A Short Note on Containment of Cores

We show that cores of ideals do not preserve the inclusion. Huneke and Swanson[2] raised the question of whether, given integrally closed ideals I ⊂ I ′ in a ring R, it is necessarily true that core(I) ⊂ core(I ). Hyry and Smith [4, Corollary 5.5.1] gave a partial answer. The purpose of this note is to show that the answer is no in general. Although the simplest counterexample is a principal ideal I which is contained in a power I ′ of a maximal ideal, we are more interested in the case when both I and I ′ have the same height. Let R = k[x, y, z, w](x,y,z,w) with k a field of characteristic zero and let m denote the maximal ideal ofR. Let I = I2+m , where I2 = (x +yw, y+zw, z+ xw). The computer algebra system Macaulay shows that I2 is radical. Then it is not hard to check that I is integrally closed (see [7, Lemma 2.1]). It follows from [8] or [3] that core(I) = J : I, where J = (x+yw, y+zw, z+xw,w) and n is the least integer such that I = JI. In fact J is a minimal reduction of I and the computer algebra system Macaulay shows I = JI. On the other hand, if we take I ′ = m so that I ⊂ I , then core(I ) = core(m) = m by [5, Theorem 1.3] or [1, Proposition 4.2]. But the computer algebra system Macaulay shows that core(I) = J : I = I 6⊂ m, therefore core(I) 6⊂ core(I ). This gives a negative answer to the above question. More generally, we have the following conjecture. Conjecture 1. Let R = k[x1, ..., xn](x1,...,xn) with k a field of characteristic zero and let m denote the maximal ideal of R. Let I = Id +m , where Id is a complete intersection ideal of s general d-forms (1 ≤ s < n). Let b = ⌊ d+1 ⌋ and a = dn− s+ 1− (d+ 1)b. Then core(I) = mI. Theorem 2. The conjecture holds true for d = 1. Proof. Let X = SpecR. By [4, Corollary 5.3.1], we have core(I) = J (X,n · I). To get the simplest log resolution μ : X2 → X1 → X of (X, I), we blow up X at the origin, and then blow up the resulting surface X1 along the intersection