A generalization of tight closure and multiplier ideals

We introduce a new variant of tight closure associated to any fixed ideal , which we call -tight closure, and study various properties thereof. In our theory, the annihilator ideal of all -tight closure relations, which is a generalization of the test ideal in the usual tight closure theory, plays a particularly important role. We prove the correspondence of the ideal and the multiplier ideal associated to (or, the adjoint of in Lipman's sense) in normal -Gorenstein rings reduced from characteristic zero to characteristic . Also, in fixed prime characteristic, we establish some properties of similar to those of multiplier ideals (e.g., a Briançon-Skoda-type theorem, subadditivity, etc.) with considerably simple proofs, and study the relationship between the ideal and the F-rationality of Rees algebras.