sqrt I=\sqrt {(a_1,\dots,a_s)}\). In the paper under review the authors consider square free monomial ideals in a polynomial ring \(K[X]\), such ideals are defined by a simplicial complex \(\Delta\) on a finite set of variables \(X\). Let \(x_0\) be a new variable, the cone over a face \(F\) of \(\Delta\) is the simplex with support \(F\cup \{x_0\}\), so we can define a new simplicial complex \(\Delta'\) where \(F\) is replaced by \(F\cup \{x_0\}\). The main result in this paper is:
Let \(K\) be algebraically closed. Then for any face \(F\) of \(\Delta\mathrm{ara}(I_{\Delta'})\leq \max(\mathrm{ara}(I_{\Delta})+1,|X|-|F|)\) and if \(\mathrm{ara}(I_{\Delta})= \mathrm{projdim} (K[X]/I_{\Delta}) \) then \(\mathrm{ara}(I_{\Delta'})= \mathrm{projdim} (K[X\cup \{x_0\}]/I_{\Delta'}) .\)