Castelnuovo-Mumford regularity of products of monomial ideals (Q2800991)

Let \(R = k[x_1, \dots, x_N]\) be a polynomial ring over a field \(k\). Let \(I\) and \(J\) be ideals generated by powers of subsets of the variables, and let \(K\) be a monomial complete intersection ideal in \(R\). The main result of this paper shows that NEWLINE\[NEWLINE\mathrm{reg}(I^mJ^nK) \leq m\,\mathrm{reg}(I) + n\,\mathrm{reg}(J) + \mathrm{reg}(K).NEWLINE\]NEWLINENEWLINENEWLINEThis study was motivated by a question of \textit{A. Conca} and \textit{J. Herzog} [Collect. Math. 54, No. 2, 137--152 (2003; Zbl 1074.13004)], which asked if for any set of complete intersections \(I_1, I_2, \dots, I_d\), one has NEWLINE\[NEWLINE\mathrm{reg}(I_1I_2 \dots I_d) \leq \mathrm{reg}(I_1) + \dots +\mathrm{reg}(I_d).NEWLINE\]NEWLINE This question has a negative answer in general, but was proved for \(d = 2\) in [\textit{M. Chardin} et al., Proc. Am. Math. Soc. 135, No. 6, 1597--1606 (2007; Zbl 1160.13005)]. The proof of the main result extends that of Chardin et al. [loc. cit.].