### 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 2004 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

*Proceedings of the American Mathematical Society*,

*132*(11), 3177-3187. https://doi.org/10.1090/S0002-9939-04-07525-2

**Characterizing Cohen-Macaulay local rings by Frobenius maps.** / Takahashi, Ryo; Yoshino, Yuji.

Research output: Contribution to journal › Article

*Proceedings of the American Mathematical Society*, vol. 132, no. 11, pp. 3177-3187. https://doi.org/10.1090/S0002-9939-04-07525-2

}

TY - JOUR

T1 - Characterizing Cohen-Macaulay local rings by Frobenius maps

AU - Takahashi, Ryo

AU - Yoshino, Yuji

PY - 2004/11

Y1 - 2004/11

N2 - Let R be a commutative noetherian local ring of prime characteristic. Denote by eR the ring R regarded as an R-algebra through e-times composition of the Frobenius map. Suppose that R is F-finite, i.e., 1R is a finitely generated R-module. We prove that R is Cohen-Macaulay if and only if the R-modules eR have finite Cohen-Macaulay dimensions for infinitely many integers e.

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

KW - CM-dimension

KW - Flat dimension

KW - Frobenius map

KW - G-dimension

KW - Injective dimension

UR - http://www.scopus.com/inward/record.url?scp=7444223587&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=7444223587&partnerID=8YFLogxK

U2 - 10.1090/S0002-9939-04-07525-2

DO - 10.1090/S0002-9939-04-07525-2

M3 - Article

AN - SCOPUS:7444223587

VL - 132

SP - 3177

EP - 3187

JO - Proceedings of the American Mathematical Society

JF - Proceedings of the American Mathematical Society

SN - 0002-9939

IS - 11

ER -