Derivations and the integral closure of ideals

Abstract

Let (R,m) be a complete local domain containing the rationals. Then there exists an integer I such that for any ideal I C R, if f E m, f In, then there exists a derivation 6 of R with S(f) 5 In+l. This note grew out of a question raised by C. Huneke some years ago: Suppose (R, m) is a complete local domain with perfect residue class field k which it contains. Call an element x E R derivationally constant for R/k if 6(x) = 0 for all k-derivations 6 E Derk(R), and denote by C(R/k) the subring of derivationally constant elements of R/k. In this context Craig Huneke asked: (1) If I C R is an ideal, does there exist a constant I = l(R, I) E N with the following property: If x E R with 6(x) E In+l for all 6 E Derk(R), then there exists a c E C(R/k) with x c E In? (2) Is it possible to bound I in (1) in some uniform way? Here (2) is meant to cover the situation arising via the techniques of tight closure and reduction mod p: If I arises via reduction mod p from an ideal 3 given in an algebra 91 containing the integers, then Huneke asks for a bound I = 1(3) working for the reductions modulo all but finitely many primes p. Originally this question arose in the study of rational and F-rational singularities and the relations between them (cf. [F]), and interest in it has been revived by recent attempts to prove Kodaira vanishing with tight closure techniques (cf. [HS]). In [FHH] a positive answer to (1) was given in case char(k) = 0 and I C R is an m-primary and one-fibred ideal. Some partial results in positive characteristics are available, for instance in the graded case ([F]) and for regular local rings. In this note we will prove the following generalization of [FHH], (1.6): Theorem. Let k be a field, char(k) = 0 and let (R, m)/k be a local domain such that the universally finite derivation dR/k : R -/k exists. Then there exists a constant I = I(R) with the following property: If I C R is an ideal and r E m is an element with 6(r) E In+l for all 6 E Derk(R), then r E In. This completely answers (1), and it also gives a uniform bound (as asked for in (2)), depending on the ring only. A first attempt to solve this problem would be Received by the editors November 20, 1997 and, in revised form, February 24, 1998. 1991 Mathematics Subject Classification. Primary 13N05, 13J10.