The aim of this paper is to settle a question about the tight closure of Ž 2 2 2 . w x Ž 3 3 3. the ideal x , y , z in the ring R s K X, Y, Z r X q Y q Z where Ž K is a field of prime characteristic p / 3. Lower case letters denote the . images of the corresponding variables. M. McDermott has studied the tight closure of various irreducible ideals in R and has established that Ž 2 2 2 . w x xyz...

This paper deals with a question regarding tight closure in characteristic zero which we now review. Let R be a commutative ring of prime characteristic p and let I ⊆ R be an ideal. Recall that for e ≥ 0, the e-th Frobenius power of I, denoted I [p e], is the ideal of R generated by all p-th powers of elements in I. We say that f ∈ I∗, the tight closure of I, if there exists a c not in any mini...

Test ideals play a crucial role in the theory of tight closure developed by Melvin Hochster and Craig Huneke. Recently, Karen Smith showed that test ideals are closely related to certain multiplier ideals that arise in vanishing theorems in algebraic geometry. In this paper we develop a generalization of the notion of test ideals: for complete local rings R and S, where S is a module-6nite exte...

In this paper, we consider positively graded isolated non F -rational singularities (R, m) with d = dimR over the field K of positive characteristic. We give a representation of lower local cohomologies H m (R) (i < d) in terms of tight closure and limit closure of certain type of parameters. As an application to isolated singularities, we show a relation between non-vanishing of the tight clos...