## Abstract

Let R be a reduced ring that is essentially of finite type over an excellent regular local ring of prime characteristic. Then it is shown that the test ideal of R commutes with localization and, if R is local, with completion, under the additional hypothesis that the tight closure of zero in the injective hull E of the residue field of every local ring of R is equal to the finitistic tight closure of zero in E. It is conjectured that this latter condition holds for all local rings of prime characteristic; it is proved here for all Cohen-Macaulay singularities with at most isolated non-Gorenstein singularities, and in general for all isolated singularities. In order to prove the result on the commutation of the test ideal with localization and completion, a ring of Frobenius operators associated to each R-module is introduced and studied. This theory gives rise to an ideal of R which defines the non-strongly F-regular locus, and which commutes with localization and completion. This ideal is conjectured to be the test ideal of R in general, and shown to equal the test ideal under the hypothesis that 0*_{E} = 0_{E}^{fg}* in every local ring of R.

Original language | English (US) |
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Pages (from-to) | 3149-3180 |

Number of pages | 32 |

Journal | Transactions of the American Mathematical Society |

Volume | 353 |

Issue number | 8 |

DOIs | |

State | Published - 2001 |

## Keywords

- Frobenius action
- Localisation
- Test ideal
- Tight closure