On the Regularity of Operations of Ideals

Let I be a homogeneous ideal of a polynomial ring K[x1,…, xn] over a field K, and denote the Castelnuovo–Mumford regularity of I by reg(I). When I is a monomial complete intersection, it is proved that reg(Im) ≤ mreg(I) holds for any m ≥ 1. When n = 3, for any homogeneous ideals I and J of K[x1, x2, x3], one has that reg(I ⊗ J), reg(IJ) and reg(I ∩ J) are all upper bounded by reg(I) +reg(J), while reg(I + J) ≤reg(I) +reg(J) −1.