A generalization of tight closure and multiplier ideals

Abstract

We introduce a new variant of tight closure associated to any fixed ideal $\a$, which we call $\a$-tight closure, and study various properties thereof. In our theory, the annihilator ideal $\tau(\a)$ of all $\a$-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 $\tau(\a)$ and the multiplier ideal associated to $\a$ (or, the adjoint of $\a$ in Lipman's sense) in normal $\Q$-Gorenstein rings reduced from characteristic zero to characteristic $p \gg 0$. Also, in fixed prime characteristic, we establish some properties of $\tau(\a)$ similar to those of multiplier ideals (e.g., a Brian\c{c}on-Skoda type theorem, subadditivity, etc.) with considerably simple proofs, and study the relationship between the ideal $\tau(\a)$ and the F-rationality of Rees algebras.