Tight closure and strong test ideals
This is in the spirit of the author's work on the uniform Artin-Rees theorem [see \textit{C. Huneke}, Invent. Math. 107, No. 1, 203-223 (1992; Zbl 0756.13001)]), in which he proved some uniform properties of tight closure, integral closure and powers of ideals. The main result in this paper is that in reasonably good rings, there exists a positive integer \(M\) such that for every ideal \(I\) and every element \(x\) in the tight closure of \(I\), \(x\) satisfies an integral equation over \(I\) of degree at most \(M\). The good rings for which the author proves the existence of this uniform bound \(M\) on the integral degrees contain what the author terms strong test ideals. The author proves the existence of strong test ideals in certain rings with desingularizations. The paper ends with a section of related and interesting results.
scientific article; zbMATH DE number 1097606