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Characterizing Cohen-Macaulay local rings by Frobenius maps


Authors: Ryo Takahashi and Yuji Yoshino
Journal: Proc. Amer. Math. Soc. 132 (2004), 3177-3187
MSC (2000): Primary 13A35, 13D05, 13H10
DOI: https://doi.org/10.1090/S0002-9939-04-07525-2
Published electronically: May 12, 2004
MathSciNet review: 2073291
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Abstract: Let $R$ be a commutative noetherian local ring of prime characteristic. Denote by ${{}^e\hspace{-1.6pt}{}} R$ the ring $R$ regarded as an $R$-algebra through $e$-times composition of the Frobenius map. Suppose that $R$is F-finite, i.e., ${{}^1\hspace{-2pt}{}} R$ is a finitely generated $R$-module. We prove that $R$ is Cohen-Macaulay if and only if the $R$-modules ${{}^e\hspace{-1.6pt}{}} R$ have finite Cohen-Macaulay dimensions for infinitely many integers $e$.

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Additional Information

Ryo Takahashi
Affiliation: Graduate School of Natural Science and Technology, Okayama University, Okayama 700-8530, Japan
Address at time of publication: Faculty of Science, Okayama University, Okayama 700-8530, Japan
Email: takahasi@math.okayama-u.ac.jp

Yuji Yoshino
Affiliation: Faculty of Science, Okayama University, Okayama 700-8530, Japan
Email: yoshino@math.okayama-u.ac.jp

DOI: https://doi.org/10.1090/S0002-9939-04-07525-2
Keywords: Frobenius map, CM-dimension, G-dimension, flat dimension, injective dimension
Received by editor(s): May 15, 2002
Received by editor(s) in revised form: April 9, 2003, and August 7, 2003
Published electronically: May 12, 2004
Communicated by: Bernd Ulrich
Article copyright: © Copyright 2004 American Mathematical Society