Tight closure of parameter ideals
Let \(R\) be a locally excellent Noetherian domain of positive prime characteristic, and let \(R^ +\) be the integral closure of \(R\) in an algebraic closure of its fraction field. The main result of the paper under review (theorem 5.1) answers a question of \textit{M. Hochster} and \textit{C. Huneke}; it states that the tight closure of a parameter ideal \(I\) in \(R\) coincides with the contracted expansion \(IR^ + \cap R\) of \(I\) from \(R^ +\) [Hochster and Huneke could only handle the case when \(I\) was generated by at most three parameters (preprint)]. (Theorem 5.1 support the underlying conjecture that the description of the tight closure as contracted expansion from \(R^ +\) holds for any ideal in \(R\). The proof of theorem 5.1 depends heavily on the result of \textit{M. Hochster} and \textit{C. Huneke} that \(R^ +\) is a balanced big Cohen- Macaulay module over \(R\) [Ann. Math., II. Ser. 135, No. 1, 53-89 (1992; Zbl 0753.13003)] (see the proof of the key-lemma 5.2). In the final section 6, applications to \(F\)-rational rings and pseudo-rational singularities are the main aim.
scientific article; zbMATH DE number 549838
scientific article