### Abstract

Let R be a commutative noetherian local ring of prime characteristic. Denote by ^{e}R the ring R regarded as an R-algebra through e-times composition of the Frobenius map. Suppose that R is F-finite, i.e., ^{1}R is a finitely generated R-module. We prove that R is Cohen-Macaulay if and only if the R-modules ^{e}R have finite Cohen-Macaulay dimensions for infinitely many integers e.

Original language | English |
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Pages (from-to) | 3177-3187 |

Number of pages | 11 |

Journal | Proceedings of the American Mathematical Society |

Volume | 132 |

Issue number | 11 |

DOIs | |

Publication status | Published - Nov 1 2004 |

### Keywords

- CM-dimension
- Flat dimension
- Frobenius map
- G-dimension
- Injective dimension

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

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## Cite this

Takahashi, R., & Yoshino, Y. (2004). Characterizing Cohen-Macaulay local rings by Frobenius maps.

*Proceedings of the American Mathematical Society*,*132*(11), 3177-3187. https://doi.org/10.1090/S0002-9939-04-07525-2