Computing the core of ideals in arbitrary characteristic

Let R be a local Gorenstein ring with infinite residue field of arbitrary characteristic. Let I be an R-ideal with g=htI>0, analytic spread ℓ, and let J be a minimal reduction of I. We further assume that I satisfies Gℓ and depthR/Ij⩾dimR/I−j+1 for 1⩽j⩽ℓ−g. The question we are interested in is whether core(I)=Jn+1:∑b∈In(J,b) for n≫0. In the case of analytic spread Polini and Ulrich show that this is true with even weaker assumptions [C. Polini, B. Ulrich, A formula for the core of an ideal, Math. Ann. 331 (2005) 487–503, Theorem 3.4]. We give a negative answer to this question for higher analytic spreads and suggest a formula for the core of such ideals.