## 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 τ(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 τ(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 ≫ 0. Also, in fixed prime characteristic, we establish some properties of τ(a) 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 τ(a) and the F-rationality of Rees algebras.

Original language | English |
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Pages (from-to) | 3143-3174 |

Number of pages | 32 |

Journal | Transactions of the American Mathematical Society |

Volume | 355 |

Issue number | 8 |

DOIs | |

Publication status | Published - 2003 Aug |

Externally published | Yes |

## ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics