Castelnuovo-Mumford regularity of products of monomial ideals

Let R=k[x_1,x_2,\cdots, x_N] be a polynomial ring over a field k. We prove that for any positive integers m, n, \text{reg}(I^mJ^nK)\leq m\text{reg}(I)+n\text{reg}(J)+\text{reg}(K) if I, J, K\subseteq R are three monomial complete intersections (I, J, K are not necessarily proper ideals of the polynomial ring R), and I, J are of the form (x_{i_1}^{a_1}, x_{i_2}^{a_2}, \cdots, x_{i_l}^{a_l}).

Castelnuovo-Mumford regularity of products of monomial ideals Keywords:

Castelnuovo-Mumford regularity , complete intersections , ideals of Borel type