Computing the core of ideals in arbitrary characteristic

Abstract

Let R be a local Gorenstein ring with infinite residue field of arbitrary characteristic. Let I be an R-ideal with g = ht I > 0 , analytic spread ℓ, and let J be a minimal reduction of I. We further assume that I satisfies G ℓ and depth R / I j ⩾ dim R / I − j + 1 for 1 ⩽ j ⩽ ℓ − g . The question we are interested in is whether core ( I ) = J n + 1 : ∑ b ∈ I ( J , b ) n 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.