Vanishing of differentials along ideals and non-archimedean approximation

Abstract

Dedicated to E. Kunz on the occasion of his sixty-fifth birthday One of the major new developments in commutative algebra over the last decade or so was the introduction of the theory of tight closure of an ideal by Hochster and Huneke. It proved to be an extremely useful technique to study ideals, and it also turned out to be closely related to many geometric questions. Fedder [F] used derivations in positive characteristic to obtain characterizations of 2 dimensional graded rational singularities in terms of F-purity and F-injectivity. In an attempt to generalize these techniques and to relate rational singularities with F-rationality, Craig Huneke raised the following problem (c.f [FHH]): Let R be a regular local ring, containing a perfect field k, over which R is essentially of finite type, and let C(R/k) be the subring of derivationally constant elements of R/k (i.e. C(R/k) = {x ∈ R : δ(x) = 0 for all δ ∈ Der k (R)}). Then Huneke asked: (1) If I ⊆ R is an ideal, does there exist a constant l = l(R, I) ∈ N with the following property: If x ∈ R with δ(x) ∈ I n+l then there exists a c ∈ C(R/k) with x − c ∈ I n. (2) If an l as in (1) exists, is it possible to bound it in a way useful for reduction mod p techniques, i.e. if char(k) = 0, does there exist a model R/A, I ⊆ R of R/k, I with A/Z of finite type and a constant l(I) such that l(R/mR, I + m/m) ≤ l(I) for all m ∈ Max(A). A result of the above type has been used successfully by Fedder [F] to relate rationality and F-rationality for two-dimensional graded rings, and a positive answer to the above questions would allow to extend these techniques and results to higher dimension. The