Abstract
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 ...
Abstract
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 two-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-rationaIity, Craig Huneke raised the following problem (cf. [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 is an element of R : delta(x) = 0 for all delta is an element of Der(k)(R))). Then Huneke asked: (1) If subset of or equal to R is an ideal, does there exist a constant l = l(R, I) is an element of N with the following property: If x is an element of R with delta (x) is an element of In+l then there exists a c is an element of C(R/k) with x - c is an element of 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,L subset of or equal to R of R/k, I with A/Z of finite type and a constant I (L) such that l(R/mR, L + m/m) less than or equal to l(L) for all m is an element of Max(A).
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