On Castelnuovo-Mumford regularity of products of monomial ideals (Q812540)

In [Collect. Math. 54, No. 2, 137--152 (2003; Zbl 1074.13004)] \textit{A. Conca} and \textit{J. Herzog} asked whether \(\text{reg}(I_1I_2\cdots I_d)\leq \text{reg}(I_1)+\cdots+ \text{reg}(I_d)\) in case each \(I_i\) is generated by a regular sequence. The author proves that this is the case if \(d=2\), \(I_1\) and \(I_2\) are monomial ideals and \(I_1\) is generated by two monomials. The problem can be reduced to the case of square free monomial ideals by passing to the polarization. Then the regularity can be expressed in terms of the reduced homology of the associated simplicial complexes, which is calculated via the Maier-Vietoris sequence.